upstream/ipython Commit - r4552:bee9c065

Updating notebook examples to current format.

Brian E. Granger -

r4552:bee9c065

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docs/examples/notebooks/basic_quantum.ipynb

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	1	{"cells":[{"cell_type":"text","text":"<h1>Basic Symbolic Quantum Mechanics</h1>"},{"code":"%load_ext sympyprinting","cell_type":"code","prompt_number":1},{"code":"from sympy import sqrt, symbols, Rational\nfrom sympy import expand, Eq, Symbol, simplify, exp, sin\nfrom sympy.physics.quantum import \nfrom sympy.physics.quantum.qubit import \nfrom sympy.physics.quantum.gate import \nfrom sympy.physics.quantum.grover import \nfrom sympy.physics.quantum.qft import QFT, IQFT, Fourier\nfrom sympy.physics.quantum.circuitplot import circuit_plot","cell_type":"code","prompt_number":2},{"cell_type":"text","text":"<h2>Bras and Kets</h2>"},{"cell_type":"text","text":"Create symbolic states"},{"code":"phi, psi = Ket('phi'), Ket('psi')\nalpha, beta = symbols('alpha beta', complex=True)","cell_type":"code","prompt_number":3},{"cell_type":"text","text":"Create a superposition"},{"code":"state = alphapsi + betaphi; state\n","cell_type":"code","prompt_number":4},{"cell_type":"text","text":"Dagger the superposition and multiply the original"},{"code":"ip = Dagger(state)state; ip\n","cell_type":"code","prompt_number":5},{"cell_type":"text","text":"Distribute"},{"code":"qapply(expand(ip))\n","cell_type":"code","prompt_number":6},{"cell_type":"text","text":"<h2>Operators</h2>"},{"cell_type":"text","text":"Create symbolic operators"},{"code":"A = Operator('A')\nB = Operator('B')\nC = Operator('C')","cell_type":"code","prompt_number":7},{"cell_type":"text","text":"Test commutativity"},{"code":"AB == BA\n","cell_type":"code","prompt_number":8},{"cell_type":"text","text":"Distribute A+B squared"},{"code":"expand((A+B)2)","cell_type":"code","prompt_number":9},{"cell_type":"text","text":"Create a commutator"},{"code":"comm = Commutator(A,B); comm\n","cell_type":"code","prompt_number":10},{"cell_type":"text","text":"Carry out the commutator"},{"code":"comm.doit()","cell_type":"code","prompt_number":11},{"cell_type":"text","text":"Create a more fancy commutator"},{"code":"comm = Commutator(AB,B+C); comm","cell_type":"code","prompt_number":12},{"cell_type":"text","text":"Expand the commutator"},{"code":"comm.expand(commutator=True)","cell_type":"code","prompt_number":13},{"cell_type":"text","text":"Carry out and expand the commutators"},{"code":"_.doit().expand()\n","cell_type":"code","prompt_number":14},{"cell_type":"text","text":"Take the dagger"},{"code":"Dagger(_)","cell_type":"code","prompt_number":15},{"code":"%notebook save basic_quantum.ipynb","cell_type":"code","prompt_number":16},{"code":"%notebook load quantum_computing.ipynb","cell_type":"code","prompt_number":19}]} No newline at end of file	1	<?xml version="1.0" encoding="utf-8"?>
			2	<notebook>
			3	<name>basic_quantum</name>
			4	<nbformat>2</nbformat>
			5	<worksheets>
			6	<worksheet>
			7	<cells>
			8	<markdowncell>
			9	<source><h1>Basic Symbolic Quantum Mechanics</h1></source>
			10	</markdowncell>
			11	<codecell>
			12	<input>%load_ext sympyprinting</input>
			13	<language>python</language>
			14	<prompt_number>18</prompt_number>
			15	<collapsed>1</collapsed>
			16	<outputs />
			17	</codecell>
			18	<codecell>
			19	<input>from sympy import sqrt, symbols, Rational
			20	from sympy import expand, Eq, Symbol, simplify, exp, sin
			21	from sympy.physics.quantum import *
			22	from sympy.physics.quantum.qubit import *
			23	from sympy.physics.quantum.gate import *
			24	from sympy.physics.quantum.grover import *
			25	from sympy.physics.quantum.qft import QFT, IQFT, Fourier
			26	from sympy.physics.quantum.circuitplot import circuit_plot</input>
			27	<language>python</language>
			28	<prompt_number>19</prompt_number>
			29	<collapsed>1</collapsed>
			30	<outputs />
			31	</codecell>
			32	<markdowncell>
			33	<source><h2>Bras and Kets</h2></source>
			34	</markdowncell>
			35	<markdowncell>
			36	<source>Create symbolic states</source>
			37	</markdowncell>
			38	<codecell>
			39	<input>phi, psi = Ket('phi'), Ket('psi')
			40	alpha, beta = symbols('alpha beta', complex=True)</input>
			41	<language>python</language>
			42	<prompt_number>20</prompt_number>
			43	<collapsed>1</collapsed>
			44	<outputs />
			45	</codecell>
			46	<markdowncell>
			47	<source>Create a superposition</source>
			48	</markdowncell>
			49	<codecell>
			50	<input>state = alphapsi + betaphi; state
			51	</input>
			52	<language>python</language>
			53	<prompt_number>21</prompt_number>
			54	<collapsed>0</collapsed>
			55	<outputs>
			56	<output>
			57	<output_type>pyout</output_type>
			58	<text>α⋅❘ψ⟩ + β⋅❘φ⟩</text>
			59	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUY4QUFBQVhDQVlBQUFCdFI1UDBBQUFBQkhOQ1NWUUlD
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			88	</png>
			89	<latex>$$\alpha {\left\|\psi\right\rangle } + \beta {\left\|\phi\right\rangle }$$</latex>
			90	<prompt_number>21</prompt_number>
			91	</output>
			92	</outputs>
			93	</codecell>
			94	<markdowncell>
			95	<source>Dagger the superposition and multiply the original</source>
			96	</markdowncell>
			97	<codecell>
			98	<input>ip = Dagger(state)*state; ip
			99	</input>
			100	<language>python</language>
			101	<prompt_number>22</prompt_number>
			102	<collapsed>0</collapsed>
			103	<outputs>
			104	<output>
			105	<output_type>pyout</output_type>
			106	<text>⎛⎽ ⎽ ⎞
			107	⎝α⋅⟨ψ❘ + β⋅⟨φ❘⎠⋅(α⋅❘ψ⟩ + β⋅❘φ⟩)</text>
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			161	Vk9SSzVDWUlJPQo=
			162	</png>
			163	<latex>$$\left(\overline{\alpha} {\left\langle \psi\right\|} + \overline{\beta} {\left\langle \phi\right\|}\right) \left(\alpha {\left\|\psi\right\rangle } + \beta {\left\|\phi\right\rangle }\right)$$</latex>
			164	<prompt_number>22</prompt_number>
			165	</output>
			166	</outputs>
			167	</codecell>
			168	<markdowncell>
			169	<source>Distribute</source>
			170	</markdowncell>
			171	<codecell>
			172	<input>qapply(expand(ip))
			173	</input>
			174	<language>python</language>
			175	<prompt_number>23</prompt_number>
			176	<collapsed>0</collapsed>
			177	<outputs>
			178	<output>
			179	<output_type>pyout</output_type>
			180	<text>⎽ ⎽ ⎽ ⎽
			181	α⋅α⋅⟨ψ❘ψ⟩ + α⋅β⋅⟨φ❘ψ⟩ + β⋅α⋅⟨ψ❘φ⟩ + β⋅β⋅⟨φ❘φ⟩</text>
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			231	QUFBU1VWT1JLNUNZSUk9Cg==
			232	</png>
			233	<latex>$$\alpha \overline{\alpha} \left\langle \psi \right. {\left\|\psi\right\rangle } + \alpha \overline{\beta} \left\langle \phi \right. {\left\|\psi\right\rangle } + \beta \overline{\alpha} \left\langle \psi \right. {\left\|\phi\right\rangle } + \beta \overline{\beta} \left\langle \phi \right. {\left\|\phi\right\rangle }$$</latex>
			234	<prompt_number>23</prompt_number>
			235	</output>
			236	</outputs>
			237	</codecell>
			238	<markdowncell>
			239	<source><h2>Operators</h2></source>
			240	</markdowncell>
			241	<markdowncell>
			242	<source>Create symbolic operators</source>
			243	</markdowncell>
			244	<codecell>
			245	<input>A = Operator('A')
			246	B = Operator('B')
			247	C = Operator('C')</input>
			248	<language>python</language>
			249	<prompt_number>24</prompt_number>
			250	<collapsed>1</collapsed>
			251	<outputs />
			252	</codecell>
			253	<markdowncell>
			254	<source>Test commutativity</source>
			255	</markdowncell>
			256	<codecell>
			257	<input>AB == BA
			258	</input>
			259	<language>python</language>
			260	<prompt_number>25</prompt_number>
			261	<collapsed>0</collapsed>
			262	<outputs>
			263	<output>
			264	<output_type>pyout</output_type>
			265	<text>False</text>
			266	<prompt_number>25</prompt_number>
			267	</output>
			268	</outputs>
			269	</codecell>
			270	<markdowncell>
			271	<source>Distribute A+B squared</source>
			272	</markdowncell>
			273	<codecell>
			274	<input>expand((A+B)**2)</input>
			275	<language>python</language>
			276	<prompt_number>26</prompt_number>
			277	<collapsed>0</collapsed>
			278	<outputs>
			279	<output>
			280	<output_type>pyout</output_type>
			281	<text>2 2
			282	A⋅B + A + B⋅A + B</text>
			283	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQU1BQUFBQVpDQVlBQUFCbjdTSGdBQUFBQkhOQ1NWUUlD
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			315	</png>
			316	<latex>$$A B + \left(A\right)^{2} + B A + \left(B\right)^{2}$$</latex>
			317	<prompt_number>26</prompt_number>
			318	</output>
			319	</outputs>
			320	</codecell>
			321	<markdowncell>
			322	<source>Create a commutator</source>
			323	</markdowncell>
			324	<codecell>
			325	<input>comm = Commutator(A,B); comm
			326	</input>
			327	<language>python</language>
			328	<prompt_number>27</prompt_number>
			329	<collapsed>0</collapsed>
			330	<outputs>
			331	<output>
			332	<output_type>pyout</output_type>
			333	<text>[A,B]</text>
			334	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUM0QUFBQVdDQVlBQUFDL2tLNzNBQUFBQkhOQ1NWUUlD
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			346	R2ZjdWpzS24KUytJZHd2bGpNQ1B2cFBEeXFpdUpOMkUvUnJBanA4Y3h3Zm5DR1dFMEdudm96L05j
			347	d0FtOGpQeXcwTDNDd2FjRlQzRGYyS2srZzVzeApaeVJlSi9Gelczak5IeloySFZWa2ZLcXdld3Ex
			348	aXBGUTNWa2xMNlpUZTlyRWUzRm5tclF4M3ZpSThHY2lhNnZLcTlrdC9adzlHV3dWCnZBMVBzZTUv
			349	L052NEJmcHZkNXROTFJsdUFBQUFBRWxGVGtTdVFtQ0MK
			350	</png>
			351	<latex>$$\left[A,B\right]$$</latex>
			352	<prompt_number>27</prompt_number>
			353	</output>
			354	</outputs>
			355	</codecell>
			356	<markdowncell>
			357	<source>Carry out the commutator</source>
			358	</markdowncell>
			359	<codecell>
			360	<input>comm.doit()</input>
			361	<language>python</language>
			362	<prompt_number>28</prompt_number>
			363	<collapsed>0</collapsed>
			364	<outputs>
			365	<output>
			366	<output_type>pyout</output_type>
			367	<text>A⋅B - B⋅A</text>
			368	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUU0QUFBQVNDQVlBQUFEMTV1aVJBQUFBQkhOQ1NWUUlD
			369	QWdJZkFoa2lBQUFBaTVKUkVGVQpXSVh0MXp0b1ZFRVlCZURQR0VHTGxRUU1haFFpcEJVRjBTcVl3
			370	bWRockN4Rk8rMEYwVUt4VVZTMEVMRU5hV09oUWhvckZmR0JsWStnCklwcE9nbURBS2tyVXFMR1lR
			371	ZGRrN3U1OUpBRmhEeXgzZHM0L2M4N1p1WFBuTGkyVXdwS0M5VmN3aVRNTmFnYXhEWnZ3RlUvd1BY
			372	S3IKMEluck9JZXBndnFOc0JDNmVmSTJ4VmI4eEZDTzJnMll3YVVFdHk5eUkxWE1MSUp1a2J5WmFN
			373	UGRLSHc3Ui8yaFdMc3JneCtMcG1wVgpUQzJnYnRPOGJUa05IY1VOVEdOTmp2cCtZWnM4VG5BMXJN
			374	Y0VQdWZVejR2NTBpMmFONGt1NGRkdnd6Zys1Qmp6Rmc4enVCUENTaDRwCmEyaUJkY3ZrVFdJSWZi
			375	SDlGRDgwdmxOWFI0Tm5aL1YzeEw0SkhDNXJaaEYwYytWdGJ6SkpINWI2ZSt0L2pOKzdZanVGL25q
			376	dHhjWFkKcm1FQTcrS2NZMDN0RjhkODZKYkpPd2Z0ZUlTMWRYMUR3cXB1YmpEdUdyNWhlWUk3THh6
			377	dnUvT2FLSUNxdW1YenpzRXhISi9WZHlGTwp0TGZCdUZIcGh6T3NFQjY0NzRzWXlZbXF1b1h5Wm0z
			378	VjdqakpHLzhlN1QzeG1uWFNkR0FqTG1mdzA4SmRzVTdZUnBQQ1ZqcWRVWi9DCmlCQ29xbTQ5eXVh
			379	ZGcySHNTUFFmRUZiZ1pNYTQvWkVmeU9DM1IvNU9YaU01VVZXM2NON1U2YmdUbjNBdndZM0hhOVlL
			380	OUVlaDFKWloKaGxPeFBaZ3h2aXlxNkZiSit3ZGJoTk9qTTRQdmpRYUhNL2puZUpubzc4Rk5mTUhC
			381	WmlaS29LeHUxYnk2OFFxL1l1RnI0YjJvSHJlRQpsOEVaNFUvMEErekJTbUcxWGtSdVNqaWQ3c2ZQ
			382	S0o3aHF2QWNtaTlVMGEyU3Q0VVdXdmgvOEJ0MXY4VEcwN0FDaFFBQUFBQkpSVTVFCnJrSmdnZz09
			383	Cg==
			384	</png>
			385	<latex>$$A B - B A$$</latex>
			386	<prompt_number>28</prompt_number>
			387	</output>
			388	</outputs>
			389	</codecell>
			390	<markdowncell>
			391	<source>Create a more fancy commutator</source>
			392	</markdowncell>
			393	<codecell>
			394	<input>comm = Commutator(A*B,B+C); comm</input>
			395	<language>python</language>
			396	<prompt_number>29</prompt_number>
			397	<collapsed>0</collapsed>
			398	<outputs>
			399	<output>
			400	<output_type>pyout</output_type>
			401	<text>[A⋅B,B + C]</text>
			402	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUdBQUFBQVdDQVlBQUFBLzQ1bmtBQUFBQkhOQ1NWUUlD
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			425	</png>
			426	<latex>$$\left[A B,B + C\right]$$</latex>
			427	<prompt_number>29</prompt_number>
			428	</output>
			429	</outputs>
			430	</codecell>
			431	<markdowncell>
			432	<source>Expand the commutator</source>
			433	</markdowncell>
			434	<codecell>
			435	<input>comm.expand(commutator=True)</input>
			436	<language>python</language>
			437	<prompt_number>30</prompt_number>
			438	<collapsed>0</collapsed>
			439	<outputs>
			440	<output>
			441	<output_type>pyout</output_type>
			442	<text>[A,B]⋅B + [A,C]⋅B + A⋅[B,C]</text>
			443	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQU44QUFBQVdDQVlBQUFCQUhrbFFBQUFBQkhOQ1NWUUlD
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			451	N0lvSXMKYzRIMWNhUEhQbS90WjVmS1RRYWZQbzRHVGdtc1cyQUM2Mm5vdUFYWUN6eUt6QVl1SHJm
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			474	c0NZCkVCSG1BQUFBQUVsRlRrU3VRbUNDCg==
			475	</png>
			476	<latex>$$\left[A,B\right] B + \left[A,C\right] B + A \left[B,C\right]$$</latex>
			477	<prompt_number>30</prompt_number>
			478	</output>
			479	</outputs>
			480	</codecell>
			481	<markdowncell>
			482	<source>Carry out and expand the commutators</source>
			483	</markdowncell>
			484	<codecell>
			485	<input>_.doit().expand()
			486	</input>
			487	<language>python</language>
			488	<prompt_number>31</prompt_number>
			489	<collapsed>0</collapsed>
			490	<outputs>
			491	<output>
			492	<output_type>pyout</output_type>
			493	<text>2
			494	A⋅B⋅C + A⋅B - B⋅A⋅B - C⋅A⋅B</text>
			495	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQU84QUFBQVpDQVlBQUFEWnUzbTlBQUFBQkhOQ1NWUUlD
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			502	WGZBeHdMZENSYTdVWVc3UmtsTHo5eUpTcy9MRnp2Wk9BWDRMR0VkUFhBTWNBbzhETFFDTlNxN1NC
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			531	SzdBQUFBQUVsRlRrU3VRbUNDCg==
			532	</png>
			533	<latex>$$A B C + A \left(B\right)^{2} - B A B - C A B$$</latex>
			534	<prompt_number>31</prompt_number>
			535	</output>
			536	</outputs>
			537	</codecell>
			538	<markdowncell>
			539	<source>Take the dagger</source>
			540	</markdowncell>
			541	<codecell>
			542	<input>Dagger(_)</input>
			543	<language>python</language>
			544	<prompt_number>32</prompt_number>
			545	<collapsed>0</collapsed>
			546	<outputs>
			547	<output>
			548	<output_type>pyout</output_type>
			549	<text>2
			550	† † † † † † ⎛ †⎞ † † † †
			551	- B ⋅A ⋅B - B ⋅A ⋅C + ⎝B ⎠ ⋅A + C ⋅B ⋅A</text>
			552	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQVd3QUFBQWFDQVlBQUFDVGsyYlJBQUFBQkhOQ1NWUUlD
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			596	U3pRQUFBQUFTVVZPUks1Q1lJST0K
			597	</png>
			598	<latex>$$- B^{\dagger} A^{\dagger} B^{\dagger} - B^{\dagger} A^{\dagger} C^{\dagger} + \left(B^{\dagger}\right)^{2} A^{\dagger} + C^{\dagger} B^{\dagger} A^{\dagger}$$</latex>
			599	<prompt_number>32</prompt_number>
			600	</output>
			601	</outputs>
			602	</codecell>
			603	</cells>
			604	</worksheet>
			605	</worksheets>
			606	</notebook>

docs/examples/notebooks/decompose.ipynb

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	1	{"cells":[{"cell_type":"text","text":"<h1>Gate Decomposition</h1>"},{"code":"%load_ext sympyprinting","cell_type":"code","prompt_number":1},{"code":"from sympy import sqrt, symbols, Rational\nfrom sympy import expand, Eq, Symbol, simplify, exp, sin\nfrom sympy.physics.quantum import \nfrom sympy.physics.quantum.qubit import \nfrom sympy.physics.quantum.gate import \nfrom sympy.physics.quantum.grover import \nfrom sympy.physics.quantum.qft import QFT, IQFT, Fourier\nfrom sympy.physics.quantum.circuitplot import circuit_plot","cell_type":"code","prompt_number":2},{"cell_type":"text","text":"<h2>Example 1</h2>"},{"cell_type":"text","text":"Create a symbolic controlled-Y gate"},{"code":"CY10 = CGate(1, Y(0)); CY10\n","cell_type":"code","prompt_number":3},{"cell_type":"text","text":"Decompose it into elementary gates and plot it"},{"code":"CY10.decompose()\n","cell_type":"code","prompt_number":4},{"code":"circuit_plot(CY10.decompose(), nqubits=2)","cell_type":"code","prompt_number":5},{"cell_type":"text","text":"<h2>Example 2</h2>"},{"cell_type":"text","text":"Create a controlled-Z gate"},{"code":"CZ01 = CGate(0, Z(1)); CZ01\n","cell_type":"code","prompt_number":6},{"cell_type":"text","text":"Decompose and plot it"},{"code":"CZ01.decompose()\n","cell_type":"code","prompt_number":7},{"code":"circuit_plot(CZ01.decompose(), nqubits=2)","cell_type":"code","prompt_number":8},{"cell_type":"text","text":"<h2>Example 3</h2>"},{"cell_type":"text","text":"Create a SWAP gate"},{"code":"SWAP10 = SWAP(1, 0); SWAP10\n","cell_type":"code","prompt_number":9},{"cell_type":"text","text":"Decompose and plot it"},{"code":"SWAP10.decompose()","cell_type":"code","prompt_number":10},{"code":"circuit_plot(SWAP10.decompose(), nqubits=2)","cell_type":"code","prompt_number":11},{"cell_type":"text","text":"<h2>All together now</h2>"},{"code":"gates = [CGate(1,Y(0)), CGate(0,Z(1)), SWAP(1, 0)]","cell_type":"code","prompt_number":12},{"code":"for g in gates:\n dg = g.decompose()\n display(Eq(g, dg))\n circuit_plot(g, nqubits=2)\n circuit_plot(dg, nqubits=2) ","cell_type":"code","prompt_number":13},{"code":"%notebook save decompose.ipynb","cell_type":"code","prompt_number":30},{"code":"%notebook load gate_rules.ipynb","cell_type":"code","prompt_number":53}]} No newline at end of file	1	<?xml version="1.0" encoding="utf-8"?>
			2	<notebook>
			3	<name>decompose</name>
			4	<nbformat>2</nbformat>
			5	<worksheets>
			6	<worksheet>
			7	<cells>
			8	<markdowncell>
			9	<source><h1>Gate Decomposition</h1></source>
			10	</markdowncell>
			11	<codecell>
			12	<input>%load_ext sympyprinting</input>
			13	<language>python</language>
			14	<prompt_number>1</prompt_number>
			15	<collapsed>1</collapsed>
			16	<outputs />
			17	</codecell>
			18	<codecell>
			19	<input>from sympy import sqrt, symbols, Rational
			20	from sympy import expand, Eq, Symbol, simplify, exp, sin
			21	from sympy.physics.quantum import *
			22	from sympy.physics.quantum.qubit import *
			23	from sympy.physics.quantum.gate import *
			24	from sympy.physics.quantum.grover import *
			25	from sympy.physics.quantum.qft import QFT, IQFT, Fourier
			26	from sympy.physics.quantum.circuitplot import circuit_plot</input>
			27	<language>python</language>
			28	<prompt_number>2</prompt_number>
			29	<collapsed>1</collapsed>
			30	<outputs />
			31	</codecell>
			32	<markdowncell>
			33	<source><h2>Example 1</h2></source>
			34	</markdowncell>
			35	<markdowncell>
			36	<source>Create a symbolic controlled-Y gate</source>
			37	</markdowncell>
			38	<codecell>
			39	<input>CY10 = CGate(1, Y(0)); CY10
			40	</input>
			41	<language>python</language>
			42	<prompt_number>3</prompt_number>
			43	<collapsed>0</collapsed>
			44	<outputs>
			45	<output>
			46	<output_type>pyout</output_type>
			47	<text>C ⎛Y ⎞
			48	1⎝ 0⎠</text>
			49	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQURjQUFBQVlDQVlBQUFCZUlXV2xBQUFBQkhOQ1NWUUlD
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			69	dWJDa2Y3Z3o0QUFBQUEKU1VWT1JLNUNZSUk9Cg==
			70	</png>
			71	<latex>$$C_{1}{\left(Y_{0}\right)}$$</latex>
			72	<prompt_number>3</prompt_number>
			73	</output>
			74	</outputs>
			75	</codecell>
			76	<markdowncell>
			77	<source>Decompose it into elementary gates and plot it</source>
			78	</markdowncell>
			79	<codecell>
			80	<input>CY10.decompose()
			81	</input>
			82	<language>python</language>
			83	<prompt_number>4</prompt_number>
			84	<collapsed>0</collapsed>
			85	<outputs>
			86	<output>
			87	<output_type>pyout</output_type>
			88	<text>S ⋅CNOT ⋅S ⋅Z
			89	0 1,0 0 0</text>
			90	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUlnQUFBQWNDQVlBQUFDNlRmY0hBQUFBQkhOQ1NWUUlD
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			124	clIzSkNOTnJvckJlNmpOcEJ2a0U3S1JOZWp6ZWxES2VyU2hpcVpGdmpTNUtnYnZvVGJMRWhyZGlq
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			128	cHF6UUFBQUFCSlJVNUVya0pnZ2c9PQo=
			129	</png>
			130	<latex>$$S_{0} CNOT_{1,0} S_{0} Z_{0}$$</latex>
			131	<prompt_number>4</prompt_number>
			132	</output>
			133	</outputs>
			134	</codecell>
			135	<codecell>
			136	<input>circuit_plot(CY10.decompose(), nqubits=2)</input>
			137	<language>python</language>
			138	<prompt_number>5</prompt_number>
			139	<collapsed>0</collapsed>
			140	<outputs>
			141	<output>
			142	<output_type>pyout</output_type>
			143	<text>&lt;sympy.physics.quantum.circuitplot.CircuitPlot object at 0x2c85f10&gt;</text>
			144	<prompt_number>5</prompt_number>
			145	</output>
			146	<output>
			147	<output_type>display_data</output_type>
			148	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQVE4QUFBQ09DQVlBQUFBdnRJMzNBQUFBQkhOQ1NWUUlD
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			206	QUFBQQpBRWxGVGtTdVFtQ0MK
			207	</png>
			208	</output>
			209	</outputs>
			210	</codecell>
			211	<markdowncell>
			212	<source><h2>Example 2</h2></source>
			213	</markdowncell>
			214	<markdowncell>
			215	<source>Create a controlled-Z gate</source>
			216	</markdowncell>
			217	<codecell>
			218	<input>CZ01 = CGate(0, Z(1)); CZ01
			219	</input>
			220	<language>python</language>
			221	<prompt_number>6</prompt_number>
			222	<collapsed>0</collapsed>
			223	<outputs>
			224	<output>
			225	<output_type>pyout</output_type>
			226	<text>C ⎛Z ⎞
			227	0⎝ 1⎠</text>
			228	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQURrQUFBQVlDQVlBQUFCQTZGVVdBQUFBQkhOQ1NWUUlD
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			250	</png>
			251	<latex>$$C_{0}{\left(Z_{1}\right)}$$</latex>
			252	<prompt_number>6</prompt_number>
			253	</output>
			254	</outputs>
			255	</codecell>
			256	<markdowncell>
			257	<source>Decompose and plot it</source>
			258	</markdowncell>
			259	<codecell>
			260	<input>CZ01.decompose()
			261	</input>
			262	<language>python</language>
			263	<prompt_number>7</prompt_number>
			264	<collapsed>0</collapsed>
			265	<outputs>
			266	<output>
			267	<output_type>pyout</output_type>
			268	<text>H ⋅CNOT ⋅H
			269	1 0,1 1</text>
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			305	</png>
			306	<latex>$$H_{1} CNOT_{0,1} H_{1}$$</latex>
			307	<prompt_number>7</prompt_number>
			308	</output>
			309	</outputs>
			310	</codecell>
			311	<codecell>
			312	<input>circuit_plot(CZ01.decompose(), nqubits=2)</input>
			313	<language>python</language>
			314	<prompt_number>8</prompt_number>
			315	<collapsed>0</collapsed>
			316	<outputs>
			317	<output>
			318	<output_type>pyout</output_type>
			319	<text>&lt;sympy.physics.quantum.circuitplot.CircuitPlot object at 0x472d550&gt;</text>
			320	<prompt_number>8</prompt_number>
			321	</output>
			322	<output>
			323	<output_type>display_data</output_type>
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			371	</png>
			372	</output>
			373	</outputs>
			374	</codecell>
			375	<markdowncell>
			376	<source><h2>Example 3</h2></source>
			377	</markdowncell>
			378	<markdowncell>
			379	<source>Create a SWAP gate</source>
			380	</markdowncell>
			381	<codecell>
			382	<input>SWAP10 = SWAP(1, 0); SWAP10
			383	</input>
			384	<language>python</language>
			385	<prompt_number>9</prompt_number>
			386	<collapsed>0</collapsed>
			387	<outputs>
			388	<output>
			389	<output_type>pyout</output_type>
			390	<text>SWAP
			391	1,0</text>
			392	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUU4QUFBQWNDQVlBQUFBZ0x1TGZBQUFBQkhOQ1NWUUlD
			393	QWdJZkFoa2lBQUFCRGRKUkVGVQphSUh0Mkh1SVYxVVFCL0NQN3VyNjNEUnJlLzBSdUVUMlFIcFEw
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			419	CkFBQUFBRWxGVGtTdVFtQ0MK
			420	</png>
			421	<latex>$$SWAP_{1,0}$$</latex>
			422	<prompt_number>9</prompt_number>
			423	</output>
			424	</outputs>
			425	</codecell>
			426	<markdowncell>
			427	<source>Decompose and plot it</source>
			428	</markdowncell>
			429	<codecell>
			430	<input>SWAP10.decompose()</input>
			431	<language>python</language>
			432	<prompt_number>10</prompt_number>
			433	<collapsed>0</collapsed>
			434	<outputs>
			435	<output>
			436	<output_type>pyout</output_type>
			437	<text>CNOT ⋅CNOT ⋅CNOT
			438	1,0 0,1 1,0</text>
			439	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQU9RQUFBQWNDQVlBQUFCeGxoUDVBQUFBQkhOQ1NWUUlD
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			472	d2I2YnpNQUFBQUFTVVZPUks1Q1lJST0K
			473	</png>
			474	<latex>$$CNOT_{1,0} CNOT_{0,1} CNOT_{1,0}$$</latex>
			475	<prompt_number>10</prompt_number>
			476	</output>
			477	</outputs>
			478	</codecell>
			479	<codecell>
			480	<input>circuit_plot(SWAP10.decompose(), nqubits=2)</input>
			481	<language>python</language>
			482	<prompt_number>11</prompt_number>
			483	<collapsed>0</collapsed>
			484	<outputs>
			485	<output>
			486	<output_type>pyout</output_type>
			487	<text>&lt;sympy.physics.quantum.circuitplot.CircuitPlot object at 0x7f082c973650&gt;</text>
			488	<prompt_number>11</prompt_number>
			489	</output>
			490	<output>
			491	<output_type>display_data</output_type>
			492	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQU5jQUFBQ09DQVlBQUFDaS9KMktBQUFBQkhOQ1NWUUlD
			493	QWdJZkFoa2lBQUFBQWx3U0ZsegpBQUFMRWdBQUN4SUIwdDErL0FBQUNybEpSRUZVZUp6dDNXdG9V
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			560	</png>
			561	</output>
			562	</outputs>
			563	</codecell>
			564	<markdowncell>
			565	<source><h2>All together now</h2></source>
			566	</markdowncell>
			567	<codecell>
			568	<input>gates = [CGate(1,Y(0)), CGate(0,Z(1)), SWAP(1, 0)]</input>
			569	<language>python</language>
			570	<prompt_number>12</prompt_number>
			571	<collapsed>1</collapsed>
			572	<outputs />
			573	</codecell>
			574	<codecell>
			575	<input>for g in gates:
			576	dg = g.decompose()
			577	display(Eq(g, dg))
			578	circuit_plot(g, nqubits=2)
			579	circuit_plot(dg, nqubits=2) </input>
			580	<language>python</language>
			581	<prompt_number>16</prompt_number>
			582	<collapsed>0</collapsed>
			583	<outputs>
			584	<output>
			585	<output_type>display_data</output_type>
			586	<text>C ⎛Y ⎞ = S ⋅CNOT ⋅S ⋅Z
			587	1⎝ 0⎠ 0 1,0 0 0</text>
			588	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQU5NQUFBQWNDQVlBQUFEN3VZcTRBQUFBQkhOQ1NWUUlD
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			639	QVNBTWNBQUFBQVNVVk9SSzVDCllJST0K
			640	</png>
			641	<latex>$$C_{1}{\left(Y_{0}\right)} = S_{0} CNOT_{1,0} S_{0} Z_{0}$$</latex>
			642	</output>
			643	<output>
			644	<output_type>display_data</output_type>
			645	<text>C ⎛Z ⎞ = H ⋅CNOT ⋅H
			646	0⎝ 1⎠ 1 0,1 1</text>
			647	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQU1rQUFBQWNDQVlBQUFETE5adVpBQUFBQkhOQ1NWUUlD
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			669	V3BTZm9pNC9ha2c5eHltZko5NmszVkhwY0FpMnpQUmltZFZyQ2RDendRSngrbTVFNUJieng5ClZO
			670	a3lCM21Ca3R2am5UQUp5eFpWL0lKczd3ME55dVo0RndIRUoxNWpVQzVObGZzTDZHbVQ5VUFNYmtJ
			671	Mk1nak5ETXNId1JSVlozOEgKK1Naa1hQMmU4YVJxaitIQTdiWm5HY2dpZEFnWjg2bEExemo1WktR
			672	dlN3M3FMRlZseHpuSWx5bTU2VndKZ3JCc1VjVnU0R2VEY3MyQgpPWVk2SjFQZDU5V3hCR244VUky
			673	c0ZFbnJtZkk3d2U2R1hwUWhMa3NqalN3RDJmcDNKYUUzVlhzc0FscG9ubHVyNlRERU52SFpyakZL
			674	Ck5zK2p6ZzZJdmQ2bGVyYk1Zai9pTnRja1lkbWlpZ1hJQU5sUFhPM01KSEZsejBYZTRFbFVUN3RO
			675	QTE3MzBQZVlxbGQzSU5pQnhFQ3oKSHJKQ3RYZlJkeGc1QkhNaUYvalE1NmZRUVZjYWtscmQ3aUMz
			676	WEpJRkRuSzMvaVJyRHd1bm1MRXRNc0hYSTBGNlBPMVVuUlV1OWFVagpoNytWU0ZaTFJ5dWx4OHYy
			677	Ym5QSGpwZnR3N1JGRmIyUlE1Z3BMbVhHa3hpRU4wZUNaZXMwZGlGeWJjSmlITzR4eVVna0tIMVdJ
			678	MHRICkRIMlgrajRXT1RNNWlmT2hXaVl5RUoxZDZneVNIcXErK1E1eXkzMjVSU1B6Nms4eTlyRElB
			679	cDUzK2QwcTFhNUpHdGxPSld1bGtkVUIKWGtGMkNDZVhCbUt4endTWE1sNXpKeDRUMjRkcGl3UnVR
			680	STd6SHlUUkYrNEl6S1Y2d0RPRFJQOTFBSEpXWVpHdjlPbTI1R3VRU2JDUwp4QlJudW1wb21XcjB4
			681	YmJmblhEcFNDOWtJRXpPYjRKZ0dqSFhSVWM1M2pHU1czLzgyc05pSHU0bnlqZXBkdlhTeUpvaG1i
			682	QWlFcytjCnVpSlhoTmFpZCtQaUtWYjZuUTZRd1h2dTZIQWJxN0J0a1VCUDVHM2NqRnlnV3c0OGdU
			683	N2pWSXgwMXFLVGFvaDFueWNMbWVodGJiL0wKUUE2c3ppQTd6WGIxK1F6eEthMExhT3Q5ZG1RMDV0
			684	bTBWSmlEK0xiVzViMnZrTEdxanl3SXE1R0padldqQXYxdUNkNkc4V09QaWNRdQpHZjZESEp4ZG9D
			685	bFhWN1ZmZDVZQTRyNU5SOXpNbDVFWDVoa2tablNLUVdZaE1jb2VZdjMrVWoyYnJTbnZOWGQwNk1h
			686	cUptMlJGT1hJCk5tYlJSalVrZnFYWkNnd09zRTYzanN4Q1Z0aHppVkFNY3c1Z01uZnNoRDFXV3Yw
			687	bUo3bHVIRUE2WnRHWTJMVjJpNlZVVDBXR1FUcHcKSXhKOFJkUitUT1pPclNEVmw2U0N4RGUvSmZB
			688	RGlhZTRSVWhLMWkzakZBU1BJaTZiN3ZRM292WmhNbmRxQmFtK0pEdEp6SElNcG5wVwo1VFNTNVpw
			689	UGtuOXVNU0FIR0VIc0pEbWk5dU0yZDlKSjRaWnUwS1I1RjNIbENKSXR1Qjc1MDA0TEpFaXpyd1pI
			690	a0tCOEJCS2pKRU0rCmN0VWhEN25DMHBSWWtENFgrYy9FMFNSMW53M2Mrbk0rNERaM0xrZHVOZTlB
			691	L3JBVjlsaTU2bmZLVlBpbERwSXU5Tm9xNnlOWmwyVEkKSW5Fbk9nWDhHSURlczRWYmY4NG5uT1pP
			692	SnJLN2xCTCtXRVcyaURnbkdZRHNLR2VkVkdPU2lJZ3dxSXZjbXRoL3Roc1NFVkdiY1RybwpqSWlJ
			693	aUlpSWlJaUlpSWlJT0svNEgyR1pJUFRKVVRpYkFBQUFBRWxGVGtTdVFtQ0MK
			694	</png>
			695	<latex>$$C_{0}{\left(Z_{1}\right)} = H_{1} CNOT_{0,1} H_{1}$$</latex>
			696	</output>
			697	<output>
			698	<output_type>display_data</output_type>
			699	<text>SWAP = CNOT ⋅CNOT ⋅CNOT
			700	1,0 1,0 0,1 1,0</text>
			701	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQVVZQUFBQWNDQVlBQUFBZHRSYlZBQUFBQkhOQ1NWUUlD
			702	QWdJZkFoa2lBQUFDTVZKUkVGVQplSnp0bkhtUUZjVWR4ejhybDRBdUlJcEdUVUFwU3dJSjVSbE5Q
			703	RERlNGxWR3E2S0ZRU0pSUzBRdGcwbU1XbW9PUWp6S1JCQkZRVW5FCkk1U2F5NFFqRUxNQ0doUzB2
			704	TkR5S1BFSUtzR05paGVzbU0wZjM1NjhtWjZlbVg2N0QvY2QvYWw2TmU5MTkvVHhuZDkwOS95NjUw
			705	RWcKRUFnRXZPa09YQURjQ2Z3TytDTXdIdWdMekFjV0F4dUJkbUFUc0J6WTJweDdDZkNhaVdzSG5n
			706	VEd4dkllQi96YnhIMEkvTUtqUG44QQp6c3lKN3dFc0JGNDIrYjRITEFJV0FFdUJWNEMvQUh0NmxG
			707	VXQ5QVpPQjZZRFMxRDdXb0JUVFB3UndBOWk2YWVoNjdBSitBd1k2Y2h6ClB2QXgwdWhkNEhwSG1t
			708	SEFMY0M5d0N4Z0hqQVgrR3BHUGM4dytXMEVWaUxObjZkMEhSYWJ1ajhPL0JkNE1hZk5uYUVXOUFw
			709	YTFaNVcKLzZjMzZrUXVCN2FJaGY4WWVBRjFsQUJYbTByT2N1UXgxTVRkbjFIRzE0Qi9BbHQ1MU9k
			710	YkpxOHJQTkllWjlKZWFJWDNBRzRGTmdDSApldVRUMWV5UHRKNEpIQUwwTk9GYkF0Y0J2MGFEeW9I
			711	V2Vkc0NUeUVOWm1Ua1BRNE5ORTFXK0JiQXplWjgyL0JQUUVZL2xqUUwwYlhwCkdRdUxqSHFpbyt5
			712	L1o5U3JNOVNLWGtHcjJ0SXF3YjFJTkJkdkF0OHozeWVoU3Q3b1NEZmN4RDJRa2M5a1lKQkhYZm9D
			713	ejVxOGJ2SkkKSDNYV0l4eHhYelJ4ZDN2azA1Vk1CdHJRak4xRk4yQUY4QUhxOE9PY0JJd0JWcG40
			714	clVuemZXQnZLNndKWGZQNTZBWnhjVFd3SHRneApGallRMll2TmJVanIzYTN3TDZFYnBKTFVpbDVC
			715	cTlyU0trRS9KTnlSR2ZFTGdWM005M0ZrZDR3elROd2pqcml2a0gxaGJINkpPdUoyCjRQY2U2WmNE
			716	cmFSSExKQ1k3Y0EvUE12dUNpYWdPcDVla0c0eThGZEgrUFhBenNENUpwOEpqalIzb0JzZ3psUmdE
			717	Wm9WWkhHWXlmUEsKV05oWndGR090QzhCYnp2Qzl3WXV5aW1qWEdwSnI2QlZOdFdtVllwalVRWFB5
			718	NGkvUFBiOUJOd2Q0K0ZvTnJrZVRkbHRwcE1lalZ4OApHWTBhUFpBUHdkWEp4dWtMZkFyOE9TTity
			719	S252ZUkreXU0SVI2SkdpeFNQdGVOeUdFSTJ5L1lDUGdHZXMrQ1pnamhYMk02VExpUVZsCkRpYnRI
			720	am1EcExzRk5PcTM0eDd4UndEN0ZwVGpTNjNwRmJUS3BwcTBjckl0Y3JCdVJQNkdBOG51eEE0ZzNU
			721	SDJRdjdKbnNCcVlKMTEKemtuQWFNKzZ6QU4yTmQ5YjBRSktIa2VZK3Z6UUVkY05PVzVYNDM0RXFB
			722	WldvUHJidmgwWCt3RkRyTEJta3Y3ZW1ZNzhScUxGc1lnZAowUlBDdjBpUDlEYUhtdnptRjZRNzFh
			723	UTd2eUJkWjZrSHZZSldvcHEwU3ZYSUFPK2dSUmJRckc4cFdtR2FEdlMzMHJZNnp2OFJXblZxCk0v
			724	RURLRDNXOWtJQ3pQT28yeGkwMmhSMWhtdUJIUXJPT2RnY2wxbmhPNkdSYUJXd0IvS1BWSkxSYU9X
			725	OW5NOTBLNC90Z1gyQXh4ejEKZC9FbzhLb1ZkZ0R3Y094MzVITTVOeFkyQ3ExQ1JreEFBOThjdE5x
			726	WXgvQlkyWGxFMStHaGduU2RvVjcwQ2xxSmF0R3FrR0hBVDFCRAoyMUJQZlorVlpoREpHZU5Ra28r
			727	eEMwMzhBUE43RW1tbnFZdG1KRkNmV05pREpxOStPZWN0UVJkZ0Z2Snh6Z0JtbTNZYzdsRnVWM0l5
			728	CmFsOW5ITWhUMERXSXN4ek4vcmN6djM5TDhnbmdJVlB1c1I3NXp6VnBYYjZmT004Qy84SHQ1NjBV
			729	OWFKWDBFcFVpMVpsTVFKNEhmbnYKNHF0SzNVbDJqQStRWEEyKzI4UVBCYjZBT2lnZmJxQzBueXJp
			730	SHR5clVSRmJvcTA0S3p6THFEYXVJZCszRzJjN1NvTk5uTG1Pc0doNwpRL1NJYzQ4VnY4YkVEeWVm
			731	d2VqNnYwMitLMklnOGdkbitYa3JSVDNvRmJRUzFhUVZrSDZVdnNTWlNvK2dkNkZHeG52clRjRDc1
			732	dnNwCmFGUDNxbGo4TytZNEVDM2p1elo4MnV3Qm5JWldvaGZFUGw4MzhWbVAwL3VoUi9VV2p6S3Fr
			733	ZGZNOFUyUHRKY2lRNHJUR3puWGJlYWkKVWZZY1pLRDJZdGhhYzN5ZmZDNUNBK0ZaNUxzaURrSTJz
			734	aVFuVFNXb0I3MkNWcUthdEFLU0hlTldhTE5uRnMzQUU4QW5WbmdyNnVXdgpJTG1OSTRvRE9BYUpW
			735	aVJRRStvOFJ3RkhXNTlwSmsxV3h4ajVIMW9LeXVnQjNFNXBVYWVJWWNqbk9oRzRGaG1KemRGb1oz
			736	NDVuNmxXCkhsRzlpOTdNR1k0V2tENjB3dmZIN1ovWkFQd0dPZE9ua0Rhc3BlWjRXRTZaK3lDam5V
			737	MzJ2dFNJNkRwa0dYQ2w5Rzh4eDFyV3EwZ3IKOExPL2lDeHRXOHl4M3JVcXg3YThkVDBHelFCdFB3
			738	TElsN2dXK0tZamJnWHlNN2oySlU1RXM4emx1QmQ2Yk1hVGZBMHB6cmR4djlFUwpzUmpWdnprbi96
			739	T0JuNXAwd3p6cTB3MTREam12UWJQWTJ6M082d2hOU011blNmcFc0K3lPRExHWEkrNHF0TDNKeFc3
			740	b01hU050QUVjCmpQUndQU3FCZEZxSC9NcytieW10UktOK2QwZGNKZld2QjczeXRJTHk3QzlQMjBi
			741	UXFoemJLdXUrdmhadG5yeVBwRTlnVjlTeFRjbzQKYjRFcHhGWGgwMUJuZGxCQlJVR3pydGZKWHRh
			742	UE5vQk9jY1ExbzMxVkt6M0tBWTEwUGpmbXlTVDNUZzVDTStZczQrb3NBNUFCUDRoVwowaVA2bzFY
			743	NktTUmZqNHJYNncxS1RuQVhpOGplQnpvR0dWMThaTytEMnY4Q3laWEhQQVlqdzF4Y2tLNVMrdGV5
			744	WGo1YWRjVCtzclN0CmQ2MGlmR3lyTEYzbkFOdWdXZUVpSk9JQzlON2pOM0lLbVVuMld6S2pTRy80
			745	dE5rTGRjanRsR2FYOFU2Mk4zcFRwZFhFcnplL1I1clAKVW5UaDJ0RUZXSVl1Umg2K04rWmxwRGVU
			746	YmlMOXlsTWw2UWRjRFB3SmJhKzRBejEydS9aKzlrTHZtMjlBN1YrRHRrdTVPQkg0ZVU2NQpSNkl0
			747	UkxQUk5iMFREWWE3RmRSM0NLVS9JRmhINmFYL1pjRGYwUFcxcWFUK3RhVFhFTXJUcWlQMmw2ZHRQ
			748	V3NWNFdOYlhYRmYxd1MrCk4rYXYwQmFFT0I5UUczOUNVYzBFL2Yzb1NQdDl0YTFYZk5wZnFLdVAz
			749	NitSV1kxR3FqaDkwRXBjWVBQVDZQbzNldnMzRjRXNmhvNHgKbnlkSS9nUFFJT1RJZnFOcnF0TndO
			750	THIramQ3K3pVWFFOUVBYZEhza3liL1RBcTJVdlVYSktmc2RzbGZZQXY0RS9mM0lhNzlMTHdpUAow
			751	ajYyVldoWFJTOTIxeHVqMFYrbEhZTEU2WS8rT0JPMFQvSlVrb3RGYldnVU9SdTl1Yk12ZW52bnZj
			752	K2x0dlZIMEw4ODh0cHY2NVduCmJTTlFqbTAxdWwybEdJQTJpRWVmK0JhRWJzQjNjODUxN2U4S2xF
			753	ZlF2K1BZN2JmMXl0TzJFZWlvYlRudEttdXpaTDN5Yms3Y1orVHIKc2JIQ2RXbEVndjRkeDI2L3JW
			754	ZWV0bzFBUjIycjBlMnFrT1BSUnRKQTF4RDBMNCtnbHo5QnEwN1FhTFBuYWlQb1h4NUJMMytDVm9G
			755	QQpJQkFJQkFLQlFDQVFDQVFDZ1VEZ2MrVi9xazJaZlg2Uys5WUFBQUFBU1VWT1JLNUNZSUk9Cg==
			756	</png>
			757	<latex>$$SWAP_{1,0} = CNOT_{1,0} CNOT_{0,1} CNOT_{1,0}$$</latex>
			758	</output>
			759	<output>
			760	<output_type>display_data</output_type>
			761	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUo4QUFBQ09DQVlBQUFBdmtkcmVBQUFBQkhOQ1NWUUlD
			762	QWdJZkFoa2lBQUFBQWx3U0ZsegpBQUFMRWdBQUN4SUIwdDErL0FBQUJFSkpSRUZVZUp6dDNiMUxJ
			763	MXNBaHZGM2todkJ4SXVhUW0xc2JRU3hFUEVUc1JNRXNSRUxDOEZHCnJOTUlGbmFDTmxZV1ZoYlJ5
			764	azcvQTIzOHJBU3hFd3ZGbEVZaUdORFpRdHg3Wlc4Vzc2N3VhMmFlSDB5VHhNT1o4REJubklGTUVJ
			765	WmgKVWRMZkF2NndJQXpEMEQwSnhGUENQUUhFRi9IQmh2aGdRM3l3SVQ3WUVCOXNpQTgyeEFjYjRv
			766	TU44Y0dHK0dCRGZMQWhQdGdRSDJ5SQpEemJFQnh2aWd3M3h3WWI0WUVOOHNDRSsyQkFmYklnUE5z
			767	UUhHK0tERGZIQmh2aGdRM3l3SVQ3WUVCOXNpQTgyeEFjYjRvTk43T0lyCkZvczZPenZUL2YyOWV5
			768	cXhGNXY0d2pEVXdzS0NtcHViMWQvZnIrYm1aaTB1TG9wZkJmYUp6Vzh5YjIxdGFYWjJWcVZTNmZ0
			769	cm1VeEcKR3hzYm1waVlNTTRzdm1JVFgwOVBqdzRQRDM5NGZXQmdRUHY3KzRZWm9XSjhRUkQ4NmJr
			770	Z1ppcWU4NFZoR0tsdGVYbFp0YlcxYi9ZeApuVTVyZFhYVlByZTRickZaZGg4Zkh6VTZPcXFEZ3dP
			771	VlNpVmxNaGtOREF4b1oyZEhOVFUxN3VuRlVtemlrMTZPNWljbkorcnU3dGJ4CjhiRzZ1cnJjVTRx
			772	MVdNWDNLZ2dDeFhDM3Y1ellYT2ZEMTBOOHNDRSsyQkFmYklnUE5zUUhHK0tERGZIQmh2aGdRM3l3
			773	SVQ3WUVCOXMKaUE4MnhBY2I0b01OOGNHRytHQkRmTEFoUHRnUUgyeUlEemJFQnh2aWd3M3h3WWI0
			774	WUVOOHNDRSsyQkFmYklnUE5zUUhHK0tERGZIQgpodmhnUTN5d0lUN1lFQjlzaUE4MnhBY2I0b01O
			775	OGNFbU12RmxzMWtGUWZDdVRkSzdQNXZOWnMxN0ZsMlJlUUxSWnoxVmlLY1ZmWjdJCkhQbFFmU0la
			776	MytucHFmcjcrMVZUVTZORUlxRzZ1am9ORHcvcjRlSGh6ZWNLaFlLeTJhd1NpWVNTeWFRNk96dDFl
			777	M3RybW5VTWhSSHgKWDd1U3orZkRJQWpDNmVucGluK1h6K2ZEM3Q3ZThQcjYrdDNqNG1ORStwenY0
			778	ZUZCVFUxTlNxVlNLaFFLUHp4WDkvTHlVdlB6ODlyYwozRlFxbFhyM3VQZ1lrVngyWDZYVGFZMlBq
			779	K3Z1N2s2N3U3dHYzaXNXaThybGNscGZYNjhZSGo1WHBPT1RwS21wS1VuUzF0Ylc5OWVlCm5wNDBO
			780	emVubFpVVk5UUTB1S1lXZTVGZWRxV1gwRnBhV25SL2Y2OUNvYUQ2K25ybGNqbU5qWTFwYUdqb2w4
			781	ZkY3NHY4a1MrWlRHcHkKY2xMbGNsbmIyOXRhVzF0VGUzdjd1OExENTRwOGZOSS9TKy9TMHBKdWJt
			782	NDBNek5qbmhHa255eTdyN2VocWttbDVmSDUrVm5wZEZvZApIUjA2T2pyNlgyTlc0L2RRTGY2cTlF
			783	YTFuZWY4TEpMejgzT1Z5MldOakl6ODB0alY5bDFVaTFnc3UzdDdlNUxFZWQ0WEU1djRVcW1VCit2
			784	cjYzRlBCdjhUaVVrdHJhNnRhVzF0MWVIajRZZVBpOTBYMnlIZDFkYVdob1NHMXRiV3BVQ2pvNHVK
			785	Q2c0T0RieTQyd3l2eVI3NnYKT2k0aWZPVEQxMGQ4c0NFKzJCQWZiQ3JlNGFoR24zRXJyTEd4OGNQ
			786	SHhJdkl4TWQvcE5XSFpSYzJ4QWNiNG9NTjhjR0crR0JEZkxBaApQdGdRSDJ5SUR6YkVCeHZpZ3cz
			787	eHdZYjRZRU44c0NFKzJCQWZiSWdQTnNRSEcrS0REZkhCaHZoZ1EzeXdJVDdZRUI5c2lBODJ4QWNi
			788	CjRvTU44Y0dHK0dCRGZMQWhQdGdRSDJ5SUR6YkVCNXR2WUJZc3hJKzRoTXdBQUFBQVNVVk9SSzVD
			789	WUlJPQo=
			790	</png>
			791	</output>
			792	<output>
			793	<output_type>display_data</output_type>
			794	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQVE4QUFBQ09DQVlBQUFBdnRJMzNBQUFBQkhOQ1NWUUlD
			795	QWdJZkFoa2lBQUFBQWx3U0ZsegpBQUFMRWdBQUN4SUIwdDErL0FBQUNUNUpSRUZVZUp6dDNFdElW
			796	SDhZeHZIMzJEQ2psZUNGTGhob0dCUkJFR0ZFRUZtTFNNelNMTHU0CmtHaFRVRkl0dWkyc0lJcWlG
			797	a0VVRmFVdXdxZzJicnFIMVNLNi9CZWhkTEVMMnQwd0tteTZwL2IrZDVIbHFQTjY5TXlNM3cvTTVw
			798	d3oKUDU3ZmVlTnBqb002cWhvVWtVUUJnREE0cXFwZWh3QVFmZUs4RGdBZ09sRWVBRXdvRHdBbWxB
			799	Y0FFOG9EZ0FubEFjQ0U4Z0JnUW5rQQpNS0U4QUpoUUhnQk1LQThBSnBRSEFCUEtBNEFKNVFIQWhQ
			800	SUFZRUo1QURDaFBBQ1lVQjRBVENnUEFDYVVCd0FUeWdPQUNlVUJ3SVR5CkFHQkNlUUF3b1R3QW1G
			801	QWVBRXdvRHdBbWxBY0FFOG9EZ0FubEFjQ0U4Z0JnUW5rQU1LRThBSmhRSG9ncXo1OC9sMGVQSG9t
			802	cWVoMWwKd0tNOEVCV2FtcHBrOHVUSk1uNzhlTW5LeXBMUm8wZkxuVHQzdkk0MW9EbEtoU01LWkdW
			803	bFNWMWRuYlMzdC84K2xweWNMRTFOVFJJZgpIKzloc29HTFR4NkllRStmUHBYNit2b094U0VpMHQ3
			804	ZUxoY3ZYdlFvRlVKKzhuQWNwNyt6QUlnaXZsQW5lSnBCcFBqMTY1ZWtwNmZMCjY5ZXZPeHhQU0Vp
			805	UVY2OWVTVXBLaWtmSkJqWWVXeER4NHVMaXBMcTZXcEtTa2lReE1WRkVST0xqNDZXeXNwTGk4QkEv
			806	TUVYVStQcjEKcTF5NGNFRVdMbHdvNzk2OWs5VFVWSzhqRFdpVUI2S080emc4VmtjQUhsc0FtRkFl
			807	QUV3b0R3QW1sQWNBRThvRGdBbmxBY0NFOGdCZwpRbmtBTUtFOEFKaFFIZ0JNS0E4QUpwUUhBQlBL
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			965	</output>
			966	<output>
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			1036	</png>
			1037	</output>
			1038	</outputs>
			1039	</codecell>
			1040	</cells>
			1041	</worksheet>
			1042	</worksheets>
			1043	</notebook>

docs/examples/notebooks/dense_coding.ipynb

0 +726 -1

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	@@ -1,1 +1,726 b''
	1	{"cells":[{"cell_type":"text","text":"<h1>Dense Coding\n</h1>"},{"code":"%load_ext sympy_printing","cell_type":"code","prompt_number":1},{"code":"from sympy import sqrt, symbols, Rational\nfrom sympy import expand, Eq, Symbol, simplify, exp, sin\nfrom sympy.physics.quantum import \nfrom sympy.physics.quantum.qubit import \nfrom sympy.physics.quantum.gate import \nfrom sympy.physics.quantum.grover import \nfrom sympy.physics.quantum.qft import QFT, IQFT, Fourier\nfrom sympy.physics.quantum.circuitplot import circuit_plot","cell_type":"code","prompt_number":2},{"code":"psi = Qubit('00')/sqrt(2) + Qubit('11')/sqrt(2); psi\n","cell_type":"code","prompt_number":3},{"code":"circuits = [H(1)CNOT(1,0), H(1)CNOT(1,0)X(1), H(1)CNOT(1,0)Z(1), H(1)CNOT(1,0)Z(1)X(1)]","cell_type":"code","prompt_number":4},{"code":"for circuit in circuits:\n circuit_plot(circuit, nqubits=2)\n display(Eq(circuitpsi,qapply(circuitpsi)))","cell_type":"code","prompt_number":5},{"code":"%notebook save dense_coding.ipynb","cell_type":"code","prompt_number":28},{"code":"%notebook load teleportation.ipynb","cell_type":"code","prompt_number":181}]} No newline at end of file	1	<?xml version="1.0" encoding="utf-8"?>
			2	<notebook>
			3	<name>dense_coding</name>
			4	<nbformat>2</nbformat>
			5	<worksheets>
			6	<worksheet>
			7	<cells>
			8	<markdowncell>
			9	<source><h1>Dense Coding
			10	</h1></source>
			11	</markdowncell>
			12	<codecell>
			13	<input>%load_ext sympyprinting</input>
			14	<language>python</language>
			15	<prompt_number>2</prompt_number>
			16	<collapsed>1</collapsed>
			17	<outputs />
			18	</codecell>
			19	<codecell>
			20	<input>from sympy import sqrt, symbols, Rational
			21	from sympy import expand, Eq, Symbol, simplify, exp, sin
			22	from sympy.physics.quantum import *
			23	from sympy.physics.quantum.qubit import *
			24	from sympy.physics.quantum.gate import *
			25	from sympy.physics.quantum.grover import *
			26	from sympy.physics.quantum.qft import QFT, IQFT, Fourier
			27	from sympy.physics.quantum.circuitplot import circuit_plot</input>
			28	<language>python</language>
			29	<prompt_number>3</prompt_number>
			30	<collapsed>1</collapsed>
			31	<outputs />
			32	</codecell>
			33	<codecell>
			34	<input>psi = Qubit('00')/sqrt(2) + Qubit('11')/sqrt(2); psi
			35	</input>
			36	<language>python</language>
			37	<prompt_number>4</prompt_number>
			38	<collapsed>0</collapsed>
			39	<outputs>
			40	<output>
			41	<output_type>pyout</output_type>
			42	<text>⎽⎽⎽ ⎽⎽⎽
			43	╲╱ 2 ⋅❘00⟩ ╲╱ 2 ⋅❘11⟩
			44	────────── + ──────────
			45	2 2</text>
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			77	OXBPCjJrQSsrcmgraklwOUh0c2tkcW4wNlFkK1pHTGhmeG9pL05XNko3TDRHRVMrUmMyQzFjWUFO
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			81	YXU1ejl2CjByak9nS2lER3dBQUFBQkpSVTVFcmtKZ2dnPT0K
			82	</png>
			83	<latex>$$\frac{1}{2} \sqrt{2} {\left\|00\right\rangle } + \frac{1}{2} \sqrt{2} {\left\|11\right\rangle }$$</latex>
			84	<prompt_number>4</prompt_number>
			85	</output>
			86	</outputs>
			87	</codecell>
			88	<codecell>
			89	<input>circuits = [H(1)CNOT(1,0), H(1)CNOT(1,0)X(1), H(1)CNOT(1,0)Z(1), H(1)CNOT(1,0)Z(1)X(1)]</input>
			90	<language>python</language>
			91	<prompt_number>5</prompt_number>
			92	<collapsed>1</collapsed>
			93	<outputs />
			94	</codecell>
			95	<codecell>
			96	<input>for circuit in circuits:
			97	circuit_plot(circuit, nqubits=2)
			98	display(Eq(circuitpsi,qapply(circuitpsi)))</input>
			99	<language>python</language>
			100	<prompt_number>6</prompt_number>
			101	<collapsed>0</collapsed>
			102	<outputs>
			103	<output>
			104	<output_type>display_data</output_type>
			105	<text>⎛ ⎽⎽⎽ ⎽⎽⎽ ⎞
			106	⎜╲╱ 2 ⋅❘00⟩ ╲╱ 2 ⋅❘11⟩⎟
			107	H ⋅CNOT ⋅⎜────────── + ──────────⎟ = ❘00⟩
			108	1 1,0 ⎝ 2 2 ⎠</text>
			109	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQVdVQUFBQW1DQVlBQUFBY1Bvck5BQUFBQkhOQ1NWUUlD
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			186	eS84QlhNMmZXWUlsVDVRQUFBQUFTVVZPUks1Q1lJST0K
			187	</png>
			188	<latex>$$H_{1} CNOT_{1,0} \left(\frac{1}{2} \sqrt{2} {\left\|00\right\rangle } + \frac{1}{2} \sqrt{2} {\left\|11\right\rangle }\right) = {\left\|00\right\rangle }$$</latex>
			189	</output>
			190	<output>
			191	<output_type>display_data</output_type>
			192	<text>⎛ ⎽⎽⎽ ⎽⎽⎽ ⎞
			193	⎜╲╱ 2 ⋅❘00⟩ ╲╱ 2 ⋅❘11⟩⎟
			194	H ⋅CNOT ⋅X ⋅⎜────────── + ──────────⎟ = ❘01⟩
			195	1 1,0 1 ⎝ 2 2 ⎠</text>
			196	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQVgwQUFBQW1DQVlBQUFBb1IwdlJBQUFBQkhOQ1NWUUlD
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			278	</png>
			279	<latex>$$H_{1} CNOT_{1,0} X_{1} \left(\frac{1}{2} \sqrt{2} {\left\|00\right\rangle } + \frac{1}{2} \sqrt{2} {\left\|11\right\rangle }\right) = {\left\|01\right\rangle }$$</latex>
			280	</output>
			281	<output>
			282	<output_type>display_data</output_type>
			283	<text>⎛ ⎽⎽⎽ ⎽⎽⎽ ⎞
			284	⎜╲╱ 2 ⋅❘00⟩ ╲╱ 2 ⋅❘11⟩⎟
			285	H ⋅CNOT ⋅Z ⋅⎜────────── + ──────────⎟ = ❘10⟩
			286	1 1,0 1 ⎝ 2 2 ⎠</text>
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			368	VC8KTDVLMlhIQlkzbllBQUFBQVNVVk9SSzVDWUlJPQo=
			369	</png>
			370	<latex>$$H_{1} CNOT_{1,0} Z_{1} \left(\frac{1}{2} \sqrt{2} {\left\|00\right\rangle } + \frac{1}{2} \sqrt{2} {\left\|11\right\rangle }\right) = {\left\|10\right\rangle }$$</latex>
			371	</output>
			372	<output>
			373	<output_type>display_data</output_type>
			374	<text>⎛ ⎽⎽⎽ ⎽⎽⎽ ⎞
			375	⎜╲╱ 2 ⋅❘00⟩ ╲╱ 2 ⋅❘11⟩⎟
			376	H ⋅CNOT ⋅Z ⋅X ⋅⎜────────── + ──────────⎟ = ❘11⟩
			377	1 1,0 1 1 ⎝ 2 2 ⎠</text>
			378	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQVpFQUFBQW1DQVlBQUFBRm1hWXVBQUFBQkhOQ1NWUUlD
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			465	L0E2bjMvdnVXdlNFQUFBQUFTVVZPUks1Q1lJST0K
			466	</png>
			467	<latex>$$H_{1} CNOT_{1,0} Z_{1} X_{1} \left(\frac{1}{2} \sqrt{2} {\left\|00\right\rangle } + \frac{1}{2} \sqrt{2} {\left\|11\right\rangle }\right) = {\left\|11\right\rangle }$$</latex>
			468	</output>
			469	<output>
			470	<output_type>display_data</output_type>
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			642	</png>
			643	</output>
			644	<output>
			645	<output_type>display_data</output_type>
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			719	</png>
			720	</output>
			721	</outputs>
			722	</codecell>
			723	</cells>
			724	</worksheet>
			725	</worksheets>
			726	</notebook>

docs/examples/notebooks/formatting.ipynb

0 +1 -93

@@ -1,7 +1,6 b''
1	<?xml version="1.0" encoding="utf-8"?>	1	<?xml version="1.0" encoding="utf-8"?>
2	<notebook>	2	<notebook>
3	<name>formatting</name>	3	<name>formatting</name>
4	<id>3eaf4ff7-e2c8-46f2-acd3-be16df73aee6</id>
5	<nbformat>2</nbformat>	4	<nbformat>2</nbformat>
6	<worksheets>	5	<worksheets>
7	<worksheet>	6	<worksheet>
@@ -49,50 +48,6 b' And another paragraph.'
49	## Heading (h2)	48	## Heading (h2)
50		49
51	</source>	50	</source>
52	<rendered><h1>Title (h1)</h1>
53
54	<h2>Heading (h2)</h2>
55
56	<h3>Heading (h3)</h3>
57
58	<p>Here is a paragraph of text.</p>
59
60	<ul>
61	<li>One.
62	<ul><li>Sublist
63	<ul><li>Here we go</li></ul></li>
64	<li>Sublist
65	<ul><li>Here we go</li>
66	<li>Here we go</li></ul></li></ul></li>
67	<li>Two.
68	<ul><li>Sublist</li></ul></li>
69	<li>Three.
70	<ul><li>Sublist</li></ul></li>
71	</ul>
72
73	<p>Now another list:</p>
74
75	<hr>
76
77	<ol>
78	<li>Here we go
79	<ol><li>Sublist</li>
80	<li>Sublist</li></ol></li>
81	<li>There we go</li>
82	<li>Now this</li>
83	</ol>
84
85	<p>And another paragraph.</p>
86
87	<h3>Heading (h3)</h3>
88
89	<h4>Heading (h4)</h4>
90
91	<h5>Heading (h5)</h5>
92
93	<h6>Heading (h6)</h6>
94
95	<h2>Heading (h2)</h2></rendered>
96	</markdowncell>	51	</markdowncell>
97	<markdowncell>	52	<markdowncell>
98	<source># Heading (h1)	53	<source># Heading (h1)
@@ -113,25 +68,6 b' Now for a simpel code example:'
113	print i	68	print i
114		69
115	Now more text</source>	70	Now more text</source>
116	<rendered><h1>Heading (h1)</h1>
117
118	<h2>Heading (h2)</h2>
119
120	<h3>Heading (h3)</h3>
121
122	<h4>Heading (h4)</h4>
123
124	<h5>Heading (h5)</h5>
125
126	<h6>Heading (h6)</h6>
127
128	<p>Now for a simpel code example:</p>
129
130	<pre><code>for i in range(10):
131	print i
132	</code></pre>
133
134	<p>Now more text</p></rendered>
135	</markdowncell>	71	</markdowncell>
136	<markdowncell>	72	<markdowncell>
137	<source>## Heading (h2)	73	<source>## Heading (h2)
@@ -161,39 +97,11 b' Here is text'
161	<td>row 2, cell 2</td>	97	<td>row 2, cell 2</td>
162	</tr>	98	</tr>
163	</table></source>	99	</table></source>
164	<rendered><h2>Heading (h2)</h2>
165
166	<p>Here is text.</p>
167
168	<blockquote>
169	<p>This is a <em>block</em> quote. This is a block quote. This is a block quote. <br>
170	This is a <strong>block</strong> quote. This is a block quote. This is a block quote. <br>
171	This is a <code>block</code> quote. This is a block quote. This is a block quote. <br>
172	This is a block quote. This is a block quote. This is a block quote. <br>
173	This is a block quote. This is a block quote. This is a block quote. <br>
174	This is a block quote. This is a block quote. This is a block quote. </p>
175	</blockquote>
176
177	<p>Here is text</p>
178
179	<table>
180	<tbody><tr>
181	<th>Header 1</th>
182	<th>Header 2</th>
183	</tr>
184	<tr>
185	<td>row 1, cell 1</td>
186	<td>row 1, cell 2</td>
187	</tr>
188	<tr>
189	<td>row 2, cell 1</td>
190	<td>row 2, cell 2</td>
191	</tr>
192	</tbody></table></rendered>
193	</markdowncell>	100	</markdowncell>
194	<codecell>	101	<codecell>
195	<input />	102	<input />
196	<language>python</language>	103	<language>python</language>
		104	<collapsed>1</collapsed>
197	<outputs />	105	<outputs />
198	</codecell>	106	</codecell>
199	</cells>	107	</cells>

docs/examples/notebooks/grovers.ipynb

0 +417 -1

@@ -1,1 +1,417 b''
1	{"cells":[{"cell_type":"text","text":"<h1>Grover's Algorithm</h1>"},{"code":"%load_ext sympyprinting","cell_type":"code","prompt_number":1},{"code":"from sympy import sqrt, symbols, Rational\nfrom sympy import expand, Eq, Symbol, simplify, exp, sin\nfrom sympy.physics.quantum import \nfrom sympy.physics.quantum.qubit import \nfrom sympy.physics.quantum.gate import \nfrom sympy.physics.quantum.grover import \nfrom sympy.physics.quantum.qft import QFT, IQFT, Fourier\nfrom sympy.physics.quantum.circuitplot import circuit_plot","cell_type":"code","prompt_number":2},{"code":"nqubits = 3\n","cell_type":"code","prompt_number":3},{"cell_type":"text","text":"Grover's algorithm is a quantum algorithm which searches an unordered database (inverts a function).<br/>\nProvides polynomial speedup over classical brute-force search ($O(\\sqrt{N}) vs. O(N))$ "},{"cell_type":"text","text":"Define a black box function that returns True if it is passed the state we are searching for."},{"code":"def black_box(qubits):\n return True if qubits == IntQubit(1, qubits.nqubits) else False\n","cell_type":"code","prompt_number":4},{"cell_type":"text","text":"Build a uniform superposition state to start the search."},{"code":"psi = superposition_basis(nqubits); psi\n","cell_type":"code","prompt_number":5},{"code":"v = OracleGate(nqubits, black_box)\n","cell_type":"code","prompt_number":6},{"cell_type":"text","text":"Perform two iterations of Grover's algorithm. Each iteration, the amplitude of the target state increases."},{"code":"iter1 = qapply(grover_iteration(psi, v)); iter1\n","cell_type":"code","prompt_number":7},{"code":"iter2 = qapply(grover_iteration(iter1, v)); iter2\n","cell_type":"code","prompt_number":8},{"cell_type":"text","text":"A single shot measurement is performed to retrieve the target state."},{"code":"measure_all_oneshot(iter2)\n","cell_type":"code","prompt_number":9},{"code":"%notebook save grovers.ipynb","cell_type":"code","prompt_number":28},{"code":"%notebook load qft.ipynb","cell_type":"code","prompt_number":63}]} No newline at end of file	1	<?xml version="1.0" encoding="utf-8"?>
		2	<notebook>
		3	<name>grovers</name>
		4	<nbformat>2</nbformat>
		5	<worksheets>
		6	<worksheet>
		7	<cells>
		8	<markdowncell>
		9	<source><h1>Grover's Algorithm</h1></source>
		10	</markdowncell>
		11	<codecell>
		12	<input>%load_ext sympyprinting</input>
		13	<language>python</language>
		14	<prompt_number>1</prompt_number>
		15	<collapsed>1</collapsed>
		16	<outputs />
		17	</codecell>
		18	<codecell>
		19	<input>from sympy import sqrt, symbols, Rational
		20	from sympy import expand, Eq, Symbol, simplify, exp, sin
		21	from sympy.physics.quantum import *
		22	from sympy.physics.quantum.qubit import *
		23	from sympy.physics.quantum.gate import *
		24	from sympy.physics.quantum.grover import *
		25	from sympy.physics.quantum.qft import QFT, IQFT, Fourier
		26	from sympy.physics.quantum.circuitplot import circuit_plot</input>
		27	<language>python</language>
		28	<prompt_number>2</prompt_number>
		29	<collapsed>1</collapsed>
		30	<outputs />
		31	</codecell>
		32	<codecell>
		33	<input>nqubits = 3
		34	</input>
		35	<language>python</language>
		36	<prompt_number>3</prompt_number>
		37	<collapsed>1</collapsed>
		38	<outputs />
		39	</codecell>
		40	<markdowncell>
		41	<source>Grover's algorithm is a quantum algorithm which searches an unordered database (inverts a function).<br/>
		42	Provides polynomial speedup over classical brute-force search ($O(\sqrt{N}) vs. O(N))$ </source>
		43	</markdowncell>
		44	<markdowncell>
		45	<source>Define a black box function that returns True if it is passed the state we are searching for.</source>
		46	</markdowncell>
		47	<codecell>
		48	<input>def black_box(qubits):
		49	return True if qubits == IntQubit(1, qubits.nqubits) else False
		50	</input>
		51	<language>python</language>
		52	<prompt_number>4</prompt_number>
		53	<collapsed>1</collapsed>
		54	<outputs />
		55	</codecell>
		56	<markdowncell>
		57	<source>Build a uniform superposition state to start the search.</source>
		58	</markdowncell>
		59	<codecell>
		60	<input>psi = superposition_basis(nqubits); psi
		61	</input>
		62	<language>python</language>
		63	<prompt_number>5</prompt_number>
		64	<collapsed>0</collapsed>
		65	<outputs>
		66	<output>
		67	<output_type>pyout</output_type>
		68	<text>⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽
		69	╲╱ 2 ⋅❘0⟩ ╲╱ 2 ⋅❘1⟩ ╲╱ 2 ⋅❘2⟩ ╲╱ 2 ⋅❘3⟩ ╲╱ 2 ⋅❘4⟩ ╲╱ 2 ⋅❘5⟩ ╲╱ 2 ⋅
		70	───────── + ───────── + ───────── + ───────── + ───────── + ───────── + ──────
		71	4 4 4 4 4 4 4
		72
		73	⎽⎽⎽
		74	❘6⟩ ╲╱ 2 ⋅❘7⟩
		75	─── + ─────────
		76	4</text>
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		156	</png>
		157	<latex>$$\frac{1}{4} \sqrt{2} {\left\|0\right\rangle } + \frac{1}{4} \sqrt{2} {\left\|1\right\rangle } + \frac{1}{4} \sqrt{2} {\left\|2\right\rangle } + \frac{1}{4} \sqrt{2} {\left\|3\right\rangle } + \frac{1}{4} \sqrt{2} {\left\|4\right\rangle } + \frac{1}{4} \sqrt{2} {\left\|5\right\rangle } + \frac{1}{4} \sqrt{2} {\left\|6\right\rangle } + \frac{1}{4} \sqrt{2} {\left\|7\right\rangle }$$</latex>
		158	<prompt_number>5</prompt_number>
		159	</output>
		160	</outputs>
		161	</codecell>
		162	<codecell>
		163	<input>v = OracleGate(nqubits, black_box)
		164	</input>
		165	<language>python</language>
		166	<prompt_number>6</prompt_number>
		167	<collapsed>1</collapsed>
		168	<outputs />
		169	</codecell>
		170	<markdowncell>
		171	<source>Perform two iterations of Grover's algorithm. Each iteration, the amplitude of the target state increases.</source>
		172	</markdowncell>
		173	<codecell>
		174	<input>iter1 = qapply(grover_iteration(psi, v)); iter1
		175	</input>
		176	<language>python</language>
		177	<prompt_number>7</prompt_number>
		178	<collapsed>0</collapsed>
		179	<outputs>
		180	<output>
		181	<output_type>pyout</output_type>
		182	<text>⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽
		183	╲╱ 2 ⋅❘0⟩ 5⋅╲╱ 2 ⋅❘1⟩ ╲╱ 2 ⋅❘2⟩ ╲╱ 2 ⋅❘3⟩ ╲╱ 2 ⋅❘4⟩ ╲╱ 2 ⋅❘5⟩ ╲╱ 2
		184	───────── + ─────────── + ───────── + ───────── + ───────── + ───────── + ────
		185	8 8 8 8 8 8
		186
		187	⎽ ⎽⎽⎽
		188	⋅❘6⟩ ╲╱ 2 ⋅❘7⟩
		189	───── + ─────────
		190	8 8</text>
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		271	</png>
		272	<latex>$$\frac{1}{8} \sqrt{2} {\left\|0\right\rangle } + \frac{5}{8} \sqrt{2} {\left\|1\right\rangle } + \frac{1}{8} \sqrt{2} {\left\|2\right\rangle } + \frac{1}{8} \sqrt{2} {\left\|3\right\rangle } + \frac{1}{8} \sqrt{2} {\left\|4\right\rangle } + \frac{1}{8} \sqrt{2} {\left\|5\right\rangle } + \frac{1}{8} \sqrt{2} {\left\|6\right\rangle } + \frac{1}{8} \sqrt{2} {\left\|7\right\rangle }$$</latex>
		273	<prompt_number>7</prompt_number>
		274	</output>
		275	</outputs>
		276	</codecell>
		277	<codecell>
		278	<input>iter2 = qapply(grover_iteration(iter1, v)); iter2
		279	</input>
		280	<language>python</language>
		281	<prompt_number>8</prompt_number>
		282	<collapsed>0</collapsed>
		283	<outputs>
		284	<output>
		285	<output_type>pyout</output_type>
		286	<text>⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽
		287	╲╱ 2 ⋅❘0⟩ 11⋅╲╱ 2 ⋅❘1⟩ ╲╱ 2 ⋅❘2⟩ ╲╱ 2 ⋅❘3⟩ ╲╱ 2 ⋅❘4⟩ ╲╱ 2 ⋅❘5⟩ ╲
		288	- ───────── + ──────────── - ───────── - ───────── - ───────── - ───────── - ─
		289	16 16 16 16 16 16
		290
		291	⎽⎽⎽ ⎽⎽⎽
		292	╱ 2 ⋅❘6⟩ ╲╱ 2 ⋅❘7⟩
		293	──────── - ─────────
		294	16 16</text>
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		372	ZjEvazN4ZjU5MFgrL1pEN0FLOUd3V1Uwcjk4NkY1dmdJZnhRVFB5UWUxL2szeGY1OTBYK2ZaRi9Q
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		376	L1MKelNmU2hTUU41d0FBQUFCSlJVNUVya0pnZ2c9PQo=
		377	</png>
		378	<latex>$$- \frac{1}{16} \sqrt{2} {\left\|0\right\rangle } + \frac{11}{16} \sqrt{2} {\left\|1\right\rangle } - \frac{1}{16} \sqrt{2} {\left\|2\right\rangle } - \frac{1}{16} \sqrt{2} {\left\|3\right\rangle } - \frac{1}{16} \sqrt{2} {\left\|4\right\rangle } - \frac{1}{16} \sqrt{2} {\left\|5\right\rangle } - \frac{1}{16} \sqrt{2} {\left\|6\right\rangle } - \frac{1}{16} \sqrt{2} {\left\|7\right\rangle }$$</latex>
		379	<prompt_number>8</prompt_number>
		380	</output>
		381	</outputs>
		382	</codecell>
		383	<markdowncell>
		384	<source>A single shot measurement is performed to retrieve the target state.</source>
		385	</markdowncell>
		386	<codecell>
		387	<input>measure_all_oneshot(iter2)
		388	</input>
		389	<language>python</language>
		390	<prompt_number>9</prompt_number>
		391	<collapsed>0</collapsed>
		392	<outputs>
		393	<output>
		394	<output_type>pyout</output_type>
		395	<text>❘001⟩</text>
		396	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUNrQUFBQVRDQVlBQUFBTmdTUTlBQUFBQkhOQ1NWUUlD
		397	QWdJZkFoa2lBQUFBYWhKUkVGVQpTSW50MXJ0ckZVRWN4ZkVQUWVNcjZEWGlDM3lrc0ZIUlJpUW9p
		398	SFZDS2l1eHloOWdZeU1XVmhhQ29vS3R0WlhZaUVaUTB5aGliN0FSCmc1R0ErQ2dVSVRlK0lGck1C
		399	TWE5ZS9mTzNOWjhZWmpaOHp1N2UzWjNkblpab1lnTDNRcXJrdkVaUE1WN3RIQWRiWHpCV3R6QzI4
		400	cisKdWI1bE51SUpUbU91VW10akd6NDNYY2trUnVKNEdsY3J0VmRZWDlrbjF6ZUlFM2lPUDloWGMv
		401	NHRPTmNVTUEwNUdnKzBQNm10dzNkYwpUclJjWHd2M2hUdCtyeUVrWE1KQVRzaTdtSytwUDhKTXNw
		402	M3JTN25ZSStSeGpGZkZ1dFJIOEtaR244VWg0YkdVK0VwNEVZUDJETGtkCkN6VjZPL1k3Q24ybHpH
		403	TnZLbFJEYmhMbVZWc255MXFyd05jUER6RFJGUEkzbG1LcnNqcjJTd1crZnBqQVZGUElSZDNYdUEy
		404	eC8xRGcKNjRjUnZFdUZ1ams1ZzgwMStsRHNQeGI2U2pnbXZEei8wQzNrN2hyOXFQRG0vaWowbFRD
		405	R2h6a2hYK0lBOWlUYVZoekd0VDU4dVF6agpxeDV6ZVZLWUQ0TjRocHRKN2J6d1RWK1RhTG0rbEJ2
		406	Q1l0NnhGdUtzc0t4MWtQNWdMT0pYYktlRVQ5dHRmTU11bk1UUHhKL3JJL3hVCjdNVEJ1UDBZcjNF
		407	SFY2STJoRTlkTG02Ri80ZS91K3R5ZUYwcUx2c0FBQUFBU1VWT1JLNUNZSUk9Cg==
		408	</png>
		409	<latex>$${\left\|001\right\rangle }$$</latex>
		410	<prompt_number>9</prompt_number>
		411	</output>
		412	</outputs>
		413	</codecell>
		414	</cells>
		415	</worksheet>
		416	</worksheets>
		417	</notebook>

docs/examples/notebooks/qerror.ipynb

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	1	{"cells":[{"cell_type":"text","text":"<h1>Quantum Error Correction</h1>"},{"code":"%load_ext sympyprinting","cell_type":"code","prompt_number":1},{"code":"from sympy import sqrt, symbols, Rational\nfrom sympy import expand, Eq, Symbol, simplify, exp, sin\nfrom sympy.physics.quantum import \nfrom sympy.physics.quantum.qubit import \nfrom sympy.physics.quantum.gate import \nfrom sympy.physics.quantum.grover import \nfrom sympy.physics.quantum.qft import QFT, IQFT, Fourier\nfrom sympy.physics.quantum.circuitplot import circuit_plot","cell_type":"code","prompt_number":2},{"cell_type":"text","text":"<h2>5 qubit code</h2>"},{"code":"M0 = Z(1)X(2)X(3)Z(4); M0\n","cell_type":"code","prompt_number":3},{"code":"M1 = Z(2)X(3)X(4)Z(0); M1\n","cell_type":"code","prompt_number":4},{"code":"M2 = Z(3)X(4)X(0)Z(1); M2\n","cell_type":"code","prompt_number":5},{"code":"M3 = Z(4)X(0)X(1)Z(2); M3\n","cell_type":"code","prompt_number":6},{"cell_type":"text","text":"These operators should mutually commute."},{"code":"gate_simp(Commutator(M0,M1).doit())\n","cell_type":"code","prompt_number":7},{"cell_type":"text","text":"And square to the identity."},{"code":"for o in [M0,M1,M2,M3]:\n display(gate_simp(oo))\n","cell_type":"code","prompt_number":8},{"cell_type":"text","text":"<h2>Codewords</h2>"},{"code":"zero = Rational(1,4)(1+M0)(1+M1)(1+M2)(1+M3)IntQubit(0, 5); zero\n","cell_type":"code","prompt_number":9},{"code":"qapply(4zero)\n","cell_type":"code","prompt_number":10},{"code":"one = Rational(1,4)(1+M0)(1+M1)(1+M2)(1+M3)IntQubit(2*5-1, 5); one\n","cell_type":"code","prompt_number":11},{"code":"qapply(4one)\n","cell_type":"code","prompt_number":12},{"cell_type":"text","text":"<h2>The encoding circuit</h2>"},{"code":"encoding_circuit = H(3)H(4)CNOT(2,0)CNOT(3,0)CNOT(4,0)H(1)H(4)\\\n CNOT(2,1)CNOT(4,1)H(2)CNOT(3,2)CNOT(4,2)H(3)\\\n H(4)CNOT(4, 3)Z(4)H(4)Z(4)\n","cell_type":"code","prompt_number":13},{"code":"circuit_plot(encoding_circuit, nqubits=5, scale=0.5)","cell_type":"code","prompt_number":14},{"code":"represent(4encoding_circuit, nqubits=5)","cell_type":"code","prompt_number":14},{"code":"%notebook save qerror.ipynb","cell_type":"code","prompt_number":23},{"code":"%notebook load decompose.ipynb","cell_type":"code","prompt_number":23}]} No newline at end of file	1	<?xml version="1.0" encoding="utf-8"?>
			2	<notebook>
			3	<name>qerror</name>
			4	<nbformat>2</nbformat>
			5	<worksheets>
			6	<worksheet>
			7	<cells>
			8	<markdowncell>
			9	<source><h1>Quantum Error Correction</h1></source>
			10	</markdowncell>
			11	<codecell>
			12	<input>%load_ext sympyprinting</input>
			13	<language>python</language>
			14	<prompt_number>1</prompt_number>
			15	<collapsed>1</collapsed>
			16	<outputs />
			17	</codecell>
			18	<codecell>
			19	<input>from sympy import sqrt, symbols, Rational
			20	from sympy import expand, Eq, Symbol, simplify, exp, sin
			21	from sympy.physics.quantum import *
			22	from sympy.physics.quantum.qubit import *
			23	from sympy.physics.quantum.gate import *
			24	from sympy.physics.quantum.grover import *
			25	from sympy.physics.quantum.qft import QFT, IQFT, Fourier
			26	from sympy.physics.quantum.circuitplot import circuit_plot</input>
			27	<language>python</language>
			28	<prompt_number>2</prompt_number>
			29	<collapsed>1</collapsed>
			30	<outputs />
			31	</codecell>
			32	<markdowncell>
			33	<source><h2>5 qubit code</h2></source>
			34	</markdowncell>
			35	<codecell>
			36	<input>M0 = Z(1)X(2)X(3)*Z(4); M0
			37	</input>
			38	<language>python</language>
			39	<prompt_number>3</prompt_number>
			40	<collapsed>0</collapsed>
			41	<outputs>
			42	<output>
			43	<output_type>pyout</output_type>
			44	<text>Z ⋅X ⋅X ⋅Z
			45	1 2 3 4</text>
			46	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUZzQUFBQVlDQVlBQUFDVitvRmJBQUFBQkhOQ1NWUUlD
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			64	alMyRG0wU1M4SVRZcG9Tb1ZoWk5jUXMyU1BPcGlmRlJqK04xeXR3eDhWK2NvNklvczRTCi9uNkpY
			65	eXR3bTFUbnlCN1JmTkEvb2ZyQjFZQ0lvR1kzeUZWaURiNnlJamV2WVltWGtaTnhtemo5K2hBM200
			66	OFVVbWk5R01FenVXdDMKSXZhUU9FUFpMTUszQ2VsUysvUHduTmpNeCtVT3c2bzAwTkFlZis1VGJZ
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			68	QUFBQUJKUlU1RXJrSmdnZz09Cg==
			69	</png>
			70	<latex>$$Z_{1} X_{2} X_{3} Z_{4}$$</latex>
			71	<prompt_number>3</prompt_number>
			72	</output>
			73	</outputs>
			74	</codecell>
			75	<codecell>
			76	<input>M1 = Z(2)X(3)X(4)*Z(0); M1
			77	</input>
			78	<language>python</language>
			79	<prompt_number>4</prompt_number>
			80	<collapsed>0</collapsed>
			81	<outputs>
			82	<output>
			83	<output_type>pyout</output_type>
			84	<text>Z ⋅X ⋅X ⋅Z
			85	2 3 4 0</text>
			86	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUZvQUFBQVlDQVlBQUFCNk9PcGxBQUFBQkhOQ1NWUUlD
			87	QWdJZkFoa2lBQUFBNGxKUkVGVQphSUh0MkZ1b0ZXVVVCL0RmMXBQSGtpNVVXbWszZWlzaW9oZnhJ
			88	YTBzaUxDeWkxQVpkcUVNSXFtUXNxaW9TSkVvZXFnSGlTQ2pHMEVZCmxSS0JHU0ZGTnpwRUZ5Z3Fq
			89	TElpNkI1azZkSFR3NW81ZS9iZWMwN3VtVy92WHZZZkJ1YWJOZk9mTmY5WjM3Zld0eGpnZjhVczdN
			90	UVkKZHVFTGJNYXIyZkZ6WnJ0b0w3aVdZU1M3Znd3NzhFekJmZzUreld4NzhEbm1kK0Zyci9sVGF0
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			99	N3A0SmZKbUZQM0Y4Tmw2UDF4UHl3MEg0QkhlVwoyTHJSZ3FqWnR4YkdHM0ZlMlVzUHdFZjRBY2Uw
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			102	M2hRSklpMzhYZjIKekc0aFVoRVhpM28zNS8xZWJDNG13d2NpZWxQeEw4YUoyYlB0UW5lckJiRkxY
			103	Rm9ZTDhLMy8vRk5YV09lbUdaSmQwb1p6c1JqV3FPbApMdWJnaXV5OFRPZ3EyS1kxTVM0VTZ6dnFO
			104	M0pnSnU3REFySC9UNDNYUktJN0JROGw0R3ZnY2p5WmdLdUliU0xTYzh3UWVRdjFoZDVmClpOaEw4
			105	YVhPOWF3T0xzRmJtcHVPVjBSMWNXaE4zbmtpZWE4U2ErMENVWHRmVjVOM1JDVHdISWVMNWFzMnBn
			106	dG44NmJOVk9XOTRLcTQKVW12dHVsYXNtU2xtWVJGUFM3TjBMQkhOckJ3YlJBRFdScjc3S2g0clVo
			107	Qm5HQmIxOHdvaHhHYVJ2RkxpZXRHRmUwL01vRHBvNEJIYwpMQUp3alVJTG9FNHZZS2FJNGlKK0U5
			108	VkFTa3dSb3U5SXpFdE03eHlqK0NrQjU1Qm1OVFRBQUFNTVVBZi9BaVQyRkJhVnNBdFZBQUFBCkFF
			109	bEZUa1N1UW1DQwo=
			110	</png>
			111	<latex>$$Z_{2} X_{3} X_{4} Z_{0}$$</latex>
			112	<prompt_number>4</prompt_number>
			113	</output>
			114	</outputs>
			115	</codecell>
			116	<codecell>
			117	<input>M2 = Z(3)X(4)X(0)*Z(1); M2
			118	</input>
			119	<language>python</language>
			120	<prompt_number>5</prompt_number>
			121	<collapsed>0</collapsed>
			122	<outputs>
			123	<output>
			124	<output_type>pyout</output_type>
			125	<text>Z ⋅X ⋅X ⋅Z
			126	3 4 0 1</text>
			127	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUZvQUFBQVlDQVlBQUFCNk9PcGxBQUFBQkhOQ1NWUUlD
			128	QWdJZkFoa2lBQUFBM1JKUkVGVQphSUh0MlZ0b0hIVVV4L0ZQVFRTWTFyVEVlaUhlU2xFRXhVdjFR
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			144	WWFMakdsZFRER2pxK3A4a2RGNVBZZW5FN0RwWUhUZApqWnlmUlRWZWl3OUVJVWlsVlNLamRpVmtF
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			146	Y0xSNGhyK2w5cUc4N0wzL2FJTnVyK2g3MW91c2pERjBrRnNiYmEyUG9kRWtwUjFGbFZVdW5UMDFR
			147	Q080d3FjaVB2RTByRkIKRkpxVU9sWDB2T2VLSDNRQ1A5WmtiaE14THhGdDJoT2l5NmlqcGVJT3Z4
			148	Nm5pV0s3VjdiMHBkZ0xHQlIvSHhYWHZWUWFOTFBIM1oxOQpYd29OaUoyK0ZPckRFWVd4UFdZWDNn
			149	VXRhRUVMbXEvK0FTMkNFZmVpa09YakFBQUFBRWxGVGtTdVFtQ0MK
			150	</png>
			151	<latex>$$Z_{3} X_{4} X_{0} Z_{1}$$</latex>
			152	<prompt_number>5</prompt_number>
			153	</output>
			154	</outputs>
			155	</codecell>
			156	<codecell>
			157	<input>M3 = Z(4)X(0)X(1)*Z(2); M3
			158	</input>
			159	<language>python</language>
			160	<prompt_number>6</prompt_number>
			161	<collapsed>0</collapsed>
			162	<outputs>
			163	<output>
			164	<output_type>pyout</output_type>
			165	<text>Z ⋅X ⋅X ⋅Z
			166	4 0 1 2</text>
			167	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUZvQUFBQVlDQVlBQUFCNk9PcGxBQUFBQkhOQ1NWUUlD
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			178	N09LYXJiN2w0MFNwc3dMcENlNEU0b1IzQ3FYT01XNEZOYWZ4bTFmbHFYZjQ3elMxVXIxcU00bDQ4
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			180	QnpMeHIwcjlYK1BHaXZsNzFXSTFya2ozQTlpQgpyV1hFeDJFWHZzTklsMjBZRzBXdFlHMlhiVDN1
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			183	N3pmaXNORkVSTjQKcE5CZW1wNGR5Y1JQdGRCMXREZ0xGeFRhVzlPNHhsaUMyOU45VWVoYzJHYm11
			184	ajhrWHZDY2pITlVDZDBVeDR1L1lhcEsyS1NRc3c1UApOUFZvRG53dWhPMmdzK244Mk9LY09iQVFq
			185	NG1NWkh1bnM2N1FJeUxUMkNEV29xdEZBZWV1Wmo3T3dFNVJDKzVncWFpVy9aQnhqall3Ckx2YU9O
			186	OHhlM3h0amkveEx4ekpSdk9rRXd5WVZ1M2dENUY0NmJoTnJOWkZpYnM3SWJZMkl2ajNpQkpVVDYw
			187	VjAzQ3FLNjdtS1JDdEUKcHJRZmI2ZDVtbFlSeDB4dmxKM3JoWTR4UnkzZ0pMRXVTZVQ3TW5CMlkx
			188	QlU1SEpoU2JxSytGNmtiSFV4SkRLdUluN0RydzA0NXpHUAplZnkvOFE4S29RMjJ6LzN3eXdBQUFB
			189	QkpSVTVFcmtKZ2dnPT0K
			190	</png>
			191	<latex>$$Z_{4} X_{0} X_{1} Z_{2}$$</latex>
			192	<prompt_number>6</prompt_number>
			193	</output>
			194	</outputs>
			195	</codecell>
			196	<markdowncell>
			197	<source>These operators should mutually commute.</source>
			198	</markdowncell>
			199	<codecell>
			200	<input>gate_simp(Commutator(M0,M1).doit())
			201	</input>
			202	<language>python</language>
			203	<prompt_number>7</prompt_number>
			204	<collapsed>0</collapsed>
			205	<outputs>
			206	<output>
			207	<output_type>pyout</output_type>
			208	<text>0</text>
			209	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUF3QUFBQVNDQVlBQUFCdnFUOE1BQUFBQkhOQ1NWUUlD
			210	QWdJZkFoa2lBQUFBTjFKUkVGVQpLSkhOMGkxTFJGRVF4dkdmN2k2THJHRTNhZENvVWJBWkJUK0d6
			211	V0FUaTlWbTIyRDNROWd0WnFObzhXWE5naTlCVUhCZFVNTzlGOGJECmxHMU9PY3d6OHo5emVNNHda
			212	Y3dVK1JLR2VNUWJPampHYXdhM01NSnUwSTV3aHRrTTJNWVl2YUN0NEFjN0dYQ0o4MFIvd0dtVE5L
			213	UGEKV01OOUFveXdXUUlMS2dQZUUrQURBM1Fqc0JpS0dRRDlDSXpyOHpzQk9ySFdBTGY0U3BxcFhK
			214	dmdKUUlUM05SdkxXTWVUeXA3LzN6SQpGWmFMNWhiV2NaMk5QbEN0d0Z6UU51cWJ0ekpnZ0R2c0Jl
			215	MEVGN0dwWEw1VkhPS3p6bnZZeDNNMjRaL0VMOVhvS0JOSStWMHhBQUFBCkFFbEZUa1N1UW1DQwo=
			216	</png>
			217	<latex>$$0$$</latex>
			218	<prompt_number>7</prompt_number>
			219	</output>
			220	</outputs>
			221	</codecell>
			222	<markdowncell>
			223	<source>And square to the identity.</source>
			224	</markdowncell>
			225	<codecell>
			226	<input>for o in [M0,M1,M2,M3]:
			227	display(gate_simp(o*o))
			228	</input>
			229	<language>python</language>
			230	<prompt_number>8</prompt_number>
			231	<collapsed>0</collapsed>
			232	<outputs>
			233	<output>
			234	<output_type>display_data</output_type>
			235	<text>1</text>
			236	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUFzQUFBQVNDQVlBQUFDTmRTUjFBQUFBQkhOQ1NWUUlD
			237	QWdJZkFoa2lBQUFBR0JKUkVGVQpLSkh0MGJFTlFHQVFodUVuV3AxQ1pRRmJxRXhoT0kxQnRFYkFC
			238	Z29qL0FvS2tjZ2ZDWjIzdVV2dXZjdVhIQzlRWUk1SktXcU1DTmRoCmN1cExkS2l3UEluUnhpNUgr
			239	ZVhYNWV5bytkMWlqaDZUL1hzQkt3WTBUeEo4ekFaQmNneThUWnFOVGdBQUFBQkpSVTVFcmtKZ2dn
			240	PT0K
			241	</png>
			242	<latex>$$1$$</latex>
			243	</output>
			244	<output>
			245	<output_type>display_data</output_type>
			246	<text>1</text>
			247	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUFzQUFBQVNDQVlBQUFDTmRTUjFBQUFBQkhOQ1NWUUlD
			248	QWdJZkFoa2lBQUFBR0JKUkVGVQpLSkh0MGJFTlFHQVFodUVuV3AxQ1pRRmJxRXhoT0kxQnRFYkFC
			249	Z29qL0FvS2tjZ2ZDWjIzdVV2dXZjdVhIQzlRWUk1SktXcU1DTmRoCmN1cExkS2l3UEluUnhpNUgr
			250	ZVhYNWV5bytkMWlqaDZUL1hzQkt3WTBUeEo4ekFaQmNneThUWnFOVGdBQUFBQkpSVTVFcmtKZ2dn
			251	PT0K
			252	</png>
			253	<latex>$$1$$</latex>
			254	</output>
			255	<output>
			256	<output_type>display_data</output_type>
			257	<text>1</text>
			258	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUFzQUFBQVNDQVlBQUFDTmRTUjFBQUFBQkhOQ1NWUUlD
			259	QWdJZkFoa2lBQUFBR0JKUkVGVQpLSkh0MGJFTlFHQVFodUVuV3AxQ1pRRmJxRXhoT0kxQnRFYkFC
			260	Z29qL0FvS2tjZ2ZDWjIzdVV2dXZjdVhIQzlRWUk1SktXcU1DTmRoCmN1cExkS2l3UEluUnhpNUgr
			261	ZVhYNWV5bytkMWlqaDZUL1hzQkt3WTBUeEo4ekFaQmNneThUWnFOVGdBQUFBQkpSVTVFcmtKZ2dn
			262	PT0K
			263	</png>
			264	<latex>$$1$$</latex>
			265	</output>
			266	<output>
			267	<output_type>display_data</output_type>
			268	<text>1</text>
			269	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUFzQUFBQVNDQVlBQUFDTmRTUjFBQUFBQkhOQ1NWUUlD
			270	QWdJZkFoa2lBQUFBR0JKUkVGVQpLSkh0MGJFTlFHQVFodUVuV3AxQ1pRRmJxRXhoT0kxQnRFYkFC
			271	Z29qL0FvS2tjZ2ZDWjIzdVV2dXZjdVhIQzlRWUk1SktXcU1DTmRoCmN1cExkS2l3UEluUnhpNUgr
			272	ZVhYNWV5bytkMWlqaDZUL1hzQkt3WTBUeEo4ekFaQmNneThUWnFOVGdBQUFBQkpSVTVFcmtKZ2dn
			273	PT0K
			274	</png>
			275	<latex>$$1$$</latex>
			276	</output>
			277	</outputs>
			278	</codecell>
			279	<markdowncell>
			280	<source><h2>Codewords</h2></source>
			281	</markdowncell>
			282	<codecell>
			283	<input>zero = Rational(1,4)(1+M0)(1+M1)(1+M2)(1+M3)*IntQubit(0, 5); zero
			284	</input>
			285	<language>python</language>
			286	<prompt_number>9</prompt_number>
			287	<collapsed>0</collapsed>
			288	<outputs>
			289	<output>
			290	<output_type>pyout</output_type>
			291	<text>⎛1 + Z ⋅X ⋅X ⋅Z ⎞⋅⎛1 + Z ⋅X ⋅X ⋅Z ⎞⋅⎛1 + Z ⋅X ⋅X ⋅Z ⎞⋅⎛1 + Z ⋅X ⋅X ⋅Z ⎞⋅❘0⟩
			292	⎝ 1 2 3 4⎠ ⎝ 2 3 4 0⎠ ⎝ 3 4 0 1⎠ ⎝ 4 0 1 2⎠
			293	───────────────────────────────────────────────────────────────────────────
			294	4</text>
			295	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQWo0QUFBQWlDQVlBQUFCZnVkTmxBQUFBQkhOQ1NWUUlD
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			362	</png>
			363	<latex>$$\frac{1}{4} \left(1 + Z_{1} X_{2} X_{3} Z_{4}\right) \left(1 + Z_{2} X_{3} X_{4} Z_{0}\right) \left(1 + Z_{3} X_{4} X_{0} Z_{1}\right) \left(1 + Z_{4} X_{0} X_{1} Z_{2}\right) {\left\|0\right\rangle }$$</latex>
			364	<prompt_number>9</prompt_number>
			365	</output>
			366	</outputs>
			367	</codecell>
			368	<codecell>
			369	<input>qapply(4*zero)
			370	</input>
			371	<language>python</language>
			372	<prompt_number>10</prompt_number>
			373	<collapsed>0</collapsed>
			374	<outputs>
			375	<output>
			376	<output_type>pyout</output_type>
			377	<text>❘0⟩ + ❘3⟩ - ❘5⟩ + ❘6⟩ - ❘9⟩ - ❘10⟩ + ❘12⟩ - ❘15⟩ + ❘17⟩ - ❘18⟩ - ❘20⟩ - ❘23⟩ +
			378	❘24⟩ - ❘27⟩ - ❘29⟩ - ❘30⟩</text>
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			449	</png>
			450	<latex>$${\left\|0\right\rangle } + {\left\|3\right\rangle } - {\left\|5\right\rangle } + {\left\|6\right\rangle } - {\left\|9\right\rangle } - {\left\|10\right\rangle } + {\left\|12\right\rangle } - {\left\|15\right\rangle } + {\left\|17\right\rangle } - {\left\|18\right\rangle } - {\left\|20\right\rangle } - {\left\|23\right\rangle } + {\left\|24\right\rangle } - {\left\|27\right\rangle } - {\left\|29\right\rangle } - {\left\|30\right\rangle }$$</latex>
			451	<prompt_number>10</prompt_number>
			452	</output>
			453	</outputs>
			454	</codecell>
			455	<codecell>
			456	<input>one = Rational(1,4)(1+M0)(1+M1)(1+M2)(1+M3)IntQubit(2*5-1, 5); one
			457	</input>
			458	<language>python</language>
			459	<prompt_number>11</prompt_number>
			460	<collapsed>0</collapsed>
			461	<outputs>
			462	<output>
			463	<output_type>pyout</output_type>
			464	<text>⎛1 + Z ⋅X ⋅X ⋅Z ⎞⋅⎛1 + Z ⋅X ⋅X ⋅Z ⎞⋅⎛1 + Z ⋅X ⋅X ⋅Z ⎞⋅⎛1 + Z ⋅X ⋅X ⋅Z ⎞⋅❘31⟩
			465	⎝ 1 2 3 4⎠ ⎝ 2 3 4 0⎠ ⎝ 3 4 0 1⎠ ⎝ 4 0 1 2⎠
			466	────────────────────────────────────────────────────────────────────────────
			467	4</text>
			468	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQWtnQUFBQWlDQVlBQUFDazdpYTZBQUFBQkhOQ1NWUUlD
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			536	PT0K
			537	</png>
			538	<latex>$$\frac{1}{4} \left(1 + Z_{1} X_{2} X_{3} Z_{4}\right) \left(1 + Z_{2} X_{3} X_{4} Z_{0}\right) \left(1 + Z_{3} X_{4} X_{0} Z_{1}\right) \left(1 + Z_{4} X_{0} X_{1} Z_{2}\right) {\left\|31\right\rangle }$$</latex>
			539	<prompt_number>11</prompt_number>
			540	</output>
			541	</outputs>
			542	</codecell>
			543	<codecell>
			544	<input>qapply(4*one)
			545	</input>
			546	<language>python</language>
			547	<prompt_number>12</prompt_number>
			548	<collapsed>0</collapsed>
			549	<outputs>
			550	<output>
			551	<output_type>pyout</output_type>
			552	<text>-❘1⟩ - ❘2⟩ - ❘4⟩ + ❘7⟩ - ❘8⟩ - ❘11⟩ - ❘13⟩ + ❘14⟩ - ❘16⟩ + ❘19⟩ - ❘21⟩ - ❘22⟩
			553	+ ❘25⟩ - ❘26⟩ + ❘28⟩ + ❘31⟩</text>
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			620	NUNZSUk9Cg==
			621	</png>
			622	<latex>$$- {\left\|1\right\rangle } - {\left\|2\right\rangle } - {\left\|4\right\rangle } + {\left\|7\right\rangle } - {\left\|8\right\rangle } - {\left\|11\right\rangle } - {\left\|13\right\rangle } + {\left\|14\right\rangle } - {\left\|16\right\rangle } + {\left\|19\right\rangle } - {\left\|21\right\rangle } - {\left\|22\right\rangle } + {\left\|25\right\rangle } - {\left\|26\right\rangle } + {\left\|28\right\rangle } + {\left\|31\right\rangle }$$</latex>
			623	<prompt_number>12</prompt_number>
			624	</output>
			625	</outputs>
			626	</codecell>
			627	<markdowncell>
			628	<source><h2>The encoding circuit</h2></source>
			629	</markdowncell>
			630	<codecell>
			631	<input>encoding_circuit = H(3)H(4)CNOT(2,0)CNOT(3,0)CNOT(4,0)H(1)H(4)*\
			632	CNOT(2,1)CNOT(4,1)H(2)CNOT(3,2)CNOT(4,2)H(3)\
			633	H(4)CNOT(4, 3)Z(4)H(4)Z(4)
			634	</input>
			635	<language>python</language>
			636	<prompt_number>13</prompt_number>
			637	<collapsed>1</collapsed>
			638	<outputs />
			639	</codecell>
			640	<codecell>
			641	<input>circuit_plot(encoding_circuit, nqubits=5, scale=0.5)</input>
			642	<language>python</language>
			643	<prompt_number>14</prompt_number>
			644	<collapsed>0</collapsed>
			645	<outputs>
			646	<output>
			647	<output_type>pyout</output_type>
			648	<text>&lt;sympy.physics.quantum.circuitplot.CircuitPlot object at 0x38a6950&gt;</text>
			649	<prompt_number>14</prompt_number>
			650	</output>
			651	<output>
			652	<output_type>display_data</output_type>
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			863	SWlJaGt4ZVNDaUlpSVpNWGtnb2lJaUdURjVJS0lpSWhreGVTQ2lJaUkKWk1Ya2dvaUlpR1RGNUlL
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			866	cmtKZ2dnPT0K
			867	</png>
			868	</output>
			869	</outputs>
			870	</codecell>
			871	<codecell>
			872	<input>represent(4*encoding_circuit, nqubits=5)</input>
			873	<language>python</language>
			874	<prompt_number>15</prompt_number>
			875	<collapsed>0</collapsed>
			876	<outputs>
			877	<output>
			878	<output_type>pyout</output_type>
			879	<text>⎡1 0 1 0 1 0 1 0 0 -1 0 -1 0 -1 0 -1 0 -1 0 -
			880	⎢
			881	⎢0 1 0 1 0 1 0 1 -1 0 -1 0 -1 0 -1 0 -1 0 -1 0
			882	⎢
			883	⎢0 1 0 -1 0 1 0 -1 -1 0 1 0 -1 0 1 0 -1 0 1 0
			884	⎢
			885	⎢1 0 -1 0 1 0 -1 0 0 -1 0 1 0 -1 0 1 0 -1 0 1
			886	⎢
			887	⎢0 -1 0 -1 0 1 0 1 1 0 1 0 -1 0 -1 0 -1 0 -1 0
			888	⎢
			889	⎢-1 0 -1 0 1 0 1 0 0 1 0 1 0 -1 0 -1 0 -1 0 -
			890	⎢
			891	⎢1 0 -1 0 -1 0 1 0 0 -1 0 1 0 1 0 -1 0 1 0 -
			892	⎢
			893	⎢0 1 0 -1 0 -1 0 1 -1 0 1 0 1 0 -1 0 1 0 -1 0
			894	⎢
			895	⎢0 -1 0 -1 0 -1 0 -1 1 0 1 0 1 0 1 0 -1 0 -1 0
			896	⎢
			897	⎢-1 0 -1 0 -1 0 -1 0 0 1 0 1 0 1 0 1 0 -1 0 -
			898	⎢
			899	⎢-1 0 1 0 -1 0 1 0 0 1 0 -1 0 1 0 -1 0 -1 0 1
			900	⎢
			901	⎢0 -1 0 1 0 -1 0 1 1 0 -1 0 1 0 -1 0 -1 0 1 0
			902	⎢
			903	⎢1 0 1 0 -1 0 -1 0 0 -1 0 -1 0 1 0 1 0 -1 0 -
			904	⎢
			905	⎢0 1 0 1 0 -1 0 -1 -1 0 -1 0 1 0 1 0 -1 0 -1 0
			906	⎢
			907	⎢0 -1 0 1 0 1 0 -1 1 0 -1 0 -1 0 1 0 1 0 -1 0
			908	⎢
			909	⎢-1 0 1 0 1 0 -1 0 0 1 0 -1 0 -1 0 1 0 1 0 -
			910	⎢
			911	⎢0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 -1 0 -1 0
			912	⎢
			913	⎢1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 -1 0 -
			914	⎢
			915	⎢-1 0 1 0 -1 0 1 0 0 -1 0 1 0 -1 0 1 0 1 0 -
			916	⎢
			917	⎢0 -1 0 1 0 -1 0 1 -1 0 1 0 -1 0 1 0 1 0 -1 0
			918	⎢
			919	⎢-1 0 -1 0 1 0 1 0 0 -1 0 -1 0 1 0 1 0 -1 0 -
			920	⎢
			921	⎢0 -1 0 -1 0 1 0 1 -1 0 -1 0 1 0 1 0 -1 0 -1 0
			922	⎢
			923	⎢0 -1 0 1 0 1 0 -1 -1 0 1 0 1 0 -1 0 -1 0 1 0
			924	⎢
			925	⎢-1 0 1 0 1 0 -1 0 0 -1 0 1 0 1 0 -1 0 -1 0 1
			926	⎢
			927	⎢1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1
			928	⎢
			929	⎢0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0
			930	⎢
			931	⎢0 -1 0 1 0 -1 0 1 -1 0 1 0 -1 0 1 0 -1 0 1 0
			932	⎢
			933	⎢-1 0 1 0 -1 0 1 0 0 -1 0 1 0 -1 0 1 0 -1 0 1
			934	⎢
			935	⎢0 -1 0 -1 0 1 0 1 -1 0 -1 0 1 0 1 0 1 0 1 0
			936	⎢
			937	⎢-1 0 -1 0 1 0 1 0 0 -1 0 -1 0 1 0 1 0 1 0 1
			938	⎢
			939	⎢-1 0 1 0 1 0 -1 0 0 -1 0 1 0 1 0 -1 0 1 0 -
			940	⎢
			941	⎣0 -1 0 1 0 1 0 -1 -1 0 1 0 1 0 -1 0 1 0 -1 0
			942
			943	1 0 -1 0 -1 -1 0 -1 0 -1 0 -1 0 ⎤
			944	⎥
			945	-1 0 -1 0 0 -1 0 -1 0 -1 0 -1⎥
			946	⎥
			947	-1 0 1 0 0 -1 0 1 0 -1 0 1 ⎥
			948	⎥
			949	0 -1 0 1 -1 0 1 0 -1 0 1 0 ⎥
			950	⎥
			951	1 0 1 0 0 -1 0 -1 0 1 0 1 ⎥
			952	⎥
			953	1 0 1 0 1 -1 0 -1 0 1 0 1 0 ⎥
			954	⎥
			955	1 0 -1 0 1 1 0 -1 0 -1 0 1 0 ⎥
			956	⎥
			957	-1 0 1 0 0 1 0 -1 0 -1 0 1 ⎥
			958	⎥
			959	-1 0 -1 0 0 -1 0 -1 0 -1 0 -1⎥
			960	⎥
			961	1 0 -1 0 -1 -1 0 -1 0 -1 0 -1 0 ⎥
			962	⎥
			963	0 -1 0 1 -1 0 1 0 -1 0 1 0 ⎥
			964	⎥
			965	-1 0 1 0 0 -1 0 1 0 -1 0 1 ⎥
			966	⎥
			967	1 0 1 0 1 -1 0 -1 0 1 0 1 0 ⎥
			968	⎥
			969	1 0 1 0 0 -1 0 -1 0 1 0 1 ⎥
			970	⎥
			971	-1 0 1 0 0 1 0 -1 0 -1 0 1 ⎥
			972	⎥
			973	1 0 -1 0 1 1 0 -1 0 -1 0 1 0 ⎥
			974	⎥
			975	-1 0 -1 0 0 1 0 1 0 1 0 1 ⎥
			976	⎥
			977	1 0 -1 0 -1 1 0 1 0 1 0 1 0 ⎥
			978	⎥
			979	1 0 1 0 -1 -1 0 1 0 -1 0 1 0 ⎥
			980	⎥
			981	1 0 -1 0 0 -1 0 1 0 -1 0 1 ⎥
			982	⎥
			983	1 0 1 0 1 1 0 1 0 -1 0 -1 0 ⎥
			984	⎥
			985	1 0 1 0 0 1 0 1 0 -1 0 -1⎥
			986	⎥
			987	1 0 -1 0 0 1 0 -1 0 -1 0 1 ⎥
			988	⎥
			989	0 1 0 -1 1 0 -1 0 -1 0 1 0 ⎥
			990	⎥
			991	0 1 0 1 -1 0 -1 0 -1 0 -1 0 ⎥
			992	⎥
			993	1 0 1 0 0 -1 0 -1 0 -1 0 -1⎥
			994	⎥
			995	-1 0 1 0 0 1 0 -1 0 1 0 -1⎥
			996	⎥
			997	0 -1 0 1 1 0 -1 0 1 0 -1 0 ⎥
			998	⎥
			999	-1 0 -1 0 0 -1 0 -1 0 1 0 1 ⎥
			1000	⎥
			1001	0 -1 0 -1 -1 0 -1 0 1 0 1 0 ⎥
			1002	⎥
			1003	1 0 -1 0 1 -1 0 1 0 1 0 -1 0 ⎥
			1004	⎥
			1005	-1 0 1 0 0 -1 0 1 0 1 0 -1⎦</text>
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			1007	<prompt_number>15</prompt_number>
			1008	</output>
			1009	</outputs>
			1010	</codecell>
			1011	</cells>
			1012	</worksheet>
			1013	</worksheets>
			1014	</notebook>

docs/examples/notebooks/qft.ipynb

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	1	{"cells":[{"cell_type":"text","text":"<h1>Quantum Fourier Transform</h1>"},{"code":"%load_ext sympyprinting","cell_type":"code","prompt_number":1},{"code":"from sympy import sqrt, symbols, Rational\nfrom sympy import expand, Eq, Symbol, simplify, exp, sin\nfrom sympy.physics.quantum import \nfrom sympy.physics.quantum.qubit import \nfrom sympy.physics.quantum.gate import \nfrom sympy.physics.quantum.grover import \nfrom sympy.physics.quantum.qft import QFT, IQFT, Fourier\nfrom sympy.physics.quantum.circuitplot import circuit_plot","cell_type":"code","prompt_number":2},{"cell_type":"text","text":"QFT is useful for a quantum algorithm for factoring numbers which is exponentially faster than what is thought to be possible on a classical machine.<br>\nThe transform does a DFT on the state of a quantum system<br>\nThere is a simple decomposition of the QFT in terms of a few elementary gates."},{"cell_type":"text","text":"<h2>QFT Gate and Circuit</h2>"},{"cell_type":"text","text":"Build a 3 qubit QFT and decompose it into primitive gates."},{"code":"fourier = QFT(0,3).decompose(); fourier\n","cell_type":"code","prompt_number":3},{"code":"circuit_plot(fourier, nqubits=3)","cell_type":"code","prompt_number":4},{"cell_type":"text","text":"The QFT circuit can be represented in various symbolic forms."},{"code":"m = represent(fourier, nqubits=3)","cell_type":"code","prompt_number":5},{"code":"m","cell_type":"code","prompt_number":6},{"code":"represent(Fourier(0,3), nqubits=3)4/sqrt(2)\n","cell_type":"code","prompt_number":7},{"cell_type":"text","text":"<h2>QFT in Action</h2>"},{"cell_type":"text","text":"Build a 3 qubit state to take the QFT of."},{"code":"state = (Qubit('000') + Qubit('010') + Qubit('100') + Qubit('110'))/sqrt(4); state\n","cell_type":"code","prompt_number":8},{"cell_type":"text","text":"Perform the QFT."},{"code":"qapply(fourierstate)\n","cell_type":"code","prompt_number":9},{"code":"%notebook save qft.ipynb","cell_type":"code","prompt_number":23},{"code":"%notebook load density_matrix.ipynb","cell_type":"code","prompt_number":207}]} No newline at end of file	1	<?xml version="1.0" encoding="utf-8"?>
			2	<notebook>
			3	<name>qft</name>
			4	<nbformat>2</nbformat>
			5	<worksheets>
			6	<worksheet>
			7	<cells>
			8	<markdowncell>
			9	<source><h1>Quantum Fourier Transform</h1></source>
			10	</markdowncell>
			11	<codecell>
			12	<input>%load_ext sympyprinting</input>
			13	<language>python</language>
			14	<prompt_number>1</prompt_number>
			15	<collapsed>1</collapsed>
			16	<outputs />
			17	</codecell>
			18	<codecell>
			19	<input>from sympy import sqrt, symbols, Rational
			20	from sympy import expand, Eq, Symbol, simplify, exp, sin
			21	from sympy.physics.quantum import *
			22	from sympy.physics.quantum.qubit import *
			23	from sympy.physics.quantum.gate import *
			24	from sympy.physics.quantum.grover import *
			25	from sympy.physics.quantum.qft import QFT, IQFT, Fourier
			26	from sympy.physics.quantum.circuitplot import circuit_plot</input>
			27	<language>python</language>
			28	<prompt_number>2</prompt_number>
			29	<collapsed>1</collapsed>
			30	<outputs />
			31	</codecell>
			32	<markdowncell>
			33	<source>QFT is useful for a quantum algorithm for factoring numbers which is exponentially faster than what is thought to be possible on a classical machine.<br>
			34	The transform does a DFT on the state of a quantum system<br>
			35	There is a simple decomposition of the QFT in terms of a few elementary gates.</source>
			36	</markdowncell>
			37	<markdowncell>
			38	<source><h2>QFT Gate and Circuit</h2></source>
			39	</markdowncell>
			40	<markdowncell>
			41	<source>Build a 3 qubit QFT and decompose it into primitive gates.</source>
			42	</markdowncell>
			43	<codecell>
			44	<input>fourier = QFT(0,3).decompose(); fourier
			45	</input>
			46	<language>python</language>
			47	<prompt_number>3</prompt_number>
			48	<collapsed>0</collapsed>
			49	<outputs>
			50	<output>
			51	<output_type>pyout</output_type>
			52	<text>SWAP ⋅H ⋅C ⎛S ⎞⋅H ⋅C ⎛T ⎞⋅C ⎛S ⎞⋅H
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			114	RnRvQUFBQUFTVVZPUks1Q1lJST0K
			115	</png>
			116	<latex>$$SWAP_{0,2} H_{0} C_{0}{\left(S_{1}\right)} H_{1} C_{0}{\left(T_{2}\right)} C_{1}{\left(S_{2}\right)} H_{2}$$</latex>
			117	<prompt_number>3</prompt_number>
			118	</output>
			119	</outputs>
			120	</codecell>
			121	<codecell>
			122	<input>circuit_plot(fourier, nqubits=3)</input>
			123	<language>python</language>
			124	<prompt_number>4</prompt_number>
			125	<collapsed>0</collapsed>
			126	<outputs>
			127	<output>
			128	<output_type>pyout</output_type>
			129	<text>&lt;sympy.physics.quantum.circuitplot.CircuitPlot object at 0x4992910&gt;</text>
			130	<prompt_number>4</prompt_number>
			131	</output>
			132	<output>
			133	<output_type>display_data</output_type>
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			249	</png>
			250	</output>
			251	</outputs>
			252	</codecell>
			253	<markdowncell>
			254	<source>The QFT circuit can be represented in various symbolic forms.</source>
			255	</markdowncell>
			256	<codecell>
			257	<input>m = represent(fourier, nqubits=3)</input>
			258	<language>python</language>
			259	<prompt_number>5</prompt_number>
			260	<collapsed>1</collapsed>
			261	<outputs />
			262	</codecell>
			263	<codecell>
			264	<input>m</input>
			265	<language>python</language>
			266	<prompt_number>6</prompt_number>
			267	<collapsed>0</collapsed>
			268	<outputs>
			269	<output>
			270	<output_type>pyout</output_type>
			271	<text>⎡ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽
			272	⎢╲╱ 2 ╲╱ 2 ╲╱ 2 ╲╱ 2 ╲╱ 2 ╲╱ 2 ╲╱ 2
			273	⎢───── ───── ───── ───── ───── ───── ─────
			274	⎢ 4 4 4 4 4 4 4
			275	⎢
			276	⎢ ⅈ⋅π ⅈ⋅π ⅈ⋅π
			277	⎢ ─── ─── ───
			278	⎢ ⎽⎽⎽ ⎽⎽⎽ 4 ⎽⎽⎽ ⎽⎽⎽ 4 ⎽⎽⎽ ⎽⎽⎽ 4 ⎽⎽⎽
			279	⎢╲╱ 2 ╲╱ 2 ⋅ℯ ╲╱ 2 ⋅ⅈ ╲╱ 2 ⋅ⅈ⋅ℯ -╲╱ 2 -╲╱ 2 ⋅ℯ -╲╱ 2 ⋅
			280	⎢───── ────────── ─────── ──────────── ────── ─────────── ───────
			281	⎢ 4 4 4 4 4 4 4
			282	⎢
			283	⎢ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽
			284	⎢╲╱ 2 ╲╱ 2 ⋅ⅈ -╲╱ 2 -╲╱ 2 ⋅ⅈ ╲╱ 2 ╲╱ 2 ⋅ⅈ -╲╱ 2
			285	⎢───── ─────── ────── ──────── ───── ─────── ──────
			286	⎢ 4 4 4 4 4 4 4
			287	⎢
			288	⎢ ⅈ⋅π ⅈ⋅π ⅈ⋅π
			289	⎢ ─── ─── ───
			290	⎢ ⎽⎽⎽ ⎽⎽⎽ 4 ⎽⎽⎽ ⎽⎽⎽ 4 ⎽⎽⎽ ⎽⎽⎽ 4 ⎽⎽⎽
			291	⎢╲╱ 2 ╲╱ 2 ⋅ⅈ⋅ℯ -╲╱ 2 ⋅ⅈ ╲╱ 2 ⋅ℯ -╲╱ 2 -╲╱ 2 ⋅ⅈ⋅ℯ ╲╱ 2 ⋅ⅈ
			292	⎢───── ──────────── ──────── ────────── ────── ───────────── ───────
			293	⎢ 4 4 4 4 4 4 4
			294	⎢
			295	⎢ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽
			296	⎢╲╱ 2 -╲╱ 2 ╲╱ 2 -╲╱ 2 ╲╱ 2 -╲╱ 2 ╲╱ 2
			297	⎢───── ────── ───── ────── ───── ────── ─────
			298	⎢ 4 4 4 4 4 4 4
			299	⎢
			300	⎢ ⅈ⋅π ⅈ⋅π ⅈ⋅π
			301	⎢ ─── ─── ───
			302	⎢ ⎽⎽⎽ ⎽⎽⎽ 4 ⎽⎽⎽ ⎽⎽⎽ 4 ⎽⎽⎽ ⎽⎽⎽ 4 ⎽⎽⎽
			303	⎢╲╱ 2 -╲╱ 2 ⋅ℯ ╲╱ 2 ⋅ⅈ -╲╱ 2 ⋅ⅈ⋅ℯ -╲╱ 2 ╲╱ 2 ⋅ℯ -╲╱ 2 ⋅
			304	⎢───── ─────────── ─────── ───────────── ────── ────────── ───────
			305	⎢ 4 4 4 4 4 4 4
			306	⎢
			307	⎢ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽
			308	⎢╲╱ 2 -╲╱ 2 ⋅ⅈ -╲╱ 2 ╲╱ 2 ⋅ⅈ ╲╱ 2 -╲╱ 2 ⋅ⅈ -╲╱ 2
			309	⎢───── ──────── ────── ─────── ───── ──────── ──────
			310	⎢ 4 4 4 4 4 4 4
			311	⎢
			312	⎢ ⅈ⋅π ⅈ⋅π ⅈ⋅π
			313	⎢ ─── ─── ───
			314	⎢ ⎽⎽⎽ ⎽⎽⎽ 4 ⎽⎽⎽ ⎽⎽⎽ 4 ⎽⎽⎽ ⎽⎽⎽ 4 ⎽⎽⎽
			315	⎢╲╱ 2 -╲╱ 2 ⋅ⅈ⋅ℯ -╲╱ 2 ⋅ⅈ -╲╱ 2 ⋅ℯ -╲╱ 2 ╲╱ 2 ⋅ⅈ⋅ℯ ╲╱ 2 ⋅ⅈ
			316	⎢───── ───────────── ──────── ─────────── ────── ──────────── ───────
			317	⎣ 4 4 4 4 4 4 4
			318
			319	⎽⎽⎽ ⎤
			320	╲╱ 2 ⎥
			321	───── ⎥
			322	4 ⎥
			323	⎥
			324	ⅈ⋅π⎥
			325	───⎥
			326	⎽⎽⎽ 4 ⎥
			327	ⅈ -╲╱ 2 ⋅ⅈ⋅ℯ ⎥
			328	─ ─────────────⎥
			329	4 ⎥
			330	⎥
			331	⎽⎽⎽ ⎥
			332	-╲╱ 2 ⋅ⅈ ⎥
			333	──────── ⎥
			334	4 ⎥
			335	⎥
			336	ⅈ⋅π ⎥
			337	─── ⎥
			338	⎽⎽⎽ 4 ⎥
			339	-╲╱ 2 ⋅ℯ ⎥
			340	─────────── ⎥
			341	4 ⎥
			342	⎥
			343	⎽⎽⎽ ⎥
			344	-╲╱ 2 ⎥
			345	────── ⎥
			346	4 ⎥
			347	⎥
			348	ⅈ⋅π ⎥
			349	─── ⎥
			350	⎽⎽⎽ 4 ⎥
			351	ⅈ ╲╱ 2 ⋅ⅈ⋅ℯ ⎥
			352	─ ──────────── ⎥
			353	4 ⎥
			354	⎥
			355	⎽⎽⎽ ⎥
			356	╲╱ 2 ⋅ⅈ ⎥
			357	─────── ⎥
			358	4 ⎥
			359	⎥
			360	ⅈ⋅π ⎥
			361	─── ⎥
			362	⎽⎽⎽ 4 ⎥
			363	╲╱ 2 ⋅ℯ ⎥
			364	────────── ⎥
			365	4 ⎦</text>
			366	<latex>$$\left(\begin{smallmatrix}\frac{1}{4} \sqrt{2} & \frac{1}{4} \sqrt{2} & \frac{1}{4} \sqrt{2} & \frac{1}{4} \sqrt{2} & \frac{1}{4} \sqrt{2} & \frac{1}{4} \sqrt{2} & \frac{1}{4} \sqrt{2} & \frac{1}{4} \sqrt{2}\\\frac{1}{4} \sqrt{2} & \frac{1}{4} \sqrt{2} e^{\frac{1}{4} \mathbf{\imath} \pi} & \frac{1}{4} \sqrt{2} \mathbf{\imath} & \frac{1}{4} \sqrt{2} \mathbf{\imath} e^{\frac{1}{4} \mathbf{\imath} \pi} & - \frac{1}{4} \sqrt{2} & - \frac{1}{4} \sqrt{2} e^{\frac{1}{4} \mathbf{\imath} \pi} & - \frac{1}{4} \sqrt{2} \mathbf{\imath} & - \frac{1}{4} \sqrt{2} \mathbf{\imath} e^{\frac{1}{4} \mathbf{\imath} \pi}\\\frac{1}{4} \sqrt{2} & \frac{1}{4} \sqrt{2} \mathbf{\imath} & - \frac{1}{4} \sqrt{2} & - \frac{1}{4} \sqrt{2} \mathbf{\imath} & \frac{1}{4} \sqrt{2} & \frac{1}{4} \sqrt{2} \mathbf{\imath} & - \frac{1}{4} \sqrt{2} & - \frac{1}{4} \sqrt{2} \mathbf{\imath}\\\frac{1}{4} \sqrt{2} & \frac{1}{4} \sqrt{2} \mathbf{\imath} e^{\frac{1}{4} \mathbf{\imath} \pi} & - \frac{1}{4} \sqrt{2} \mathbf{\imath} & \frac{1}{4} \sqrt{2} e^{\frac{1}{4} \mathbf{\imath} \pi} & - \frac{1}{4} \sqrt{2} & - \frac{1}{4} \sqrt{2} \mathbf{\imath} e^{\frac{1}{4} \mathbf{\imath} \pi} & \frac{1}{4} \sqrt{2} \mathbf{\imath} & - \frac{1}{4} \sqrt{2} e^{\frac{1}{4} \mathbf{\imath} \pi}\\\frac{1}{4} \sqrt{2} & - \frac{1}{4} \sqrt{2} & \frac{1}{4} \sqrt{2} & - \frac{1}{4} \sqrt{2} & \frac{1}{4} \sqrt{2} & - \frac{1}{4} \sqrt{2} & \frac{1}{4} \sqrt{2} & - \frac{1}{4} \sqrt{2}\\\frac{1}{4} \sqrt{2} & - \frac{1}{4} \sqrt{2} e^{\frac{1}{4} \mathbf{\imath} \pi} & \frac{1}{4} \sqrt{2} \mathbf{\imath} & - \frac{1}{4} \sqrt{2} \mathbf{\imath} e^{\frac{1}{4} \mathbf{\imath} \pi} & - \frac{1}{4} \sqrt{2} & \frac{1}{4} \sqrt{2} e^{\frac{1}{4} \mathbf{\imath} \pi} & - \frac{1}{4} \sqrt{2} \mathbf{\imath} & \frac{1}{4} \sqrt{2} \mathbf{\imath} e^{\frac{1}{4} \mathbf{\imath} \pi}\\\frac{1}{4} \sqrt{2} & - \frac{1}{4} \sqrt{2} \mathbf{\imath} & - \frac{1}{4} \sqrt{2} & \frac{1}{4} \sqrt{2} \mathbf{\imath} & \frac{1}{4} \sqrt{2} & - \frac{1}{4} \sqrt{2} \mathbf{\imath} & - \frac{1}{4} \sqrt{2} & \frac{1}{4} \sqrt{2} \mathbf{\imath}\\\frac{1}{4} \sqrt{2} & - \frac{1}{4} \sqrt{2} \mathbf{\imath} e^{\frac{1}{4} \mathbf{\imath} \pi} & - \frac{1}{4} \sqrt{2} \mathbf{\imath} & - \frac{1}{4} \sqrt{2} e^{\frac{1}{4} \mathbf{\imath} \pi} & - \frac{1}{4} \sqrt{2} & \frac{1}{4} \sqrt{2} \mathbf{\imath} e^{\frac{1}{4} \mathbf{\imath} \pi} & \frac{1}{4} \sqrt{2} \mathbf{\imath} & \frac{1}{4} \sqrt{2} e^{\frac{1}{4} \mathbf{\imath} \pi}\end{smallmatrix}\right)$$</latex>
			367	<prompt_number>6</prompt_number>
			368	</output>
			369	</outputs>
			370	</codecell>
			371	<codecell>
			372	<input>represent(Fourier(0,3), nqubits=3)*4/sqrt(2)
			373	</input>
			374	<language>python</language>
			375	<prompt_number>7</prompt_number>
			376	<collapsed>0</collapsed>
			377	<outputs>
			378	<output>
			379	<output_type>pyout</output_type>
			380	<text>⎡1 1 1 1 1 1 1 1 ⎤
			381	⎢ ⎥
			382	⎢ 2 3 4 5 6 7⎥
			383	⎢1 ω ω ω ω ω ω ω ⎥
			384	⎢ ⎥
			385	⎢ 2 4 6 2 4 6⎥
			386	⎢1 ω ω ω 1 ω ω ω ⎥
			387	⎢ ⎥
			388	⎢ 3 6 4 7 2 5⎥
			389	⎢1 ω ω ω ω ω ω ω ⎥
			390	⎢ ⎥
			391	⎢ 4 4 4 4⎥
			392	⎢1 ω 1 ω 1 ω 1 ω ⎥
			393	⎢ ⎥
			394	⎢ 5 2 7 4 6 3⎥
			395	⎢1 ω ω ω ω ω ω ω ⎥
			396	⎢ ⎥
			397	⎢ 6 4 2 6 4 2⎥
			398	⎢1 ω ω ω 1 ω ω ω ⎥
			399	⎢ ⎥
			400	⎢ 7 6 5 4 3 2 ⎥
			401	⎣1 ω ω ω ω ω ω ω ⎦</text>
			402	<latex>$$\left(\begin{smallmatrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\1 & \omega & \omega^{2} & \omega^{3} & \omega^{4} & \omega^{5} & \omega^{6} & \omega^{7}\\1 & \omega^{2} & \omega^{4} & \omega^{6} & 1 & \omega^{2} & \omega^{4} & \omega^{6}\\1 & \omega^{3} & \omega^{6} & \omega & \omega^{4} & \omega^{7} & \omega^{2} & \omega^{5}\\1 & \omega^{4} & 1 & \omega^{4} & 1 & \omega^{4} & 1 & \omega^{4}\\1 & \omega^{5} & \omega^{2} & \omega^{7} & \omega^{4} & \omega & \omega^{6} & \omega^{3}\\1 & \omega^{6} & \omega^{4} & \omega^{2} & 1 & \omega^{6} & \omega^{4} & \omega^{2}\\1 & \omega^{7} & \omega^{6} & \omega^{5} & \omega^{4} & \omega^{3} & \omega^{2} & \omega\end{smallmatrix}\right)$$</latex>
			403	<prompt_number>7</prompt_number>
			404	</output>
			405	</outputs>
			406	</codecell>
			407	<markdowncell>
			408	<source><h2>QFT in Action</h2></source>
			409	</markdowncell>
			410	<markdowncell>
			411	<source>Build a 3 qubit state to take the QFT of.</source>
			412	</markdowncell>
			413	<codecell>
			414	<input>state = (Qubit('000') + Qubit('010') + Qubit('100') + Qubit('110'))/sqrt(4); state
			415	</input>
			416	<language>python</language>
			417	<prompt_number>8</prompt_number>
			418	<collapsed>0</collapsed>
			419	<outputs>
			420	<output>
			421	<output_type>pyout</output_type>
			422	<text>❘000⟩ + ❘010⟩ + ❘100⟩ + ❘110⟩
			423	─────────────────────────────
			424	2</text>
			425	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQVA0QUFBQWlDQVlBQUFCL2FtUUZBQUFBQkhOQ1NWUUlD
			426	QWdJZkFoa2lBQUFCUjVKUkVGVQplSnp0bkZ0b0hGVVl4My9iSk5hbXBob3JhUnZGVnEyWVFnUXQr
			427	aUJLS3lxaWd1TGxJWUpRUVgzd1FiRytGQzhnNjFNcGxHb3JvcjNnCkJYendwWWhJOVVHMGlvaGFC
			428	VVdyQmtHRldtOTRiMU1xYWMzNjhNMjRrM0V1NTh5ZU16dVovWDZ3N002Wi8zNTgvejJabVhNTktJ
			429	cWkKWkRBTTlIYzdDVVZSeW1FK3NBYjRCbGpSM1ZRVVJYSEJQQVBOQkRBRW5PVTVGMFZSS2tnTGZl
			430	SXJTaTB3ZWVJcmlsSXpldjNDUHgyNAp1TnRKbEVRdmVRWDFtMG1mUmVBbXNCWDQwektoS0FOSWdv
			431	ZUFHNEhKMlBuNXdMM0F6Y0FGd0JUd2MwSWNWN29wNEc1Z2J5RTNkc1Q5Cm1ub0FhQUFiZ1ErQTZS
			432	Uk5sYjJHdVBCaHFwc3JmazExUmZ5ZUEveVJuMzQybmZieCt4Rnpwd1RIemRqNVB1QWo0S0hnZUJI
			433	d0NiRE8Kcys0ZVlJbTVqY0kwSTU5TmN3dTFUeUsvLzJrcHNhdnNOY1NsajdyNE5kVVY5YnNSdWZn
			434	TDB3Z1M2MlJrZnpOd2J1UzRHVHQvRy9BWAowaW9JV1FjY1lYYnlyblduQXZjYmVvaGoyMkt5eVcw
			435	RWVCbDREL2lhN0QrTUtudjE0YU1PZnN1bzMySGduWXk0bWR3T1BJUGNjVjRFCjFoZUljVDd3Vkt5
			436	c0dmbmNBRDREWG9wcGxnSXp3QVpQdXBCSEtUYmVzY2RDMjdUTWJRQTRNZmk4bWZRL2pDcDdCZmMr
			437	NnVLM3JQcTkKQmRnVUQycnlnendQM0FGY0JOd0tQRzd3blRnYmdKMFo1ODhFeG9FUFkrVS9BZDhC
			438	VjNuU2hid0dYSmZwSUptRkJiNWptdHN4NEcrSAo4VUxLOUFydWZkVEZiMW4xdXh1NG5QWk5CaWhu
			439	Vkw4UHVBenBrNlN4TEhnL25IQnVDbGpwU1JmeVBuQkpSbjR1c2MzTmRid3l2ZHBRCmg3cjFnUXUv
			440	dndBM1JRdkt1UERIZ1lOSXN5U05wY0g3Vk1LNUk3UUhCRjNyb2h3RXpzakkwUlZGY25NZHJ5eXZO
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			443	eklJL0lOWm42dFRiSE56SGE5TXJ6YlVvVzU5NE1MdkFkb3RCNkNjYmJhL2t6OE4rQVZ3Cm5QYmRM
			444	Y29nOEtNblhjZ0VNbU9SUkFQWWxoTHJQR0JYUXZrTThDREpMUjNiM1BLb3NsY2I2bEMzUG5EaGR6
			445	bXhoVHhsWFBnL0FHTTUKbW1uZ0svN2ZEeHRFVnZydDg2UUxXUWs4bTVKYkM5aE9iRlEwNEVMZzZZ
			446	VHlHZEpYVE5ubWxrZVZ2ZHBRaDdyMWdRdS9xNER2b3dWbApYUGo3a2FiSE1yS2ZacDhpVTRaUjFn
			447	QW5NSHNxMExWdU5iSkdJWXY5S2VXSERiNmJoR2x1cnVOMXc2c05kYWhiSDNUcWQ0eFk2NldNClVm
			448	MFpaRzV4Vlk3dUJlQnFabmNMcmtmV0w3L2hVWGNEOEVxZUNjZVk1dVk2WGplODJsQ0h1dlZCcDM3
			449	UEJsNlBGcGc4OFJjQWR5RnIKZ0E4QXo1Rzk0U0NKVGNIcnpRek5IdUFCWk1IQlZtUTI0RXFreitK
			450	THR3aVpKamx1NmFkVFRITHJCKzVEQm1YdURNcDJJNXN3OWdKdgpXOGJybGxmWFBreDFWZmRiVnYx
			451	T0FLOFNXd2VRTjVvL0QzZ0NlQVRwMTJ4Qm1oeHJrZWE3RGR1QWoybjNQNWFRdk91cUgybXlIRVdt
			452	CklaS21NVnpwaHBEZjRKQ3BpUmh2SWF1aVRFanltNVZiSDNCdFNxeHZnYzhUeXF2bzFiVVBVMTNW
			453	L1paUnY0c1JIMnVSUVhaanhwRSsKMEVod1BJb01pRnhxRXlTUzlBNWkwd3B6bkRLMmZGYUZYdklL
			454	OWZDN0Jka25ZODBROEJqU2pBQzVBYlF3dnhQR0NmZmoxd1hiNWJWegptVjd5Q3ZYd3U4SlZvUFhJ
			455	Wm9HQlBLR2lLUFZnREhnWGFlNHJpdElEakNMOTg1T0Q0MTcvWDMyS1Vuc1dBdy9UYnQ2dkJxN3BY
			456	anFLCm9uUkszblRlU2NqRS96VHRIWGVqd0JYSTFqOUZVZVlnZVF0NEJwRUZBMUZhNkVXdktJcWlL
			457	SXFpS0lxaUtJcWlLRjNuWDdyc1VmN2kKM2xJV0FBQUFBRWxGVGtTdVFtQ0MK
			458	</png>
			459	<latex>$$\frac{1}{2} \left({\left\|000\right\rangle } + {\left\|010\right\rangle } + {\left\|100\right\rangle } + {\left\|110\right\rangle }\right)$$</latex>
			460	<prompt_number>8</prompt_number>
			461	</output>
			462	</outputs>
			463	</codecell>
			464	<markdowncell>
			465	<source>Perform the QFT.</source>
			466	</markdowncell>
			467	<codecell>
			468	<input>qapply(fourier*state)
			469	</input>
			470	<language>python</language>
			471	<prompt_number>9</prompt_number>
			472	<collapsed>0</collapsed>
			473	<outputs>
			474	<output>
			475	<output_type>pyout</output_type>
			476	<text>⎽⎽⎽ ⎽⎽⎽
			477	╲╱ 2 ⋅❘000⟩ ╲╱ 2 ⋅❘100⟩
			478	─────────── + ───────────
			479	2 2</text>
			480	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQU5BQUFBQWtDQVlBQUFENHA3UjdBQUFBQkhOQ1NWUUlD
			481	QWdJZkFoa2lBQUFCWkJKUkVGVQplSnp0M0h1b1pWVWR3UEhQSFcvam5kRWNMYTFyVHRvZmFRWTVR
			482	eFNwbzg1Y1N3SmoxQVRGQjhySWRLdXhtb2dKMG54T1lZMys0U1A2Cko2bEltSUVvUmhLTmZDR29K
			483	Q2dPZ29NamdYL0kwRU1sRll1Um9sRnVmL3pPeG5QMjNlZHh6OW1QYzYvckM1dkRYZnUzOTE1bmY4
			484	OWUKZTYzZjJ2dVNTQ1FTaVVRVFREUmRnVDdNbGJDUGNmK09TNTNqOExjUzlyUG9QWjZBcVpxUHVS
			485	c2Zxdm1ZUzVua3NHU1dEUkN6R3BkagpENmFyclU0SG44Vit2Rm5qTVpjcXlXR0RuSXUxb2p2MWlS
			486	cVB1MHZjL2hPamt4eFd4T1FBTVE5V1hvdjVuSVMzOGZjR2pyMFVTUTdICmdEcGJyMS9peEpxTzlY
			487	NGlPU3laUWNaQWRmTnhNZEI5cWVtS2xNU21waXZRQUV2TklWMDhEdEtGcTV2djQvWWU2OWRnVmd5
			488	TW44WDkKMkZjUXQ3YjErWHl1L0VwOFFReHNYOEc5K0dmQjltWEZmVUJrdi9iMytFNUxqWDRPR2Qx
			489	anhwbFlqNTkyV1Q4Mkh1dTQvUjhqMHA3ZApPQi8zNFhNaW03UUQvOEVGQmJFenJhV2RHL0ZuTEcv
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			515	Wk9ETlc0Ck1IRFZBQUFBQUVsRlRrU3VRbUNDCg==
			516	</png>
			517	<latex>$$\frac{1}{2} \sqrt{2} {\left\|000\right\rangle } + \frac{1}{2} \sqrt{2} {\left\|100\right\rangle }$$</latex>
			518	<prompt_number>9</prompt_number>
			519	</output>
			520	</outputs>
			521	</codecell>
			522	</cells>
			523	</worksheet>
			524	</worksheets>
			525	</notebook>

docs/examples/notebooks/quantum_computing.ipynb

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	1	{"cells":[{"cell_type":"text","text":"<h1>Symbolic Quantum Computing</h1>"},{"code":"%load_ext sympyprinting","cell_type":"code","prompt_number":1},{"code":"from sympy import sqrt, symbols, Rational\nfrom sympy import expand, Eq, Symbol, simplify, exp, sin\nfrom sympy.physics.quantum import \nfrom sympy.physics.quantum.qubit import \nfrom sympy.physics.quantum.gate import \nfrom sympy.physics.quantum.grover import \nfrom sympy.physics.quantum.qft import QFT, IQFT, Fourier\nfrom sympy.physics.quantum.circuitplot import circuit_plot","cell_type":"code","prompt_number":2},{"cell_type":"text","text":"<h2>Qubits</h2>"},{"cell_type":"text","text":"The state of a set of qubits (Two state systems) is the quantum state that is of interest in Quantum Computing."},{"code":"alpha, beta = symbols('alpha beta',real=True)","cell_type":"code","prompt_number":3},{"code":"psi = alphaQubit('00') + betaQubit('11'); psi\n","cell_type":"code","prompt_number":4},{"code":"Dagger(psi)\n","cell_type":"code","prompt_number":5},{"code":"qapply(Dagger(Qubit('00'))psi)\n","cell_type":"code","prompt_number":6},{"cell_type":"text","text":"SymPy supports many different types of measurements."},{"code":"for state, prob in measure_all(psi):\n display(state)\n display(prob)\n","cell_type":"code","prompt_number":7},{"cell_type":"text","text":"Qubits can be represented in the computational basis."},{"code":"represent(psi)\n","cell_type":"code","prompt_number":8},{"cell_type":"text","text":"<h2>Gates</h2>"},{"cell_type":"text","text":"Gate objects are the operators which act on a quantum state."},{"code":"g = X(0); g\n","cell_type":"code","prompt_number":9},{"code":"represent(g, nqubits=2)\n","cell_type":"code","prompt_number":10},{"code":"c = H(0)Qubit('00'); c\n","cell_type":"code","prompt_number":11},{"code":"qapply(c)\n","cell_type":"code","prompt_number":12},{"code":"for gate in [H,X,Y,Z,S,T]:\n for state in [Qubit('0'),Qubit('1')]:\n lhs = gate(0)state\n rhs = qapply(lhs)\n display(Eq(lhs,rhs))","cell_type":"code","prompt_number":13},{"cell_type":"text","text":"<h2>Symbolic gate rules and circuit simplification</h2>"},{"code":"for g1 in (Y,Z,H):\n for g2 in (Y,Z,H):\n e = Commutator(g1(0),g2(0))\n if g1 != g2:\n display(Eq(e,e.doit()))\n","cell_type":"code","prompt_number":14},{"code":"c = H(0)X(1)H(0)2CNOT(0,1)X(1)3X(0)Z(1)*2; c\n","cell_type":"code","prompt_number":15},{"code":"circuit_plot(c, nqubits=2)","cell_type":"code","prompt_number":16},{"cell_type":"text","text":"This performs a commutator/anticommutator aware bubble sort algorithm to simplify a circuit."},{"code":"gate_simp(c)\n","cell_type":"code","prompt_number":17},{"code":"circuit_plot(gate_simp(c),nqubits=2)","cell_type":"code","prompt_number":18},{"code":"%notebook save quantum_computing.ipynb","cell_type":"code","prompt_number":35},{"code":"%notebook load grovers.ipynb","cell_type":"code","prompt_number":90}]} No newline at end of file	1	<?xml version="1.0" encoding="utf-8"?>
			2	<notebook>
			3	<name>quantum_computing</name>
			4	<nbformat>2</nbformat>
			5	<worksheets>
			6	<worksheet>
			7	<cells>
			8	<markdowncell>
			9	<source><h1>Symbolic Quantum Computing</h1></source>
			10	</markdowncell>
			11	<codecell>
			12	<input>%load_ext sympyprinting</input>
			13	<language>python</language>
			14	<prompt_number>1</prompt_number>
			15	<collapsed>1</collapsed>
			16	<outputs />
			17	</codecell>
			18	<codecell>
			19	<input>from sympy import sqrt, symbols, Rational
			20	from sympy import expand, Eq, Symbol, simplify, exp, sin
			21	from sympy.physics.quantum import *
			22	from sympy.physics.quantum.qubit import *
			23	from sympy.physics.quantum.gate import *
			24	from sympy.physics.quantum.grover import *
			25	from sympy.physics.quantum.qft import QFT, IQFT, Fourier
			26	from sympy.physics.quantum.circuitplot import circuit_plot</input>
			27	<language>python</language>
			28	<prompt_number>2</prompt_number>
			29	<collapsed>1</collapsed>
			30	<outputs />
			31	</codecell>
			32	<markdowncell>
			33	<source><h2>Qubits</h2></source>
			34	</markdowncell>
			35	<markdowncell>
			36	<source>The state of a set of qubits (Two state systems) is the quantum state that is of interest in Quantum Computing.</source>
			37	</markdowncell>
			38	<codecell>
			39	<input>alpha, beta = symbols('alpha beta',real=True)</input>
			40	<language>python</language>
			41	<prompt_number>3</prompt_number>
			42	<collapsed>1</collapsed>
			43	<outputs />
			44	</codecell>
			45	<codecell>
			46	<input>psi = alphaQubit('00') + betaQubit('11'); psi
			47	</input>
			48	<language>python</language>
			49	<prompt_number>4</prompt_number>
			50	<collapsed>0</collapsed>
			51	<outputs>
			52	<output>
			53	<output_type>pyout</output_type>
			54	<text>α⋅❘00⟩ + β⋅❘11⟩</text>
			55	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUdrQUFBQVhDQVlBQUFBSXFtR0xBQUFBQkhOQ1NWUUlD
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			79	T1JLNUNZSUk9Cg==
			80	</png>
			81	<latex>$$\alpha {\left\|00\right\rangle } + \beta {\left\|11\right\rangle }$$</latex>
			82	<prompt_number>4</prompt_number>
			83	</output>
			84	</outputs>
			85	</codecell>
			86	<codecell>
			87	<input>Dagger(psi)
			88	</input>
			89	<language>python</language>
			90	<prompt_number>5</prompt_number>
			91	<collapsed>0</collapsed>
			92	<outputs>
			93	<output>
			94	<output_type>pyout</output_type>
			95	<text>α⋅⟨00❘ + β⋅⟨11❘</text>
			96	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUdnQUFBQVhDQVlBQUFEbmFBcTFBQUFBQkhOQ1NWUUlD
			97	QWdJZkFoa2lBQUFBNk5KUkVGVQphSUh0MkZ1TVhWTWN4L0dQS3BPVWlCb3oxU0pFTkc0eG9pSHRR
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			99	MEtrZEZxaVpqejgxMDczN05ubnpONHpjODZNNW55VG5iWDNXcis5MXZydnRmLy9kYUhEQWMvYTZl
			100	N0EKTk5BMm0yZFA4djJ6OFcxSi9uTGNtY3FHOFF1ZW5xQnVMUjZaWkQvcnNocExNSVJUc1FsdnBi
			101	SkdObWRjaHhXNGNadzJtdW1tek9iMQp4Zzd5NmRpRGhlbDVOajdHUFJQVURVeEZSeXZTSlQ3TTRs
			102	emVJdnlEbnZSY1pqTWNqN3V3RHk4MmFhT0tiaUM3bVZXaDA0MDRBcnRUClEzbnVFeDg2Kzh2MllS
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			105	YWdHMkZ2TFBFMzFhTTBkK2xkRmxOWFR1NQpESzhWOHJyRlQ3UUd2eW0zdWFXVXhkS1QwQzlpWlM5
			106	MllwMXdkY0kxbCtLamtuZVBTZW51UXY0dzltSitUVjI3Nk1HUHFmMWx3bXVPCkZHRnRFNTdDZWNw
			107	dGJpbEZEK3JEKy9oU2RQSnFNU0NET2UzanVCRHZsdFNYZmZpaGtySWhZWFFkWGJ0WWJyODlRK25h
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			117	eXl2NjZLN3U2cUJpVHFuQ1RNRitkaWM4UVozUE40UnV6dwpteTBJaWpiMzR4TVJ0a2ZTOVQwK05I
			118	cFBWMVZYMStiL0ZYVUdxRjhjaHM1WUpuUFVNMVBabzNxSVhDVG13ZzR6bEkzVDNZSHhPQkE5CnFD
			119	bzkybnp3MmFGRGh3NGRSdkVmTWFmaitnL1d6VDRBQUFBQVNVVk9SSzVDWUlJPQo=
			120	</png>
			121	<latex>$$\alpha {\left\langle 00\right\|} + \beta {\left\langle 11\right\|}$$</latex>
			122	<prompt_number>5</prompt_number>
			123	</output>
			124	</outputs>
			125	</codecell>
			126	<codecell>
			127	<input>qapply(Dagger(Qubit('00'))*psi)
			128	</input>
			129	<language>python</language>
			130	<prompt_number>6</prompt_number>
			131	<collapsed>0</collapsed>
			132	<outputs>
			133	<output>
			134	<output_type>pyout</output_type>
			135	<text>α</text>
			136	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUE0QUFBQU5DQVlBQUFDWjNGOS9BQUFBQkhOQ1NWUUlD
			137	QWdJZkFoa2lBQUFBTWxKUkVGVQpLSkhWMFRGT1FsRVFoZUhQMTJDQ0xSWVFHa0xySzRrTDBKQklZ
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			139	ZkRIT0NuUldyaERFK2RZNEI2ckEyYWJRam5tdUlrNnd5TmU0dzNqRkdyakJ3K0ozb3NOSFRSSytw
			140	NEQKckNaNk04QitRUFhDU3BHdjhZN2ZCUHlLZklrS3ZnK2JGekYxa05xSTJPSVRwNFZRYkp4SGMx
			141	RUNuV0dETjZ4VGNJa1hkQlBvQ2tQTQo3RStUNFpiak8xWXdRaTMrbGVFakJ1YVlZSXFuc1AxZlln
			142	Y1ByaVYrNlVjV0xBQUFBQUJKUlU1RXJrSmdnZz09Cg==
			143	</png>
			144	<latex>$$\alpha$$</latex>
			145	<prompt_number>6</prompt_number>
			146	</output>
			147	</outputs>
			148	</codecell>
			149	<markdowncell>
			150	<source>SymPy supports many different types of measurements.</source>
			151	</markdowncell>
			152	<codecell>
			153	<input>for state, prob in measure_all(psi):
			154	display(state)
			155	display(prob)
			156	</input>
			157	<language>python</language>
			158	<prompt_number>7</prompt_number>
			159	<collapsed>0</collapsed>
			160	<outputs>
			161	<output>
			162	<output_type>display_data</output_type>
			163	<text>❘00⟩</text>
			164	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUI4QUFBQVRDQVlBQUFCb2JOWkNBQUFBQkhOQ1NWUUlD
			165	QWdJZkFoa2lBQUFBV2RKUkVGVQpTSW50MWJ0TEhVRVV4L0VQRWpVK0lDYmlvekJxcTJBYUNXS2Ex
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			167	ODF1bWtESUQ0WXplODUzK0EwN1ozYjV4L1htb2NLamFENkg5L2lNTHF5aGlrczh4bHNjcDlZWDRh
			168	cm8KeFVYZURpc1lUdWE3V0VuVlBxSTl0YVlJMTQyRlBPUFlmQUszR0lscWJmaUc1U2hYbElNbE5C
			169	VXgzOEJKUm4wSEg2TG5vaHk4d25RYQp6TnJOT0E0ejhrY1lFMTVqR1E3Mmt3MzgxcndQVnhuNWFo
			170	TDdTM0oxbldBb3oveUpjRzVWamFybnVrcHdzVFl4azJkK2cxb3kwbXBPCllxMEVGMnNHVzNubTF4
			171	cnZjbDBkU1R3dHdjVWF4cWM4YzBLblBzM0lkeWJ4ckNRSGswTFQzZE5ENXM4ejhpK0ZUdjVla29N
			172	cGJCY3gKUDhBb0JxTmNEMTVnOVErNFovZ2l1ejkrcVNLY1N3djJzQjdWRm9WdmZtdVVLOHJOQzll
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			174	QUFFbEZUa1N1UW1DQwo=
			175	</png>
			176	<latex>$${\left\|00\right\rangle }$$</latex>
			177	</output>
			178	<output>
			179	<output_type>display_data</output_type>
			180	<text>2
			181	α
			182	───────
			183	2 2
			184	α + β</text>
			185	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUQwQUFBQXRDQVlBQUFBZHRHTnlBQUFBQkhOQ1NWUUlD
			186	QWdJZkFoa2lBQUFBbDFKUkVGVQphSUh0Mmp0b0ZFRWN4L0dQU1JBakJud2hDR3JRUUVRUkh5QWlH
			187	TkJHRUxGSTZTTUlpaStpVGFJV2dzaUpZR09UUXRCR0VNRXlWVnA3CkJVRlJSREVnU01vZ0VoUWx2
			188	cUxGYk9KcVl1Njh2ZDBwc2w5WWJtWnZtTjkvYm5iL08vUGJZdzR5TDZMMmVUUmhBYTVGaktNd2Rt
			189	TkgKVXI2RG5VV0tOeFVwbG1JdHVwUHlHM1JHaXFOUVdyQTRLUThxZU5DeFp2bzd4ckFWcnpFY0tZ
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			196	YnNvclg2cEhGY0RUWDRLVXd5TVBvd1FNOHpVbnZEd1p3S1ZVZkZPNnhRMWhlWjUrVkd0cDArNzNx
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			198	WjRYN0tNMEZ2TUxTRFAxV2FtaHo2cS82ZGp6TQpvRGxGdGVkY0xFZHprNUNwSnhObkd5N2pXQ002
			199	cjNaNUR5VUI5T0tEY0xsZHhFMGhzOTV0UkJBenNBSG5oSVE1aGtYQ3pJL21wRmNJClI2cDgzNU9u
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			201	bUNDCg==
			202	</png>
			203	<latex>$$\frac{\alpha^{2}}{\alpha^{2} + \beta^{2}}$$</latex>
			204	</output>
			205	<output>
			206	<output_type>display_data</output_type>
			207	<text>❘11⟩</text>
			208	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUI4QUFBQVRDQVlBQUFCb2JOWkNBQUFBQkhOQ1NWUUlD
			209	QWdJZkFoa2lBQUFBT1pKUkVGVQpTSW50MVRGT0FrRVV4dkZmaUxFa0NJa0ZGa0RDSlRUaEFCcHFD
			210	aXF1WUFXRnR6SGhCR0JqYTB3czdLeXNqRDJ4a1JZb2RvMncyZDNaClJpY3hmczI4dlBsZS9tOG1r
			211	emY4Y2MyS05tcDc4UmhubWYwNm50QUxBTXA4YTV5R09weWdtOGJIR09BQlcvUUxhcXI0V3JqT0t6
			212	N0sKeVRWd2kxZXNTcHF0Nmx2aFJITExteUxUeFBmSnYzU2ovT1JWZlJlNHlpWnJPY2FmMEdQYVFC
			213	UTR2S01UQzc3QU1CWjhpR1VzZUJkdgpNZURua2tkM29OK0NYK0l1QnJ5SkR6a0RKZ1J2cG10b05w
			214	ZjV4cGdINm8zUVR1Tjd2RWltMWhhZmVNWTBVMVBGVi9pci9TdUtkcWlvCkswNXl6bzdrQUFBQUFF
			215	bEZUa1N1UW1DQwo=
			216	</png>
			217	<latex>$${\left\|11\right\rangle }$$</latex>
			218	</output>
			219	<output>
			220	<output_type>display_data</output_type>
			221	<text>2
			222	β
			223	───────
			224	2 2
			225	α + β</text>
			226	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUQwQUFBQXdDQVlBQUFDaS9ISTNBQUFBQkhOQ1NWUUlD
			227	QWdJZkFoa2lBQUFBbmhKUkVGVQphSUh0MlUySWpWRWN4L0hQdkJTRENYZVkxQlJwUkFsWkVRMU5a
			228	bVpoZzBqWlNJbXNsR1JoeXN0a0o3c3BaU1VyeW9vc0xjWktGaExHClpoWlRTbG1SdDd3c0dCYm5t
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			230	ek0KeWxmUkcxTzhPYVpZR2F1eEx5dVBZMjJpT0tMU2dxVlorU1kyeGhSUE5kTS84QTdyOFJxak1j
			231	VmJZNHFoRy8zNGh2bm93dG5JTVVSbApPd2JLNnAvOVR1czlNUU9KbGQ0bExNZjlySDRBRXhqQlcz
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			236	Qkk5azFsdTVmT1k1RldYa20zdlZNTXF6U0w3K0xrM1hxNXFaUGNEL2VDTjUxCi93eWVQVmRqdnhM
			237	Mmx0V1hDT244czA3ZEdkT0V4YlAwcnFFYSt6WEVMNi9sbTJqR0JYd1FabmFIa05adFdDWnNHNDFp
			238	S3IvOGVaNlgKMWpMb0c0SjdPWnpWQjRSQmY4UzlQT0pWYUJkU0dDNW4xMEZzenZ2aWFvUGVJQmox
			239	Ujh2YVJyRU5qL0MrUnAzVGdrazRTWS93cjJVNQoxL0N5b3M5SVJaOG9mdmxKM0tsb080TVhmcStl
			240	OVRCVVE1K0crZVhWWm5yTW42dGpTVGdVakFtSGhEWjh6UnZFRkRUTUwyK3BjbjljCjJDYTJZSjNn
			241	WFE4TGk5bGtxazNVb2R1TEI5UGM3OFFDWEJjRytRcTMxZjQ1L1pNY3IzTC9uL1RMRzgyeDFBR2tv
			242	UERMQ3dvS0NxYmoKRndudWJ2aG5lQTVsQUFBQUFFbEZUa1N1UW1DQwo=
			243	</png>
			244	<latex>$$\frac{\beta^{2}}{\alpha^{2} + \beta^{2}}$$</latex>
			245	</output>
			246	</outputs>
			247	</codecell>
			248	<markdowncell>
			249	<source>Qubits can be represented in the computational basis.</source>
			250	</markdowncell>
			251	<codecell>
			252	<input>represent(psi)
			253	</input>
			254	<language>python</language>
			255	<prompt_number>8</prompt_number>
			256	<collapsed>0</collapsed>
			257	<outputs>
			258	<output>
			259	<output_type>pyout</output_type>
			260	<text>⎡α⎤
			261	⎢ ⎥
			262	⎢0⎥
			263	⎢ ⎥
			264	⎢0⎥
			265	⎢ ⎥
			266	⎣β⎦</text>
			267	<latex>$$\left(\begin{smallmatrix}\alpha\\0\\0\\\beta\end{smallmatrix}\right)$$</latex>
			268	<prompt_number>8</prompt_number>
			269	</output>
			270	</outputs>
			271	</codecell>
			272	<markdowncell>
			273	<source><h2>Gates</h2></source>
			274	</markdowncell>
			275	<markdowncell>
			276	<source>Gate objects are the operators which act on a quantum state.</source>
			277	</markdowncell>
			278	<codecell>
			279	<input>g = X(0); g
			280	</input>
			281	<language>python</language>
			282	<prompt_number>9</prompt_number>
			283	<collapsed>0</collapsed>
			284	<outputs>
			285	<output>
			286	<output_type>pyout</output_type>
			287	<text>X
			288	0</text>
			289	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUJrQUFBQVlDQVlBQUFBUHRWYkdBQUFBQkhOQ1NWUUlD
			290	QWdJZkFoa2lBQUFBWHhKUkVGVQpTSW5sMVQ5SVZtRVVCdkRmOXlsOGlKTkRFWWhHU0tCZ0lEUUlp
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			292	UjN1KytITDdjTFYydzBISDdod3puUE8renp2djNzdlo0Z0ZiT013UE5zWWllcXpVVzBQNzFBcFl0
			293	U0kKelNEVWthcFZzSVZIYUNnaUhtTXFtRHhPOFRPNC9hL2lkVnpBUHI2aEdyaUhHRDJOU041Uzk5
			294	Q05QbnhFUDJxWU80M0pTVEFnMmJJTgpQQ2xiUE1aNk1McGNaSEExdjBVbkRrSThYc1FrRDYxWVFw
			295	dmtmSDVJcm5acGFNRXJ5UTJEZWNtV0RaVmwwSVJGWEltNDNtRHlOcU8vCmdqSGN4elR1NUJrMDRp
			296	VjZNbXBya3ZPNW11S0g4VHpFVlh6Qk5SRVJvd1B2OFJXZk1rd1d3cXduVS93VVhvZjRRTExhdStu
			297	QnR4eC8KcHc2eGl4dXBuaGY0RS9WOHhrU28vY2IxcVBjQjNtUk1zakJxd2JRcjR1N2hRejA1eVh1
			298	U2gzM3NCS002bXZHclRCTll4Y1VvdjRUdgpaWnVzWURES2IrSlpQU24wTjh0QVV4QmRSanQrNG1s
			299	SjJuK2g5citFenhHT0FHdERTUzdLeG84WkFBQUFBRWxGVGtTdVFtQ0MK
			300	</png>
			301	<latex>$$X_{0}$$</latex>
			302	<prompt_number>9</prompt_number>
			303	</output>
			304	</outputs>
			305	</codecell>
			306	<codecell>
			307	<input>represent(g, nqubits=2)
			308	</input>
			309	<language>python</language>
			310	<prompt_number>10</prompt_number>
			311	<collapsed>0</collapsed>
			312	<outputs>
			313	<output>
			314	<output_type>pyout</output_type>
			315	<text>⎡0 1 0 0⎤
			316	⎢ ⎥
			317	⎢1 0 0 0⎥
			318	⎢ ⎥
			319	⎢0 0 0 1⎥
			320	⎢ ⎥
			321	⎣0 0 1 0⎦</text>
			322	<latex>$$\left(\begin{smallmatrix}0 & 1 & 0 & 0\\1 & 0 & 0 & 0\\0 & 0 & 0 & 1\\0 & 0 & 1 & 0\end{smallmatrix}\right)$$</latex>
			323	<prompt_number>10</prompt_number>
			324	</output>
			325	</outputs>
			326	</codecell>
			327	<codecell>
			328	<input>c = H(0)*Qubit('00'); c
			329	</input>
			330	<language>python</language>
			331	<prompt_number>11</prompt_number>
			332	<collapsed>0</collapsed>
			333	<outputs>
			334	<output>
			335	<output_type>pyout</output_type>
			336	<text>H ⋅❘00⟩
			337	0</text>
			338	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQURZQUFBQVpDQVlBQUFCNnY5MCtBQUFBQkhOQ1NWUUlD
			339	QWdJZkFoa2lBQUFBcmxKUkVGVQpXSVh0MWp1SWxVY1VCL0RmK21EQkY3NlNySVdLa0NEWUNFWkJX
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			356	QUFBU1VWT1JLNUNZSUk9Cg==
			357	</png>
			358	<latex>$$H_{0} {\left\|00\right\rangle }$$</latex>
			359	<prompt_number>11</prompt_number>
			360	</output>
			361	</outputs>
			362	</codecell>
			363	<codecell>
			364	<input>qapply(c)
			365	</input>
			366	<language>python</language>
			367	<prompt_number>12</prompt_number>
			368	<collapsed>0</collapsed>
			369	<outputs>
			370	<output>
			371	<output_type>pyout</output_type>
			372	<text>⎽⎽⎽ ⎽⎽⎽
			373	╲╱ 2 ⋅❘00⟩ ╲╱ 2 ⋅❘01⟩
			374	────────── + ──────────
			375	2 2</text>
			376	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUx3QUFBQWtDQVlBQUFBemZGQ0ZBQUFBQkhOQ1NWUUlD
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			402	dVhDeldtVk9pMDh6akl0eUtWMHF1M1l4L2lKRzF6WHJzaDdzSHNEc1JmNmxROTZYQU1OclFqRDVa
			403	CmRtajlGb1ZUQjlabkgvSEUzNG5iRlU4dFZWa2wzb0dvOG9uWVVTS2kwQzNxc0NabmYyTHJ5REly
			404	MXFvUGlGM0psOFE2TlU4L2RyY1oKL0h2TVVZM3drNklOemVyekhlRVF2OURLK3pQeGt0blB4YTR3
			405	WGZRcEUyeGFlTzgvYXQzTURhMEtyQlBycW1INGlwaXUrbVVXcTN1YwovN3VvWTV2MlUvOUpnZTNP
			406	WXVUWXF2ZUg0cjNzZGhjZFl4RG14VnA1R09kMGtyU2hXWDNhL2s0UlcxcDVEcVRQV3Z6YjlvWC9s
			407	MXNWCitGYlZqQklkYkJUZm9nNUQwcVlHWnNRNzBPM1hNMWVLUmwwL3Rob3REMGJ4dG1UU3BnRTI0
			408	V21UK2FYVTU1MU5rallqNVdqOHlYYW4KSURFNUpHMUd6R3I4VEh4c3NBSmZIRzkxRWhtU05pUG1B
			409	QkVpYWtkMGpzRzN4MWVkUklha1RVWEt3cEpmRXQ4WVpnUDdlK0p3dkZWWApwUko5a2JRWmdMSU92
			410	NGZGR3dZTElxYWFHQzlKbTBRaWtVZ2tFb2xFSXJIcytUL2lSYi9rVlg4UDZnQUFBQUJKUlU1RXJr
			411	SmdnZz09Cg==
			412	</png>
			413	<latex>$$\frac{1}{2} \sqrt{2} {\left\|00\right\rangle } + \frac{1}{2} \sqrt{2} {\left\|01\right\rangle }$$</latex>
			414	<prompt_number>12</prompt_number>
			415	</output>
			416	</outputs>
			417	</codecell>
			418	<codecell>
			419	<input>for gate in [H,X,Y,Z,S,T]:
			420	for state in [Qubit('0'),Qubit('1')]:
			421	lhs = gate(0)*state
			422	rhs = qapply(lhs)
			423	display(Eq(lhs,rhs))</input>
			424	<language>python</language>
			425	<prompt_number>13</prompt_number>
			426	<collapsed>0</collapsed>
			427	<outputs>
			428	<output>
			429	<output_type>display_data</output_type>
			430	<text>⎽⎽⎽ ⎽⎽⎽
			431	╲╱ 2 ⋅❘0⟩ ╲╱ 2 ⋅❘1⟩
			432	H ⋅❘0⟩ = ───────── + ─────────
			433	0 2 2</text>
			434	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQU9nQUFBQWtDQVlBQUFDRGczYTNBQUFBQkhOQ1NWUUlD
			435	QWdJZkFoa2lBQUFCdDVKUkVGVQplSnp0blZtTUZFVVl4Myt6RndSWERsZWppQWdvYWhSY0VnZ0NM
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			439	QTZNQy9tc3BjQUo4ZGNabG93dXRzdwpJNmllTGtCN1lGL001ZllCZGdCN1lpNDNMUmpkRFlHWVRy
			440	dzkrU3RBMXhqTFN5dEdkOHdJbWpaNkFvZUFiVWxYcE14SXJlNXBicUNqCmtQbElPZkVBTURQaE9o
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			471	SWwxQ0VMM0VIbUxvUFE3MHZkam43TFpSSzZOeUZXV0ZoS1NmY0s5UHRyZHlPLwo0QUJGZnRhcmdI
			472	K0FNMnhoVTRCM2lsVmdUQ3dqZHdScUJENU5xQzZ0aVpidXhUVzYyOUN0Z3diaENPSWg3SWowS2lC
			473	dk5PelFwbWdkClBFZXVLNzR0N2swT0JqYzN0VEM5MGIwSU5KTHJXbDlQeTh5Y3RGR0htUEdYSkYy
			474	Uk1zUG9IaEU5a0wyUC9aSDV3cHhrcXhNcEZZalgKK29ha0sxSm1HTjBqcGdiWnFCRFZDN3RwWVJw
			475	d3NUcnZrR1JGeWd5anU4R1g4ZVQrSE1palNWV2t6REM2SzZKd0VwVXE5eUF2T1Z0Ygp1ekxBdThs
			476	VnAyd3d1dHN3LzBOQ3o0VzRUYXN0Tkh1cURjWEI2RzR3R0F3R2c4RmdNQmdNQmtNQy9BZU1RQWFp
			477	cVVraThRQUFBQUJKClJVNUVya0pnZ2c9PQo=
			478	</png>
			479	<latex>$$H_{0} {\left\|0\right\rangle } = \frac{1}{2} \sqrt{2} {\left\|0\right\rangle } + \frac{1}{2} \sqrt{2} {\left\|1\right\rangle }$$</latex>
			480	</output>
			481	<output>
			482	<output_type>display_data</output_type>
			483	<text>⎽⎽⎽ ⎽⎽⎽
			484	╲╱ 2 ⋅❘0⟩ ╲╱ 2 ⋅❘1⟩
			485	H ⋅❘1⟩ = ───────── - ─────────
			486	0 2 2</text>
			487	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQU9NQUFBQWtDQVlBQUFCN1k0MUFBQUFBQkhOQ1NWUUlD
			488	QWdJZkFoa2lBQUFCc0ZKUkVGVQplSnp0bldtTUZFVVV4MzhEeTRLQ0lPQUJ3WFZCTVNZclFnSkJS
			489	UEFBVlB4Z1NNQk5sdkFCcncvaUZjVUxOUW9EWWdqeENvS3dSc1NiClF4Q01Sc1hFSXh3YW80S0F3
			490	VWlpaUM0b1JsQXhvaEtROWNPcmRudHF1bnE2ZDN1NmU1cjZKWk9lcWVxdXF2MTNWZldyVjI5bXdX
			491	S3gKV0N3V2k2VVM2QTgwUi9DeW1FbVZ4dTJpS2lqam5CMXpmU09BcTRGY0JLOUt3V3BzTWRJUk9C
			492	OVlDaXlPdWU2VndFa3gxNWtFVm1NWAo5c2xvcGcvUUZmZ3Q1bm9IQVh1QWZUSFhtd1JXWTBzb1po
			493	UHZyUDBpVUJOamZXbkFhb3g5TXFhTi9zQWhvQ25waG1TWTFHcWM1c0hZCmdLd3BqaVh1Qmg1TnVB
			494	MVoxejIxR3BzRzR4cmdNMkNMZW0wQ25sRjV0Y0NIS3MzSi94UzQyYWZ5eVQ1NTQ0QXBIdWxiZ1hx
			495	ZjY3SkcKSDZBenNNT1FYd1hjQ3J3S0xBSW1BaDFLbE9tbGV3MHdENWdMM0E5Y3FPVm5XZmU0Tkhh
			496	SXJHOFBRL1pRbHVNOWFPOVIrYmNIS0N2dgprZFlldUFveEdXWWJyak9seDBsYzY1bkhnY0dHdkdy
			497	Z1hlQUd4T0V4Q3ZnT1dJdDBMaE41N2ZPcGlOUGlVdlc1Ry9BMWNoL2N4SzE3CmxqU0dWdlp0UHpO
			498	MWlEcXVCSTU2NUE5ejVZZGxKdkErTXZOVSs1eTNGUmpZaXZLanBKcnltMjA5RVl0anN5SC9EbUFk
			499	OERUd0IyS1oKVEFiR0VzN2ttb3FzbGQ1VG53OEF6d0ZMZ0ZOYzU4V3RlNVkwTGt2ZlhvNE1RcSs5
			500	bUJ6d0MvQk53TEx5aHZUVGtLZXJhZmJvQU13SQpXRWZVOUFYbUFLdUExWWhaTjZCTWRjMENMdmJK
			501	WHcrc1FHWnNOL3VRMmR0RTN2VytHekw0NW1qbkRFTHVnZHZjaWt2M3ZtUkxZemVSCjl1MGZnVzJH
			502	dklHcW9pVUJ5OG9iMGtzMUdPQmU0SVNBOWFTUi9zQUd6S1pPRitDdEVtVjhoT2gwa1phK1JhVjNN
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			517	M0laMHNqNnV0Rks2cDVta05YWUkzYmYxbVM0SFhJSU1OcThuV3czUWorSzF4SFRnRWNRYzJJdk1J
			518	cE9RCkdFZy8ycXVqMTQzdmdqaGIvT3oxWmlyck8ybU53SDJJWjY4dXhIWGRnZGNSazhZZDA5Z0I2
			519	VUJCblJNTGdPZVJUcm9mdWQ4VEVOTnUKanpvbmlPNXBKbW1OSFZyZHQzc0RUd0p2cTBvUEFFOGgz
			520	bFFRMDNRUkVrclVqRGgzNWlIUkc2aUMzWC80UTBpd3JjTjByYjU2SlBEOApLMVhlSVNUaTV3bGE5
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			522	VlJWbkhyCm5qV05XOTIzblVkdk5kNnp5RjcxNnU1MU1SSU9WNDNNc3JXMGVGK25JYWJyR1BYNWVH
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			524	N3V4aHlXcU92dXBoYTViL3BtZHB5NloxSGpSUHQyRmZBM1lpTTd6RUJtM0VwbQpEWVd6WGlQd1FV
			525	SnR5U3BXWXhkQnZXTitIQUcrUkJhOXUxVmFEMXJXSUpYS0FncGQ0cDBvRGhpd3RBMnJjUmxvcERB
			526	U1podmlDTW9LClBSR1RUdjhpcmlVNnJNWVIwUStKRHh5SzJQd0xrMjFPcExSRHZNY1RrbTVJaHJF
			527	YVIwdzFzdW5mSSttR1JNd3M0QUwxdmx1U0Rja3cKVm1OTFNhWlErRXZYRHliVmtBeGpOVlpFNGNE
			528	SktsT1JIMnh5UXFKeXdKdkpOU2VUV0kxZDJQOFRZT1pjaWsybVhiUjRqQzF0eDJwcwpzVmdzRm92
			529	RllyRllMQlpMQVA0RE5NcmxDQ0JJTDB3QUFBQUFTVVZPUks1Q1lJST0K
			530	</png>
			531	<latex>$$H_{0} {\left\|1\right\rangle } = \frac{1}{2} \sqrt{2} {\left\|0\right\rangle } - \frac{1}{2} \sqrt{2} {\left\|1\right\rangle }$$</latex>
			532	</output>
			533	<output>
			534	<output_type>display_data</output_type>
			535	<text>X ⋅❘0⟩ = ❘1⟩
			536	0</text>
			537	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUZRQUFBQVpDQVlBQUFDdnJRbHpBQUFBQkhOQ1NWUUlD
			538	QWdJZkFoa2lBQUFBMEJKUkVGVQphSUh0MkV1SUhFVWN4L0hQSnBGbFNReUtVYklzNmlGS0RCandn
			539	WmZnUVR3b0VSVXhxUGhXVm9rbUo5RUVFcERNUVJSZkNBcWlvZ2pCCkhCUkJEZDRVRkZGUXdlZmlR
			540	UTFrVFFJUlJRK2FSRmJVemFHcTJlNmFucDdwN1ZsM2RlY0xRM1g5cS9wZnYvbFBQZjQxREZnMDNJ
			541	S3gKK1JaUlFTMTllM0FJMC9GektEckllRGJYZGd6dlk2akMzL1lTMjQxNERRL2pVZHlRdEsvRXRs
			542	NEZONlJNWDhZbUlSNHB0ZlV0d3o0aAphR3VTdGlIc3gwNHM3Y0ZYSzZsZmljTTRKU2R1RXRlVXZM
			543	ZXNSNzFOYUpYWXhyQUZmK0wxaXZjSytwWlVEUElYWG9qUE55VnRUMkNyCk1MditycFJhems3c3hT
			544	K3gvaHZld3NzWXp2WGJpNnRtNGI4cGQrTXhyTUEvRmYxcTZ6c1ZVL2pCVFBCMzRMYWFBbHU1NTR1
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			556	V3hiOGVsY0RMYlE2ZVdtMUkwcElkV1p6dG1XNDljKytQN1AwWStBd3VjNExWZGZMWnltaTQ1K0Jm
			557	UVR4WCtKTGhQU3JFVkgKbmZ5dGloRWhnQjhKLzB6OWhLZjc1SHRSTTl5OXk0QUJBd2I4YnpnT0px
			558	S2orRWxZcEh3QUFBQUFTVVZPUks1Q1lJST0K
			559	</png>
			560	<latex>$$X_{0} {\left\|0\right\rangle } = {\left\|1\right\rangle }$$</latex>
			561	</output>
			562	<output>
			563	<output_type>display_data</output_type>
			564	<text>X ⋅❘1⟩ = ❘0⟩
			565	0</text>
			566	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUZRQUFBQVpDQVlBQUFDdnJRbHpBQUFBQkhOQ1NWUUlD
			567	QWdJZkFoa2lBQUFBMEpKUkVGVQphSUh0MkZ1SVZWVVlCL0RmbUhFWTFDRURRNUdpc0RJaElRdGZl
			568	b29ldWtIUlBlZ09KbWsrUldVb1JlY2hJc2pBTGtSRkVSaENRUmVJCjNncnNvYUNDcnZOa0dwRUpS
			569	VUZCWlRWZ016MThhNXA5bHZ2c3M4L0ZJelhuRDRlMTEvOWI2MXYvK2ViNzFsNXJNOEs4d2MxWWVh
			570	eEYKVktBcmZidHhFRFBwZHpBNW1NVXpCZHNmZUE5akZmNjJWdGcyWVdjSlA0SDc2Z3J1RTJYNkxz
			571	SHJlRFQ5Tm1iMnJ2VXR4SDRSdEZXWgpiUXpmWUR1T3ErR3JXY0tkaG0yWXh0TVY4eGJXOE44dm1s
			572	bC9QWDdGS2FuZndCZTRzMlJlaTc0RkZZc2N4dlBwK2NiTXRnTmI4QWorCnJpRTR4LzNtL29pcXpI
			573	NExsL2ZndjE5c3h4NGNTUDBwdklJbmNGSmhYTmY2bGlWbjM1b0wvamJjMnFYQVpodCtzYWlBZGhr
			574	S0QzZTUKVmk5b0ZwN1BGRlZ6VnpabW5kQ2FKMWVMdnFvTWhaL3dwa2o5eTdCQlpPNnVydVQyaC8w
			575	NGZZanJyUk5WczY5RUIxeFF3ditycjFOQQo0Ym5VUG80MWVLeHJpZjNoVmR3d3hQV1dwL2IzakQr
			576	VTJoVVozNkt2em9hL0IzdXhHay8xSUxCZi9JbEZiV3hyUmJWVTdjTTVEdUNLCkN2dHNRQTlsL0RU
			577	K3dnbFYrdW9FOUt6a0RPN0FnelhtREJMalFuUVpKa1dKRGhKVHFaMHVzUjFmd3JmbzYxVHlLOFdt
			578	ZTNHYXRNRncKampGRlhDL0thbGlZYk1NM3hCSHgrNHh2MFZjVjBLV2l4RGZqdXpScEJhN3NWV21Q
			579	V0kydmhyamVsNmxkbXZHTFUvdER4cmZvYTVkdAo0K0psZEk5NDA4T3p1RjNjYkY3THhvL2hOaXdS
			580	V2YwKzNxNmp2Z1BPeFNjVjlyUHhrdTczMEtzcjdGK0wvZlBrakYrZjJtSUdkOUtICkNQSXVuRk5p
			581	KzB6c0lXZGsvSFY0SVQwdkVQL2x0UVY3czgxYW5jNmhEemsyTjZVUDhYTEc3UkRsM2lod1IrakxT
			582	MzZWeUs1SmZGNnkKOEc2UkRmbWhkNnU0TlJBQmZ4ZTN0Rk5md0ltcFhWWmlXeUsrRXh5dTRXZlFl
			583	QUJYNGRUVW44QTE0dWc0KzlLcTFIZXB1WHY3akNqego4N014TDRwU21CMVR2TnYraHZNS1krL0ZP
			584	NFgrM1ptdmpmZ1l2eFQ4N1JNZldTYlNtSnNjV1haSEM3ayt1RlpjYW5iaURYRmRMaWJnClVkUFhF
			585	QUZaVStDMmlJRE5POVM1S1hYQ2xOaGJaZ3JjSXZ3OEFOLy9PUXdpb1BDcDFxOHd5ODE5cVpsWEdG
			586	UkFQOEtGaGY1RjRwZzEKNzlETithMEs0eUtBSDRndlV6L2l5UUg1bnRkb2RCNHl3Z2dqalBDL3dU
			587	K1VES1FiK0JQTG9RQUFBQUJKUlU1RXJrSmdnZz09Cg==
			588	</png>
			589	<latex>$$X_{0} {\left\|1\right\rangle } = {\left\|0\right\rangle }$$</latex>
			590	</output>
			591	<output>
			592	<output_type>display_data</output_type>
			593	<text>Y ⋅❘0⟩ = ⅈ⋅❘1⟩
			594	0</text>
			595	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUZZQUFBQVpDQVlBQUFDcldObE9BQUFBQkhOQ1NWUUlD
			596	QWdJZkFoa2lBQUFBM2xKUkVGVQphSUh0MkYyTVhWTVVCL0JmUDhiWDBLSlN4VFRVay9CRVMrTEJF
			597	RjhSU1ltRXBobEUwSGdZUElnRTR5TnVoQ1lpOFNSQmVKS1FJVkdDCkJQV1ZOaUpGcEZJSlF0T0k3
			598	NGVXRkJNUlREMnNmVE5uOWoxbjdqMjNkMXJwM1AvTFB2dS8xOW43ZjlaWlorMjlEbjBjMEZpSEpm
			599	dGIKUkJkb3Ezc1ZQc0RmMklNSmJNSmdabmNzZmtrMms5aUc0ektiTzBybXZ4N2plQWdQNDRwc2ZD
			600	bHViZk1RczQweTNYQW8zc1hwSldNZAo2NzVXT08yWk5qWmJNRlF4M3NqNmEvRU5GcVgrRXZ5SWl6
			601	SzdCekN2RTVHemhFYldIOERaMkNoOHNxcml2bExkODdQK0JoR3RsK0hnCmtrbFdZRFhPeGZjZGlK
			602	MkhNYnlJM3hLM0M2K0xsMWRjLzIydHp0NWZHQkMrV0lOZjI5aDJyUHRaOFlhdXpQaEZlQWxIdDdt
			603	L1ViaSsKSk0xMWNXWnpWZUxQeXZnSE94RTRTMmhVOExlWU9XSXAwWjFITE9GWXVMckFMY0FUdUZQ
			604	azJFNnhNclZmWi96MjFKNlg4VC9naEJyegovMS9Rb3J2TXNSdXhFNWZpcU1ROWdxZndWYzBGbDZY
			605	Mmo0eWZTRzIrOFQySGtacHIxTVdJQ0pKeDVjL2ZEVnAwTHl3eCtnY3ZZRlNrCmc0UHdPZDdyWXNH
			606	bVl5Y3l2dGsvTXVOM20zcVpPVmJpYWZVMnVPMm1wN1RWNHVSenM4ajVGK0N0R3ZOVm9VVjNtV09K
			607	ZERDS3U4V2IKSGV0eXdiOVNPNW54QXhYODRpU3lESjhvUC9MVXdlWGk3SGtPRHNOM2V6bGZFeTI2
			608	cXo2RkxjSXB1M0RQWGl6NFdRWGZQQi8vbFBFago0ck9hTGF4UDdZM2lHYi9zMGJ3dHVxc2NlNm80
			609	YnIyaE5hcnFZRnRxODgvNzhOVCtuUEZEZWhkRlpkZ2hvbXVOMkRONmhSYmRWYWxnCk9MV2JTc2Jt
			610	aTBycUVDd1h1ZmZOaW5tYWpsMXVlblNlbWRwaVJBOWpjOFU4Y0FhZVZEL0hyczI0YThRKzhyeHc4
			611	a0x4WlhhTGRycW4KWVZ3aytieWtKU3F2eDlMMUFueUJVd3JqamN4K0orN1B1SmZ4VWNidHE4cnJZ
			612	MVBSZXArcERiWlJZZC91SE50UjVVVTRheGlmYXQzTgppWnI2bFhUOXI0alltWTVJZCtFbUVSMXdv
			613	dGlOMXhkc2poRlJzMmVHZVhxRklieUQ0OFVtbXFlakhNMkNhR25KV0VlNlR4SWh2U01aCi9vNzN4
			614	YWZUeER5eHFaMVc0TzdGcTRYK2JTVnpqNG9vZlJTdjRicHNmRjBTdVM5d2d3aU1NUnhSNEhQZEcw
			615	UXFteFQrK0JOYlRhK3kKZXFaN2NWcGtSWUc3WFkwY00xZFF0L0xZYmVxM1lST0Q2cFc1Y3dMZGxI
			616	UmJUYzgzeS9CdGIrUWNPT2pHc1IvaS9FTC9RbkVNNnFPQQpibzQzZytJbnhtYWNMS0wxOFY2S211
			617	c28reEhlUng5OTlOSEhGUDRENzBpdXlrYnJ4SzhBQUFBQVNVVk9SSzVDWUlJPQo=
			618	</png>
			619	<latex>$$Y_{0} {\left\|0\right\rangle } = \mathbf{\imath} {\left\|1\right\rangle }$$</latex>
			620	</output>
			621	<output>
			622	<output_type>display_data</output_type>
			623	<text>Y ⋅❘1⟩ = -ⅈ⋅❘0⟩
			624	0</text>
			625	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUdVQUFBQVpDQVlBQUFBb25PQjFBQUFBQkhOQ1NWUUlD
			626	QWdJZkFoa2lBQUFBNTFKUkVGVQphSUh0MlZ1TW5WTVV3UEhmdExSaVJJc2FJNllvSGx5ZWxFWjRH
			627	RUs5RUgxd1QxMFMwZlNoY1luVUpVcmlSQWpDdzBpYXFKQklKS1FSCklVSGlFaUp0aElvS1dpR3FS
			628	RnhDQk1tRUVkZU9oN1dQT1dlZjd6dVhtVG50VEhyK3lXUi8zL3JXdDcrMTlqcDc3YlgzMEtOSENh
			629	dHcKeUo0MllnL1ExTzlUOFRiK3hqakdzQW45bWQ1aCtDWHA3TUkySEo3cDNOckVpTFc0dDBBK2dP
			630	dWJ2RGNiS1BMN2ZEeUwrM0EvcnMyZQp0K1gzVldMQW4yeWhzd1ZESmM4ckJiTGprbnc4R1ZmRTNl
			631	aHJaZUFNcHBMZG40NVJFK08wSDdackRFeUQzM015aGVmRUxGbUIrUVVmClhvSUxjQ2ErYmRQWXUz
			632	Q0hDRWd6WHNlNWJmWTVHMWduZktxTzB4OTRCdXZWcDZ5Mi9INUtET0RGbWZ4QVBJK0RXN3hmS1pF
			633	UGFqNVQKNEo1V3hzMWdLalhYSjRqMHZqclRXU2JHNE5KTVh1ZDNQbE9Jb01BVk5iSzUySURieEpy
			634	U0xiN0RFVjNzZjNleFZLU2t6elA1enRTZQpsY25yL0M0S3ltdjRDZWZob0NSN0VJOWh4OVJzYmNu
			635	VFdObmxiMHlWbzhUQy9STE9LZEVaVE8xdm1Yd3N0WGx4Vk9mM1BnVWQvaU55CjN4cVJ3dWJoRTd6
			636	WnJ0VlRZTlRFRHlIbkZEeXVzMkpncDhZMFBCWDZSTGE0T2JYcjhFYUJYalVvWTVuOEx6RytDek41
			637	bmQ5RlFTRlMKMkpyMDBZMjR2UVBEcDhJQ1lXQVI3K1BrM1dSSEdTdndNbjdIY28zcHFjcWZxZDJW
			638	eWZ2RVVwREw2L3d1QzhxVzFQSFBvbkxhWGF3VQpVM21tOGhHK3h2RTRRL3hvaTloZUl0OWZCT2I3
			639	VEY3bmQxbFFUaFFsOFNzYW85cE5odkJObDcveE1FN3JRUDg2YkUzWFg2VjJ0VmhmCk41VzhzeTIx
			640	ZVNvK0lMVS9aUEk2djh1Q01wemFvby9Pd1RWaU03UllyRFd2bHZUVENjUFkzT1Q1VWp5cTh6WGw4
			641	a3gyWTRkMjVjekgKMVhnZzNSK0xMektkSFdKZnNoanYxTWlYcGJaMkpyWHkrMzgyaWlPWC9KaUYy
			642	Tkd2VDlkejhhbVl6bFVxSlgyMjJxZk1saDM5UldLeApIc0FpRS81V01yMnRlQ0tUallqTjVMd2FX
			643	Y3NkUFRIUXcvaFFZL1ZBblBHOGtLNy9GVE9sblRLMnV1a2NLSGkyU0t4ZnJYYjlNNEVoCmZJd2Z4
			644	YXpiVUtKM0p5N0JrZWwrSVM3RVE2SUtvdzIvanhiVDZNdWs5Q3Zld3BVMU9uMmlBRGdwKy9pTE5m
			645	YzNaZjNlZ1BkRWRUR2UKL2o0VHdkdzM2YXhLQnM0R0RoVjd1Ukd4MkZmSi9ZYkx4Q25JU0dwdlVU
			646	OHJwc1h2QldKUWw5VEkxbW96Si9ab2o2TDAxWXhSRTBmMwpWZnAxOStobHI2UFRvTUFINnRlRlFW
			647	Rzc5NWdtSmhPVWQzRjJ6ZjF5VWFyMm1DWW1VNEwyaTRwak00NFJzK1NSNlRTcXgrUXAraWRZCmp4
			648	NDlldlRZZS9rUDB5SzFTZ2ZuL0lFQUFBQUFTVVZPUks1Q1lJST0K
			649	</png>
			650	<latex>$$Y_{0} {\left\|1\right\rangle } = - \mathbf{\imath} {\left\|0\right\rangle }$$</latex>
			651	</output>
			652	<output>
			653	<output_type>display_data</output_type>
			654	<text>Z ⋅❘0⟩ = ❘0⟩
			655	0</text>
			656	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUZFQUFBQVpDQVlBQUFCSmhNSTNBQUFBQkhOQ1NWUUlD
			657	QWdJZkFoa2lBQUFBdmxKUkVGVQpXSVh0MkYySVZrVWN4L0hQbzY1TFdwcFM2NVpGV1VGWGlSR2lK
			658	S1lJMVJJbFFsZW0rZGFiQzRJWFFua1JxRWdoWHFrWlhvYjJSa0ZRClJDRklFWFhwblZKR2htK3dy
			659	VkVwK0JZcTduWXhzM1dhem5OMnozT2VkVk9mTHp6TW1kK2M4OHlQbWYvTStjK2h4WFZITnlhTXRJ
			660	a0MKU3ZucndDWDA0eklPWXgvMnh0OGZzZTNaT3MrUHhyb2MvUlY4aURld0ZjOGs3Vk94ZXFnbUsz
			661	QlYvSzNFUmF4Rlc5SzJDRmV3bytENQpNWGc5MFZiZ0o0eVA5UTc4aW5uSmZadUhhcklDVjhYZkoz
			662	ZzZSNStGQy9nTW8wcVlISVVmaFJuTzhqNk9vNWJSbnNCalF6WGFJTVB1CmJ5eSt5ZEh2RTJabVA4
			663	YVZOTGxJV1A1cDU4OUhmWHBHcXhuK2FHeXF2N3hvNnZUZkdabU1ML0duRUtFWFNwcCtKSmFIRS8z
			664	bldNN1AKYVAzNERiZVg3S01LbGZ6bERlSUo0U1V5UURzK3hSUThKVVJqV1RwamVTN1J6OGZ5amtU
			665	L0FFc2I2S2RSS3ZrYk04aWYxL0FPWnFNTApQelRtOFcrVDV4TjlvSDVyb3YrdWZpVE94VnNsK3or
			666	QVpRWHRsZndOTm9odllqR1c0K3RCN2kzaVlpejdFcjJ0am42YmtFYmw4UjFtClZQQ1NSeVYvUlcv
			667	WWw3RWVHN0VubzkrRUJTVk5IcXlqRDZRVHZZbStCTytWN0tNS2xmelZHOFF1dkkzZDJKUzBMY2JF
			668	Y2g0ZGlPV2sKUkw4NWxpY3pXazFZS28zc3ZZMVN5Vi9lY3A2QmovRXRYc3BwWDRVbmhWQmZKV1Qv
			669	MC9DRi9OUW9hL0p1bk03b00yT1pqWVRIL2Z2RgpsaklIMnd2YTh6Z29IQ0RxMFV4LzdrSVB2cGNm
			670	YmMvaDgzamRqUzN4dWsxSUIrNk45VFFQcStFc1hrditiNit3eDJVWmlSTkwwL3hOCkVHYWtGL2Nr
			671	YlhjSysyTWZYb3phVVR5YXVXY1BYcTFqRXRiZ0dHNko5UWVFZkxNcjZhZTd5R1NUYUtxLzdITGVo
			672	b2VFU1B3b2FtTngKdjMrK1hGd1dJckZkaUxxZXpQTy80T0Y0M1ljelNWODdZMy92NGdnZXhBdkNi
			673	QSt3VVBnQU1OejhML3hORlRMM2JDNjNRVGpWM0hBVQpwVGhGbkJTT2dQMFpiVHhPVlhaMERkTG9J
			674	RjRSOXMrT2pOWXBIQmxibEdDN3NCa1QzbTRuaEM4OU54eTF3VytweTBUc3dsZkNKbnhJCk9HZTNh
			675	SUQya1RiUW9rV0xGazNpTDRkZnRMa09FUklrQUFBQUFFbEZUa1N1UW1DQwo=
			676	</png>
			677	<latex>$$Z_{0} {\left\|0\right\rangle } = {\left\|0\right\rangle }$$</latex>
			678	</output>
			679	<output>
			680	<output_type>display_data</output_type>
			681	<text>Z ⋅❘1⟩ = -❘1⟩
			682	0</text>
			683	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUdFQUFBQVpDQVlBQUFBaGQwQVBBQUFBQkhOQ1NWUUlD
			684	QWdJZkFoa2lBQUFBdDFKUkVGVQphSUh0MkYrb0ZWVVV4L0hQdWY2NVdHQWhlVFVOTktGOE5RZ0tS
			685	QjhrU2Fvbmc4Q1VRaUZUa0JBUzZTRkpFY1ZYa2FpbnBCQ0RJS2pJCkNDSU5SUkdFSGhMMHdSZEpp
			686	OEwrVUZDQkVxZUhOVmZIT1h1T1orNDU1ODZWNWd2RDJXZnQyWHV2Myt4WmU2ODlORFFVMklyWmRU
			687	c3gKSkNwckc4TjF0SEVEbC9BVnZzeXVYN082NTB2YVQ4UHJYZnAvRTdzUzlvWFlVc1hSbXVpbXI3
			688	SzJrWktPbmhVUGVUdnV3U05ZalRWNApGL2ZqRUQ0dWFkL0NySVQ5VWV6SFhpR2t5QStaczFPZGxM
			689	NEphNXRlTXNoejRpMy92R0IvQWtjeisvYmUvTDNKUGp5SXEzZTQ3eFJXCjRtVEYvdXRrNE5wbTRw
			690	dUVmUWwreGprUkhkMllMc0l5eFdJUlpidEw2bHZpYlpyS2xPbGJiQUxhVXN2UmZER3plZWJnQy93
			691	am91VHYKbmx5ZEdHMWN3OXdoamxFWFNXMnBTZmhlYk1MampPSVR6TU16SWhxR3pWRnNtSVJ4NnFC
			692	RFc5bWVNRTRMaC9HazJKUXZETWV2RG41UgpIZ2tyUkZKUWhlL3dVbDhlRFk0T2JYZWFoUDFZaDVk
			693	eGZFaE9wWGhBcE1FcFRtSFpKUG95YURxMGxhV29zQmx2aUUzbWc1eDlGbFlOCjJyTUM2M0ZreUdQ
			694	VVJZZTJza2hZZzdmeFB2WVU2dGJoOTRHN2RvdVdDTmRoN3ozdjRMRWU3MjNqVmJHczlVTlNXMm9T
			695	bHVFamtjdSsKa3FqZmhLY3hJeXRQdzhNNEpwM2FWbVcxMnhPRElzdHhzR0tmNTdHeFlOdGFzWTlC
			696	a05SV25JU0h4TU84Z3JYaWswV2VGMFVVL0NWRQpMQkpMMWd4Y3hGTzQzS2VqSzZTUC9lT2N4dU45
			697	amxFWFNXMzVQV0cyT0F1TWlGVDBqMXpkQXZHd2orRFR6TFlUbjJYbEd6aURGM3B3ClpFNzJPNWFv
			698	VzRBZmUraGpxdEszdHZmRTJuY1ZaN1ByV3pFWjdleTZMczRMbzluL1JibjJCL0JoVmg3QmE0WCtk
			699	NGpUOXArNS9pN2kKUk82ZUxlSzcxRlNucUs4V2JRdXpnZkw1N2xzaWtob3EwaTFGN2NaUDRoTkdP
			700	MmU3RjcvMTdkSC9rSWxPd3I4aVhjdXZmZlBGSjQrRwpTZVFndG1YbGxwaUFKZlc1Yy9mUzZxUHRm
			701	ZUxBOHpXV2lvM284Q0NjYXFqT2FOME9ORFEwTkRRTWhQOEE1dHlRaEROTkdtY0FBQUFBClNVVk9S
			702	SzVDWUlJPQo=
			703	</png>
			704	<latex>$$Z_{0} {\left\|1\right\rangle } = - {\left\|1\right\rangle }$$</latex>
			705	</output>
			706	<output>
			707	<output_type>display_data</output_type>
			708	<text>S ⋅❘0⟩ = ❘0⟩
			709	0</text>
			710	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUZBQUFBQVpDQVlBQUFDbVJxa0pBQUFBQkhOQ1NWUUlD
			711	QWdJZkFoa2lBQUFBeEZKUkVGVQpXSVh0MkZ1SVZWVWN4L0hQTkdNb0UxcUJOUm9SR3BIVlEvU2dG
			712	WVNYd0VLb2ZDbXdncDU2NmlWSWlNS2dFMXBFU2hBRVBnaUNxRlRZCmcxQmdGM29wQ08zQmw2SFFM
			713	a3hSakVrM2pTWXZYY2FIL3o3TWRyWDM5dXl6ejJnajV3dUhmZFovM1g1bjdiWCsvLzg2OUxsb1dJ
			714	REgKTHJTSUNncjFEVlYwR01JVFdKWjluNDEzOENiZXduMFZmWi9HSzRudEZyeUFNWnpBYVd6Qnlh
			715	eitDRzQ0eDQvb0ZkT3VidzdleFhPNApKR2QvRm9mRklsYlJTc3J6Y0F6MzVHeTdzQzFwdHhaTE94
			716	WFpnRlpTN3JtK1BRV2QyNHpqOFpvQ04rQ0h4TFlhazdnN1p4c1V1MkM2CmFTWGxydlhsZDFlYmVX
			717	S2w5NVJNUG9xUE9oUUtsK0pKZkpEWVA4YWZ6bllGLytCM1hGNWovS1kwMGxlMGdIZGhsdkx6L29u
			718	d0U1MXkKTGVianE4UitTcnoxbFluOURUeFNZL3ltTk5KWEZFUU9pSlYrRll1d043UDlsZFZ2cWls
			719	d0pIditVVkEzSWFKYm5uRXNyRGxIRXhycApLMXJBbjBXdzJJVDEyV2NDTzRTdk9OYWx3SWtTZ1VY
			720	SDlVdmNLQUpXeW9kaXgzVEtKQjdDMTlPaHJ5eU4yU3hTbG9leENuZUlsT1pxClBOaVI3Q2xPWmM5
			721	L0MrcG1sZGlYWUdmSmVLdHJ6bjh1R3VrcjhvRnREdUY1TE1kdCtGNEVsOWsxQlk1VzFBMkwvQ3JQ
			722	QW5GTUptdk8KMHkyTjlLVUwrRXpKUUo5amQ5WnBvS2JBNzNBY1Z4VFVYWVlmRTl1ajJWem5pMGI2
			723	aHBMR0svRnl5VVJ6Y1ZCazZYQW5Wb2pnTWdjdgpLdDgxb3lMYTVabVA2L0JlemphWXpmTmJ5VGl3
			724	VDMwZnVBN2ZWTFRwaWI0MStCdlhGOVJkaGFQQ0g4S1Z3c0czWDhBV2NUMXEwMHI2CnY0NHZFdHM2
			725	RWUzejZkSUQ0dW80M2JTU2NrLzBiUmE1ME51NE9XZGZqUDBpR3JmWmdPMjU4bklSbWNvRUxzSlB1
			726	RDhyRCtCOS83MFMKYml3VDEyTmFTYmxyZmZralBJTGJjU3RlRStIN0YzRmtuOEtudWJaTDhHMnVQ
			727	QzdlMUxBSS9jZVRlY1pFUnI5ZTdQUzU0a2psZCsySQo2bVBXU3k2NHZuMGlWMnh6amZBMTV6TUIv
			728	bDlRbGNaVU1lYnNnREdjUFg5dEptZm0wZTBDSGhTQnBjMkk4Q0VuaTV0ZnZIUzdnSitaCmlzaHdM
			729	N1kybHpQekdPeXkzMUh4dDljYTNDUjI0RXVtcmtWOU9tUkEvSi9XcDArZlBqT1JNeUNPd1JpWFM5
			730	ZTRBQUFBQUVsRlRrU3UKUW1DQwo=
			731	</png>
			732	<latex>$$S_{0} {\left\|0\right\rangle } = {\left\|0\right\rangle }$$</latex>
			733	</output>
			734	<output>
			735	<output_type>display_data</output_type>
			736	<text>S ⋅❘1⟩ = ⅈ⋅❘1⟩
			737	0</text>
			738	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUZZQUFBQVpDQVlBQUFDcldObE9BQUFBQkhOQ1NWUUlD
			739	QWdJZkFoa2lBQUFBd3BKUkVGVQphSUh0MkYrb1ZGVVV4L0dQWHBPOGxrV1FYZE1IS1NVelRYeFFL
			740	cUs4UVlXUS82Q0hOSWdnUytvaEVFSHFTU0g3WjlKYkNFR2lxZFNiCmtjSU5UTHdnSWxxWUlFSUZw
			741	aUlVV21sRVVGRjVlMWpuTm1lbWM4NDBNK2NvWGVjTHc3Mno5OXByL1didGZmWmUrOUJseERNSlQx
			742	MXAKRVcyUXFYdDB3WUF4ZUJFNzhDRSt3ak1ZanoxTmdxMHQ2THNiUnpMYXY4UDBKbjZycGpUZGVZ
			743	a2RKeEk1UWN6R0UxaUtpVGlLWDVvSQo3TTFvdXhITDhHbmlKNHZQTWErSjd5b3BUWGRlWXQ4WE03
			744	RUJsMUx0citQNkpFZ3JMTUI3bUlNTEJYWjc4RmlMdnF0a2dUWjFqOGt3CnVnRkxHZzFUSE1lK0Zn
			745	VU9KaDlDN05RY3U3L3dzMWdsUDdVWW93b0d0YWs3YThYZWoydms3M2NIY0tvZGxmK1JEN0NpUXY5
			746	VlVhYzcKSzdHSHhReThqVTFxaVI1bVE1WHE4QzF1clRqR0xMeUZBZkdZbDBHZDdxekUvb0NYay8v
			747	WGlCVjZFZStJcFg0NStCcDM1UFR0eGJFVwpQbDlnV21wOEwxYUozM2hlVkQ2bDY4N2FZNG5aM0kz
			748	bDZNYzllQUczNFBFU2hlUXhBOXR6K2g3dTBQZXplRmNjeXYzWTBxRy9OUC9vCkxxcGp2OFE2UElD
			749	NU9Dc090V3RMRkpMRkpQRllEVlhrZjdjNGdCL0JGR3d0eVcrZDdzYkV2cFF6NkFSMkpvTkdsU1Fr
			750	anllVFdGWHgKVGZMM09WRTJuaTdKYjUzdTlGWnduU2dwM3NnWk9FRmNEbjVOdnQrTEIvR0h1RkM4
			751	cXZOVjFwUEV1VmhnTTRDYlcvQTVKQzQ0SjFOdApmVmlrZG9yZjN0RGZLb1c2RitMUEpFZ2pFM0ZP
			752	N0Vsd0U3NVNtNWhONnErRDZ3dEVETXBmSllzeHYyQnNXYXpCanhpTDJlSXdvMFRkCjZhM2dJVkdm
			753	dm9tWnFmYmI4REUyWW4vUzlqd09pb21ROUs4c0VEVk1qN2lBOUlwM0RvM01rMzBmTDVzcG90cTVK
			754	TjUvYkd0aTM1SHUKN1dJbDlvdVM1ak44Z2wyNEw4UDJsZFQzYWVLUkd3NjZ1c0YrZHZKRHppUjJR
			755	K0lKT0tSMnl2Zmg2VHh4SlRORDNCNDM0cTVVK3hYWApQYUJXNjhMa0pHalZoZjMvaHFKeXE0aFQ2
			756	ZytxNFpWYTlLTGlxcUxkeEI1Vi93cXREOS9qdDQ0VmpSRGFUZXdSdFFvQkhzWG16dVdNCkhIcmFI
			757	SGRPbkpJTGNhZFlzYS9oOTVKMFhmV01FclZnbHk1ZHVuVDVOMzhEaXFhbVFHZGFQZmNBQUFBQVNV
			758	Vk9SSzVDWUlJPQo=
			759	</png>
			760	<latex>$$S_{0} {\left\|1\right\rangle } = \mathbf{\imath} {\left\|1\right\rangle }$$</latex>
			761	</output>
			762	<output>
			763	<output_type>display_data</output_type>
			764	<text>T ⋅❘0⟩ = ❘0⟩
			765	0</text>
			766	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUU4QUFBQVpDQVlBQUFCdzQzTnNBQUFBQkhOQ1NWUUlD
			767	QWdJZkFoa2lBQUFBdk5KUkVGVQpXSVh0MkUySVZsVWN4L0dQbzQzanU3WVFxekZ4RkVKRk1pVmNS
			768	RVBVTW5RcjZTYlNRREFHaElTV0xoUkZhS2VRRUdSUWlSRVZMU3A3CldmaTIwWkJTQ0VJTHEwbGRw
			769	Q2xXdnFZdHpyMXhQTjc3ekIzdkhZZXg1N3M1NS83dXZmL3plODQ5TC8vejBHYkUwNGR4dzIyaUJZ
			770	WCsKUmtYMWhmZ3VxLytBbnpFVlMzRURoM0FGVS9BNHVqQU5GNk1ZNDdBYTI1TTIrdkFrVHFFVFgr
			771	R0w2Sm01ZUFwdjM4VVBHd3hENW04egpQc0NrU0Z1QVc5aVZQTHNRRndwaVRNU3JpZllLamdtZERk
			772	MDRsNW1OMlZSbXJFRWE5ZGNSMVpmZ0JWeUt0TjZzL0RKNTc3Z3dFZ2ZpCkFXekFibUhVUWo4T3V2
			773	TXJIc1hpQ2pHYnBKYS92UE9ld0FGY1QxN0lPMjkvb28vRzd4WE1yY0NqMkp2b24yTWVlaUx0RXl5
			774	dkVMTkoKYXZuTE8yODYzaW9JM2l1c0E3OG1laWMrcldCdVNWYWVTUFNUV2ZsTXBOM0EzNWhjSVc1
			775	VDFQS1hkOTVlbkU0Q3pNWEQ3aHgxY0JsNwpLcGlia1pWL0pYcCsvVkNpdjRkVkZlSTJSUzEvWTFv
			776	RXpxZnN2cnUyRnN4ZHhzMFNjMU1UdlI4elMySTloOWNIMmY0UnZEeFUvcXAwClh0SElxOHJWQW1P
			777	RWhWckJ2VzdCWUJGZlkxRU5MMFhVOHRlaG5GNWhLcDlzOGN4QUhDL1JKMlRsbVVSZktVeU5lMFV0
			778	ZjJXZDE0M1oKd2c1Y2gyTVlMMnd3TVJPejhteWtqUkZNRitXUFEwVXRmMlhUdHRXVUhZdVhoTXk4
			779	Qng4TGVWR1p1VkhDeC9ncDB2TUVOUDd5eTRSMApvSXhuc2EzRi9TSyt3ZG9XOTV2MDl4OXZDQ2VM
			780	QlFYMzFtTmpWdS9LR24wa3UwNHorRTVjdzdva3hrRjhsbWpEY2NKbzNOOTQvSWcvCjNINzJ6VG5q
			781	OWt6N2ZlRnNXR1FPWHNQM1dWekNCN21HcDZObjV1REZLdVpxMHFpL2ZOcDI0Q1BoWE51RFdabCtX
			782	SmpqYndwNTNSUmgKZS84dGluRmFPS0hBUC9nemFXT3JNSXJmRlJMdXg0VE1QbDVQbjgvYUdHcUcx
			783	ZCtjcktINHo0TXQrTEJ1NEpGSXExU2xpSDdoL0hzcgowaWJnZkdPT1JoQ0Q3Ynlyd3Zvd1BkSm00
			784	SmZHSE4zbjdNU2FyRDVhV1BQS2psVDNOVVc3NlVBOGlCM0NmM3p6OFMzZWFkTFUvNEd4CncyMmdU
			785	WnMyYlFiSnYyYzB0aEx3SEVvZ0FBQUFBRWxGVGtTdVFtQ0MK
			786	</png>
			787	<latex>$$T_{0} {\left\|0\right\rangle } = {\left\|0\right\rangle }$$</latex>
			788	</output>
			789	<output>
			790	<output_type>display_data</output_type>
			791	<text>ⅈ⋅π
			792	───
			793	4
			794	T ⋅❘1⟩ = ℯ ⋅❘1⟩
			795	0</text>
			796	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUhnQUFBQWxDQVlBQUFDSmRDMzdBQUFBQkhOQ1NWUUlD
			797	QWdJZkFoa2lBQUFCRlpKUkVGVQplSnp0Mm51b0ZIVVV3UEdQM3UycHB2bW9zSkNMRkZtWVdOR0w4
			798	aEZsWVJGbEpJbFJpQVFLRlpWRVNVVVk5b2FRNko4Qy9VZnNYYVJGCmtBUVJGb1JJRCtoQlpFYVJQ
			799	UzB5emJLYlpuK2NXZTY0ZDJidjdyMjdkL2RlNWd2RHpQN216Ry9PbkRPLzN6bm5OMHRCUVlFMXJW
			800	YWcKb0RsMDRGWjgxV3BGQ3ByTE82MVdvSzhNYjdVQ0JjMmxjUEFRcDNEd0VLZHc4QkNubzlVS0RB
			801	SnV4dFVZaGMreHA3WHFGQlNrR05acQpCWVlnSlZ5R2lYZ0tuYmdGeDFmSTdjZGkvRE9ReXJXS2E3
			802	WFB5M2Fnbjl0MUdJT1BNUW1yTUFPUFkzcHFQNlhCZW1mYXNKUTZub2puClJlSjFRTVNhdzNCSWNy
			803	NHIyWTVNWlA3RDVkaWQ2dVBRcE8zVkhDWG1KZjI5V05IK1BTN0J4bHFmcG9uMDkwVTdFeE9FTFVx
			804	NFErUTYKTTRYVHIwLzJlVFROaHJmalUxeUtjVW5iaWNMWmIrck91RXU0QVgvcm1ZV1BUQjZva21F
			805	aVVlbktPUThyOHhRYmhEeUdaYW5mQ3pGWAoyT2ZaWHE1dHFBM1REcm80MlRiaXQ2VHRuR1QvbWhp
			806	eHNBOXJzU1hWVm8xSHNRbExkYzhHV1d6SENUWDAxd282OEdTTnNpWGgwTFdZCm43UXRFVGFZSUti
			807	bmV1bXpEY3NPbm9SdCtLbmlncG5KZmxOT1o3VndqNGhCeTNxUmUwNFlwaDJaamROcWxEMEN2eWJ5
			808	TzhTei95bEMKM2c1TUZtR3VIdnBzdzNJTW5vSFZHUmZNRktQNXM0cjJFV0k2cjRWOU5jcnRFZzll
			809	cXVPYWdXQTA5dFlodjF1TTNDNThpMk5GdGt3NAoraXo4VmFjT2ZiWmgyY0hQWkFoUEVKbmVlaEdI
			810	MCt6QlEzVXFXUXNiY0NWZWFVTGZsWnlDT1RoR2xDcXI4V09HM0N3Um91cGhhK3I0CjU0cHpuOVRa
			811	VjcwY1pNTlNGY0ZxMDNPeitBalh5SGJ3K1NJUnJJY3ZjRzlHKzRNaWdWeUNuYUpXWFlrYksrUW1p
			812	MUU0bURqSWh1M20KNEZGaUdzdmlBeXl2czcvZEdXMzM0MEx4d3BSbnBnVjRLME4yS3NiaWRCd25S
			813	bnlXWER0eGtBMTdjL0F1MVd1MlJyTkFKQXBaN05YLwpmMVowNGo1UjlqMHNhczRPOGZLOGtDR2Zu
			814	cG9YYVgvblVtSERQQWVQd1RSUk11MGZBS1hLZE9LYkp2WS9KOWt2bGgxdjgxZ3U4cEVyCjhIcWps
			815	V293blZJMnpIUHdCYUtFeXB1ZXArTnNVUks4aEI4YW9OaDVlTC9LK1ViRTRITEpVbzl6NFpGa2Ez
			816	ZDYyRERQd2RYaTd6eXgKb3JKSUpDRnY2RFpjZjVpTEZWWE9iNUcvZ3BOSHBVN2I1WmNvUy9HMG5o
			817	WERZS0tIRGZNY1BGc3NSVzdKT0xjQ3Q0bXBleXQrd1ZWWQoxOHZOeTJ1OFdkK2d4K0ozMVZmR3V2
			818	Ui8rdDZBTzBVR1hZN253OFRvZkUvN083ZHVHNVlkUEJ4M2lTbDNxaWpHdThTYTZrNjhqWGR4CnVJ
			819	ak5YNmI2MkNwRzhEcmRYMVRTekJmVDYwWEo3N3ZGeVA4T0R5UnRDL1crUnRzSU51TW1zVkMvUm56
			820	Q0c1SGNlL01BM0w4V0dtckQKOUplVGMrV1A2RzBpYmsxT2prZUxESnZJUms4U3RaZmszQitwYTZk
			821	Z2ZFYWYrM1hIaS9GaWVXK2dLT0ZrOGFMK080RDNyWldXMmZBbwpNUVdNVExXdEVndnJCVzFJdlgr
			822	NjI0V3Z4UnRXNW1pTnlhSUxta0JmL2xYNUljNUkvWjZGbHh1alRrRTdNRTJzNkp3cVBtTTkwVnAx
			823	CkNwckJLRkZLamV0RnJxQ2dvS0Nnb0tDZ29HRG84VCtzQXZZaDBLcnJLZ0FBQUFCSlJVNUVya0pn
			824	Z2c9PQo=
			825	</png>
			826	<latex>$$T_{0} {\left\|1\right\rangle } = e^{\frac{1}{4} \mathbf{\imath} \pi} {\left\|1\right\rangle }$$</latex>
			827	</output>
			828	</outputs>
			829	</codecell>
			830	<markdowncell>
			831	<source><h2>Symbolic gate rules and circuit simplification</h2></source>
			832	</markdowncell>
			833	<codecell>
			834	<input>for g1 in (Y,Z,H):
			835	for g2 in (Y,Z,H):
			836	e = Commutator(g1(0),g2(0))
			837	if g1 != g2:
			838	display(Eq(e,e.doit()))
			839	</input>
			840	<language>python</language>
			841	<prompt_number>14</prompt_number>
			842	<collapsed>0</collapsed>
			843	<outputs>
			844	<output>
			845	<output_type>display_data</output_type>
			846	<text>⎡Y ,Z ⎤ = 2⋅ⅈ⋅X
			847	⎣ 0 0⎦ 0</text>
			848	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUhZQUFBQVpDQVlBQUFEa0JkcWVBQUFBQkhOQ1NWUUlD
			849	QWdJZkFoa2lBQUFCU1pKUkVGVQphSUh0Mlhuc1hGTVVCL0RQYjBGcGlRcFIwa1lwaVlvcUdnU05v
			850	SmJpRDJKZlNxd1ZxVmdpaEVvdHNhUmlpVkRVRW1JWGF4RFJXQ2NSCjRSZGliZElvc2RaU0ZSSjd0
			851	YzNQSCtlTnZubS9OL1BtVGVjM2t2YjNUVjVtN2pudjNuUGZ1ZmZjczF5R3NGcWlLOU4rQW1Qd0pV
			852	N0gKaGhuKzMvZ3BReHVHalZQdGYvQmpDM01aaWVFRjcveUJYK3J3ZXZBbWV2RXlMbWxoRHExaWZm
			853	eldZdC9oNHR2VFdJb2x5ZitzZmlXOApwV1dFVkZML1Q4U0xXSUYrek1kbE9YMm1pby9xeDllWVUw
			854	WmdDbDhrWS95TXQvRVE3a21lK1FudnFpYkcyUlp6VzV4RFdVeEhINTdECng3Z042NVljNDJnOEx3
			855	eWlYeGpGRFNuKy9saVU4SDdCSzVoVWRxS1ZITnJjWk5DVEcvUjdIRFBMQ2t0aEFwYmpjblJuZUxz
			856	SVMzMWUKV0dVUk9yV3daK0ZtSytjMENvdnhockRnc3BncDlEd3JoemNIOXlrKzBlcWlra1BiS3hI
			857	NFVwMCtNM0JHcXdJVHpNUkZPZlN4K0FIdgphdjZqT3JHdzNmak13QVU4VTdFUjFNTW9MTVBuYWwz
			858	a1ViaWloZkZxVU1taGRlRXJZVkdiWlhnSDRycFZGWW9YREZUU1NDeElaR2ZsCk5rSW5Gbllic1lC
			859	dlorZ1RFL3BETFk3N1pOSi9hdEtlb2tYWGxqMzI4dENQUjhXUmMyeUt2aDJtYVUrUWNyZmE0R050
			860	UENNVzlHQjgKM3dZWjdjUWlZVm5aQUtZLytjMzYyVjdONmJxNkljL0V6c0x5ejIxdGlyV28xS0h2
			861	SUNiOWJ0TGVCRTlqUkR1RVp0Q0ZoMFV3TWFXRgovczFZYkxkUWRyTlBubS9QbzUwazlIUkJpbll3
			862	cmhmV3ZXUEJ2THJFRWI5TWJPeXlnZGgvNkczeXZZOUVaRHBKTFBJc3NaTitiMVZ3CkExeU40OFZ1
			863	ZlhVUXh1L0ZhMWl2Uko4Vk9FenR5YkVpNTcxelJHWndaOUxlSEx2aVF1RnVUdEhZQXZ2RjZUVWJI
			864	K0N2RW5Oc2lFb0QKM3NXSjRLOUVwRG9ZT0MyUmNlVXFqTkhKZENlTjAwWDBQamxGdXhhYmlzRHZO
			865	NUVlTmNKd1BDWHFCZDlvTGd0b0NwVUd2Q2xDNmF1UwoxalRDQWVJSWVpQ0hkMUtKY2Y2UGhaMGdV
			866	cDE5TXZScTRlRTBzZWdiTkJoakxUeUM4U0w0NnNlaDdacGdwUUh2MGtUWVlGanJSUHdxCmpzaTFN
			867	N3lkY0UySnNUcmxZNnZZVkxpcXlRM2U2Uk41YUQxMDRTN3NuclQzRkxwK3NlQTc2cUpaSDB2a3M3
			868	L2p2UnplRmpoVitLQ2QKY1l2d3lWVk14RUw1UG1PMDhEK0xjTGdJbXRLWUlUNjZXVmxGNkJFbHg3
			869	SSs5Z2lSVTZleG5najBwcXROZmM0VHhRc2lKdGtWNTRzRgozTWxBSGM3R3MzZ3JhYjhwS2xrSFlF
			870	dFJsY3VpbEI0cWRlZzl3cUxtMWVFL2kzMlQvK05FS3JCVzBoNHJQK2NqY3RjUGhjTEc1dkQzCkZs
			871	RmlPbUZ2Skl2T0hjWGRlRkJZVnhvamhKS3J1QUdmSlA4UHdpR1o5eStXWDlDWUlmUTJ1NDc4SWoz
			872	VW9GS0hQamtSa2xjcjNrcFkKY3RyNkYxcnBiNFlKQzNsZnJZL3BGUnZsRDdYSCt6cllYa1NZUzNG
			873	SENWbDBibUZ2RXFkTUpmUE1GM092WWk3dUZmNzI5aFI5REI3RApPM1hHMzFMb2ZMR0Jsd1NGZWlo
			874	S21vOFNsdllBdnNWeGVGMG9yNHB4b25DL1BFVmJMQmFIaVBEMkY5RmUraWcrSVhubk85eWF5T2tU
			875	CnUzdWVLREV1RVhYb1ptVjFDdU9GYm1EcnpMT2hPRWFydUZGRXUyZUxOSEc0K002M2hNR01NakFM
			876	T0ZiNDEyOUZRTmtuYnQ2cUtOUkQKa1k5OUlqTmdIamJDbnhuYTN3YVdDTmROSmxuRi9jbFRCczNL
			877	R213c0VCYlhERDdGTVJuYWJnVjlIa3VlZWlqVVF6TmxyaUlzTlBCZQpkNFRhZTl2OTFPN2l3WlMx
			878	SnFCUUQrMVkyUGxpQjZVeFdqanpLaVpwdkFQYktXdE5RS0VlMnJHd3k0UmYzQ05wVHhBWHd1bHlZ
			879	RHR1CmdKcVZ0U2FnVUE5Wkg3dEVGUHEveEpFbEJFMFRCWXdkUkFBeHpjcWJqbmFqbnF3ZUVXVDBx
			880	SitXclU3b3BNNkhNSVFoREdFSVF4aEMKZ244Qjh3aEN5UDVGK0g4QUFBQUFTVVZPUks1Q1lJST0K
			881	</png>
			882	<latex>$$\left[Y_{0},Z_{0}\right] = 2 \mathbf{\imath} X_{0}$$</latex>
			883	</output>
			884	<output>
			885	<output_type>display_data</output_type>
			886	<text>⎽⎽⎽
			887	-⎡H ,Y ⎤ = -╲╱ 2 ⋅ⅈ⋅⎛- X + Z ⎞
			888	⎣ 0 0⎦ ⎝ 0 0⎠</text>
			889	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQVBRQUFBQWhDQVlBQUFEdTNJWmlBQUFBQkhOQ1NWUUlD
			890	QWdJZkFoa2lBQUFDTnRKUkVGVQplSnp0bkh0d1hVVWR4ejlKK3JqVXBKUVVuSlJLTzFJWlVXbEJv
			891	STZvMUxZQ0xZSVVLdEtLcjBKQkhNZWlZbm1vS0tnVVI5Q0JzWXlGCmlBVmxtSWpHSWtKNU9DaXhC
			892	ZXV6Z0tEVVJ5MzRtTkppa1RvUmlqYkVQNzU3ekxrblo4OXpUMjZTMmMvTW1YdnZ2czd2N081djk3
			893	ZS8KM1hQQjQvRjRQQjZQeHpPOGVCWFE3K0R5ZUR3TnBnMjRvOUZDZUR3ZU4xd0l2TDNSUW5nOG52
			894	S01CKzVwdEJBZWo4Y05Id0tXTkZvSQpqOGRUbmhiZ1B2UHBHV1dNY1ZUT2E0SHp6ZmMxd0tPT3lt
			895	MEVwd0lMemZkUEFQOXVvQ3hWc0JUb0J2b2FMWWpIUGE0VWVocndJbkFUCjhHUWs3aXBnRGxBenYv
			896	dUJmeUNIekV2QVhjRCt3RmdUM3dkc0FENWVRSTRMZ0hlSHlub0p1Qis0S0pKdUtYQko2UGZEd0Zu
			897	bSs4L04KTTF3TjdNUG9VdWdtNEgzQUtRbHBGcGxyUExBUldBZnNyRjYwUWt3QTdnVGEwYk1CL0Jm
			898	NEdQQ1ErWDB2MEJISzh5dmduS0VTTUlaRApnTzh3SUc4ZjBvZGcrMjh5NnI5bkE1dUhYRHJEUXVv
			899	VkpNb0VKUGhqcUtORU9RRTkwRnFndWFRc0x3TTJtZkpPdGFScEJ0YWp4bjRkCkE1VWJwaHNOTktP
			900	SlJXalFzM0cxdWFZQ2h5SEgyVGEwWDEyR2c0QWpTcGFSeEZHb2YrMEdKa1hpMmszY1ZjQXJLNVFo
			901	S3hjaGVhNUUKZFZ3THhaMEU3QVd1VDhoL012SDkxU2xwQ3YwbXBHQmZzTVIveHNTZjVFaWVjMDE1
			902	cXkzeHg2S2xRZEk2Y2pRcTlIcWcxUkozREJyZwp3clNqMmZtSkV2YzhFT2pFblRWbzQyN1U1c2RH
			903	d3IrQ0JqTFh0QmZNdDRGNEMra29vQmM5UjFLL25JTUdnMHBKVStoTFVHWFB0OFQvCkNJMWErenFT
			904	WnorMEJOako0STQwQTdndEpqekthRlBvK2NCbENmR2ZSOHVObVpId2J0UjIwd3Zjc3huTjhrTlJq
			905	NmNnT1c4TmhYMEsKTFRHcTRQNENlZG90K2FZRDI5SFN6emJnaHVsRU94V1ZrYWJROXdCNzBKbzB5
			906	bmpnQmR5dkYyNUhEUncrUExHdkNZK2FaWEdNSklWdQpCYjRMdkNVaHplMGt6eXFyVUgxOU5oSity
			907	UWsvdVlCY1p5UHJheWhvQWY2Ryt0bGt0RTVlV2VIOWVncmttUSs4TFJJMkNmZ3Q4RmRrCnpXUmhF
			908	ckNWYlAyNEVFa0szWUxXTmoreHhMOFZkWmhySE12MEx1cEg3REZBRjluWGd5TkZvYWNCbDZQUi9U
			909	WkxtdG5BbDFQS2FVVjEKMWhZSnZ3dlY0MEdXZkJPQmk0R3ZSZEkwQVg5aWFPdndjaVRybmNnWFVD
			910	VTlEc29ZaTZ6VDNjQ3NuSGsvaXB4L2xaQ2swRWVqU3Q2SwpsQ1I2YlRieHB6bVdxWVlxcWhjNXlx
			911	NUY2NCtzcENuMGU5R2FNODlWUlFNRUh2MnprSWQzYWt5YUxyS1AvbUZlRGp5UDNieHNCcjZPCnZN
			912	ZzNBdGVGNG1ZajUrUlE4Z3JrVk5wRmVlZHFHajBPeXZnbWFyUGpDK1NkQzJ4eElFTXNTUXA5QVFN
			913	T3I5YVlheVBhWHBwY2dWdzMKbVh2L0VGaVdNMithUXRkUVI4NXpUY2dwUXg1cWFQc2o2bmg4RGNs
			914	ZTB5UldBLzhFRHJiRW44NUFaM3dVK0dvbzdqTFNyUUxYSEFvOAppOXI4dUlydjFWTXkvK2VRbk1z
			915	TDV1OHcrYWVWbENPV0pJVytBNDJhY1l2OXNXaWY5L0VxaEVLemZqLzFNMGRXUm9ySkhlYUx3QTVn
			916	ClhDaHNMY1cyblJZZzVaaWRrR1l1TXExZmorbzVuUGJid0ljTDNMY29VMUdiTFRheWRGZDh2NTRT
			917	ZVpjaEdWZEZ3dHZJNTN6Y1RjU3IKSC9iMEhvWTZSTlk5cmkya094MmFrS1BtRVdUNlJqa1N6VnEy
			918	OVhWWlpwalByb3JLZDRITGVsK0QzcUphQXR5Q09zY1l0SmJOdzZ2UgptbmdCOE11RWREM204NFBB
			919	YnlKcE85RHNIb2ZydnJZZnNpYk9RMWJLVnJSVjFRRThuZkVlY2JTaWdTbHVSMlFXZzdmNVFDYjA2
			920	V2lYCkpZN2prSmU2QzdnMEVuY3U4QXZncVl6eS9SNjE4Y1lnSUN6bzQyamt5RnJKTDJSSU14TjVW
			921	bSt4eEwvWmZHNnd4QitNVExvMnBQUkoKblN1T09VaE9XNzR5NWIrSC9Gc2k5NksxZkJpWDlmNFha
			922	Qkd0UUhXK0V1M0Q1bUYvdExZN0RTa3B3QlMwRy9Ga1RQb0p3Sm5BcDgzdgo4YWd6dDZIT0hZZkxa
			923	OTRIdUFFZDAzM0doSFVDWDBLZTdpdGk4bVJ0OTE0alo1eENyeU4rR2JjWHV6TFBCTDRIL0F6NVBN
			924	Si9EdEdFCkJvS2dmMlNSY1Fkd2dPVmVwYkNaM0I5QlFyL1RrbStkaWUrSWladU9IaUpZYS8rVXda
			925	N0FwQTdSaEV6R0J5enhhZVdubWR6alRIeWUKcXhaYmtsdm1vVHA5QjlyS3lrTU5lWWlqcDdxV0FX
			926	KzA1UGtBVXJoSlNQR0RXZWMra3JjeVhkQUNmQXM0UEJKK0FGS3FweGpzSE12UwpyN0xRa3pQOWdX
			927	akEzVUw4OXVGU3REektJK1B2Z1BlSEE2cjJCTTR6bncvR3hMV2dHZnFQeEp0Rm53UytqMGJKWG5U
			928	S2FVVW8vZ3IwClFMYmpmRE9SS1dhYi9kUEtUK00veUx6TGMrM0pVWDVSSGtBellCZURyWUVrbW9D
			929	YjBWN3UwV2gyT3dlWjA4dXhlMVNQUk9lbW4wUDcKemplWThLZVJrNm9xT3BERnM1N0JMd005ZzJi
			930	Q2FRdytmVmkyM1l2UWlyYi9hdWhjeExPUitPWElvZ3IrUVNhTGpDMW9TYm1qQ29IRApNM1FMMmdP
			931	OEZaMys2Z2UrZ1dacmtBS3VSbHNod1lzYTF3RW5Sc3I4TlhCRzZQY1NwTUFCYTVGNUUzM1FCYWhU
			932	UFdMS2Z4RHRjVWUzCmJkTEtINGxPc1lEenNBOWtOaTdHL3A5aFNaM21HR1NhcjZMKzdQd0taRnE2
			933	WmpIcTRNOFoyWDVNdmRuWmdyenR1eGlRdlJONGc0bFAKYS9lczlPUklHNXkyZXdqdE9GeVBsaldi
			934	a0oraEgyMFBCcnNnV1dROEJPbFgzZTVRRmVkcis1QVpGQ2gyd0c3enVSMHBlSlMvUjM1UApvZDZS
			935	MWt1OThNdlIxc2laa1h3UG8wWmNFd3JyWjJCOWxiWDhrY3pONVBmeS9nQWQvWXdqenFFWnNBbVo0
			936	LzNVcjNYdlJzZEpYYk1KCitIUG9kei8xenJjK3REZStsbnEybWM5R3RQdWx4Sy9sdy9RaXBZWnNN
			937	aDZLSEdpN3dvRlZIWmgvTENGdUQ1bzkwOWlKOXFjRGF0Uy8KeWhpY1hvcWFXenZKOXJwZld2a2pt
			938	UmV4TzJac1BFSHhsekNlanduYkN2d0JyY2V6dEhkV3Rwc3JpYVQrNTZyZDkrWkltL2NBU0JZWgo1
			939	MUovYmgyb2ZnMWRoczNVbjFWdFovQXN2Z2c1REtvcTMxT09DNUY1T1p6Nm1hdDJQeU05U1dIU1pK
			940	eUYxdUtkMFl6RHFhS2piRWJPCm1ZQjUxSjlWUGhHdFNmNVZVZm1lOG14QXM5UDVhUW1IRUZmdEhu
			941	VnN1U1JKeG1ZMFNLNUVqdGxLV0lqcytSdHg5eUw3Uk9Ud09CNXQKalhSVGZ3SnFZa1hsTDBiUHNZ
			942	MlI2eFFiVG94aDRKRExjQ0N0WHcwSGttUmNRWFd2aFA2Zk5xVElSK0R1bldiUWFIUTRGWjFYdFpR
			943	LwpoWUZuOFgrazU0WngyTi9XYWdSVjl5c1gyR1NjRVpQVzQvRjRQQjZQeCtQeGVEd2VUMDcrQjl2
			944	dE01YzJvTmlJQUFBQUFFbEZUa1N1ClFtQ0MK
			945	</png>
			946	<latex>$$- \left[H_{0},Y_{0}\right] = - \sqrt{2} \mathbf{\imath} \left(- X_{0} + Z_{0}\right)$$</latex>
			947	</output>
			948	<output>
			949	<output_type>display_data</output_type>
			950	<text>-⎡Y ,Z ⎤ = -2⋅ⅈ⋅X
			951	⎣ 0 0⎦ 0</text>
			952	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUpVQUFBQVpDQVlBQUFBNDBHenNBQUFBQkhOQ1NWUUlD
			953	QWdJZkFoa2lBQUFCVmhKUkVGVQphSUh0Mm5lb0hGVVV3T0V2UG1PTXNVWkRGQldOWFZHeFlSUjdR
			954	Nk5pdzk1ckZJSmRKUGFLaW9vWUN4S0pMZGl3dHlTSUlyWllzTVVTClMreUN3UzdZRW8weC9uRm1l
			955	Zk1tdTdNeisrYkZMTzRQQm5idXpEM243cGt6NTU1Nzd0Q2hROFgwcTBqT3RyZ1J2K0pNZklJRk12
			956	ZDgKaTFtWnRxSG9uenIvSG4rMk9JWUZzV3lCKzZiam4wemJpVGdFUzJOai9OTGlHUDRMRmhOMmI3
			957	WHZFcG0yR2ZneCtUMVEyQ1ROZC9pcgpSWDJsMkFXams5K0RjUmUrd2h6TXhBTllvMDYvRjVKN1p1
			958	QjViTktMTVJ5ZnlKcUZhWGdDNDVMajd1VGFkMWdxUjhZaldMSVhZNWhYCjlNTVplQmtUOFRhdUZD
			959	OVdHWTdFSlB3dDdETWRGNld1N3lXQ3dSejhnQ2V4ZGkvR1hZcTBVOVVZbGd6bXc1eCttMkFLVnE1
			960	Z0RCUHcKRnRiTXRDOGduT1VQREc4aW8xMmM2anljcjNzMldGVkUxd2tZMElLOEs4V3pPckhPdFR0
			961	eGZZdHllMFU5cDRLWHhHQTNxbk50Q0I3RwpvaFhvWHdTZmluQ2RaUXhtWTU4Q2N0ckJxUmJEUjNx
			962	bURYQ09zUFh1TGNoY1RhUUViMlhhajhGcExjaXJoRVpPTlVyODBXc3k3UVBFCmxMaGlSZnIzRU5O
			963	QmxsTVMvVVVOMHc1T3RibjRUeE15N1RzazdkZTJLUGZwcFA5bXlmbWV1S0pGV1pYUXlLbUdpQnhu
			964	dXA2SisxaHMKV3FIK0hjUWJuR1p2RWFGdUtDR25IWnhxR1dIUEJ6UHQyd3VuR0p0cHowYTBSdXli
			965	OUw4TlcrRm0xUzNrV3FLUlV4R0o1QnpzbUp5ZgpqZjM2ZUR6RFJRNzFHTHBLOUd2bVZQMUVNbHpt
			966	NklzSFUrOC9uU2JzZkVTcWJYK1JMNzBtOHE0OCt1TWJZYmQ3RlhmR3VTaTdXbWlGCnV6QkNMTmxy
			967	SzYvNysxRGZLbmdjNytNZ0VhMnE0amlNTE5ublB2RmdxeVQ3bjdwRXFqRVY5eVJ0YTJGNVVlSjVG
			968	UWZqa2h5WnMwU1UKR2kxV2xkbnl6enduTDFJTndtL0prUTNOVlROWXJEYS9VS3htbGFVZHByOTZq
			969	TVpQV0QvVk5rWXNnb1lLQjJtMlVGbEtSUGJaNG9WcwptWFNrV2srNWVYUUtUaWh3MysvNEFDdmhw
			970	RktqSzhjQTRSVExZZ3NSeW10c0xlcG1YL1NoL2pKVWFlc3RjU3Ayd2p1cDlndkZpendLClA0dm8z
			971	WWlCNG9VL1NlUyt1MkViUEZkd2ZIMUNYcVFhS0tyazJjU3lTdnFKc1ArWFNGaXpQSVNGQzhocGw1
			972	eXF4akF4NVcyUU05NlAKY1ZXT2pDNk0xeDNsZGhPNTJUME5lelJoWHVSVW0yRWhVVEhQNHppeFpU
			973	QklPR0Y2T2JzNFZ0QTRMRitPQTBXRitKbk10ZFZGRFdabQpBVDNOT0FiSGxyaWZLSjFjWGJKUEVa
			974	YkVIU0labjVwcVAwVjNXV0U3VVlNYUo1N0JtbmczSTJjTWJ0RWQ1U2JoU3pGZERoRmJaL1hvCmpS
			975	MExrUmVwemhlZTMraHRJcGF3RTFQbkR3a25xWEcveUF1MnJkTjNaQ0wvb2pyWEJvcDZ6dUVGOWJS
			976	TFRyV1FjTmIxTSswcjR1TFUKK2UyWW5QdysxTnc3Q3BlcG4ydlZDcW1Obm1rek8xWkNubE85S09i
			977	MjdBWnptZ2YxM0NJWXFXZHg3MmhobkxNeS9VWUlaeHVmYVY5TwpUSU12aTMydDJxWm9NejN0NGxS
			978	MzRITThtemsrRXF1OEdnK0lsZWRRRVpGcURCTTUxbE1ONUc4Z25Pb3o5WGM4Y3UyWTk2QjdRNWZZ
			979	CmZIeFA3T3Y5TEp6aXdnYjNyeVkyTG10OGkzVlQ1N2VLN1ljWkdSMVhKZmV1ZzFlU1k2cW95NHdY
			980	Yis0anVuZmRtK2xwQjdZUXhkNysKNHYra2owRjZUb1dYaWdYU1VUaFhUR2R2aUkzOERjVjBlSHBH
			981	L3JHaURQSzFpSWl2Qy91bnliVmpYK1ZVczdGemlmc0hpNkpialpubQpycEFQRndaSjZ5anJFRVgw
			982	ek85TUZ2bGxFYWJnZ05UNXIrTFRuanhxWDNia2tXdkh2b3BVWlptbTV5cHBVZkdwUlkwdUVhbGU2
			983	bU05CkhZcVJhOGY1eGFuZTFMTll1WUtZejJ2c0x2YndlbHNkYjZhblF6Rnk3VGkvT05Wa2tYVFgy
			984	RlhQM2ZaSDVYK1hWWldlRHNYSXRXT1YKbnhQZkpLcm5wMnV0RW51Qm1LY0hpQS9PcnF0b2JFWDBu
			985	SXpEeE1wdkkrMzFPZkYveGJ4NlhoMDZkT2pRb1VPSER2OUwvZ1VWSWowVAowdXBoZXdBQUFBQkpS
			986	VTVFcmtKZ2dnPT0K
			987	</png>
			988	<latex>$$- \left[Y_{0},Z_{0}\right] = - 2 \mathbf{\imath} X_{0}$$</latex>
			989	</output>
			990	<output>
			991	<output_type>display_data</output_type>
			992	<text>⎽⎽⎽
			993	-⎡H ,Z ⎤ = ╲╱ 2 ⋅ⅈ⋅Y
			994	⎣ 0 0⎦ 0</text>
			995	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUp3QUFBQWhDQVlBQUFBczdNTG1BQUFBQkhOQ1NWUUlD
			996	QWdJZkFoa2lBQUFCblJKUkVGVQplSnp0MjMrc0hVVVZ3UEZQWDB0L2hWS3AxWUlvdFZoaVZCUi9v
			997	RWFzTkRSUmk1aUlVcW1TR0JUUm9GWUxSR01OYVl5eGl1SnZ3VmdWClVXekVpQllzYWxyalQ2cGlv
			998	bExiUUJRMVVyV1JLajlzRUtUYTlMM25IMmV2ZDkrKzNidDc3OXY3N3RYdU45bmN0ek03TTJkbnpz
			999	dzUKWjJZZkRRME5EUTBOdzhwOEhJL3hHcTYrTTJzNkdtbm9LNWZoWjVneGFFRWEvditaajI4UFdv
			1000	aUdJNGYxZVBtZ2hXZzRNamdLTy95UAptZEtSUVF2UTBET3Z3ZldteWRrZk5wNk16Y2wxNm9CbHFZ
			1001	dTEydTgwYk1IVkNMNWorT1FxcFM2QlQ4Uy84UVg4TVpOM0pjN0EzT1IrCkhQZmhKUmpEdDdCWW1B
			1002	Z1l4VTVjV29OY1orTjlxZnREK0h2cWZrblMzaXI4STFQMkovZ3RQaVg2NlhBTjh0VEZ1YmhadVV5
			1003	TGNRbWUKZ2p2eFhmeWd5N1pXWVpNSVVJanh1MGVNMzJqcXVaTnhnN2FKZnhndnhvTmR0bGVKMWRq
			1004	UUlYOStJdHp0bUpPVC95THhJdGVxMTh4ZgpqMy9pemFKRDBoTnNROUxteFNWMTdOQ2VMTVBDZHN3
			1005	cmVXWTVmb2lYNGhoaGdnL2h2VDIwTjRKUGlQNzZlSWZuM29rLzRFek03cUdkCnlwUXAzT2xDMktL
			1006	WDNaamtuMTJqVExPd0QwL0t5WHVWV0YydnJGRFBzQ25jYXJ5cnduUGJSYituYWZYejJoN2FQVGtw
			1007	ZTN0Qi9oTGMKaEVVOTFOMDFaUXJYV2sxV0ZlUi9YNnlBQzJ1VWFhWDgyZmdDL012RTViOFR3Nlp3
			1008	TjRzVnF4TnpoSXZ3bmt6NktXSWN2dGhqMno5UAp5ajh0a3o0WFczRkNqL1YyVFpuQ2JSZURuR2NH
			1009	NXVBZ2R0VXMwOFY0WENidGliZ2ZQMVZkaWFaVDRjNFNFNkdJNStQOUZlcFpJUHIwCllSTmRtRWNJ
			1010	aGZsbGovS3RUOHBuTGNNMWVGYVBkZlpFSjRXYmlRZHdTMEgrU3ZFU0grdURYR2tlTGZ5TDN3dG51
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			1014	MWNSNWI1UUtPZ1NFWWpjV0ZHbTJ1aTB3bDJtSFJBY25YUDlXRGp3ait5VGJDT2lRdzRLazlRdFpT
			1015	dmNQREh6dTdrNlJaaHYKeDBNbSs3TmJzTFI3OGYvTDBxVGVUMmJTWjRpVmpUZ21HOFdqT3RSemdS
			1016	alBuZmowRk9TWkVwMFVicHRZaG8vT3lUdEtiRnZjMFNlNQppTUJoRE9kbDBrOVU3bnd6L1VIRHNh
			1017	SlBMa21sTFpkdkJxc3lTN2cwVzB6ZWRscWc3WDl0VS80eHdMRkM0WGJwWWVzamJWSlB3UldxCm44
			1018	M2RLV1pqSjJaZ0JYYUwyWlhsbVdLUHJzaS9teXJyazJ1RHljNzR1NVhMUHdnTzRNdDRpL2JlMXp2
			1019	dzRTblVlUlh1RXVaL0xKUDMKSUc3RFk0UVZlbVZKWFU5SWZyY0tjOW9WYVlXN0E2OVZYZUVPVm5q
			1020	bXFXSmZaa3RCZnN2RUZmbHZKd2wvWVlGUXlqeS9wSWh6OEZGOApGaC9NNUMwV0h5MGVxS0dkdW55
			1021	NE5GZmhEU0pxM1NOVzR0OTAyVWFMUzhXWVhxaDk3bm82YnMwOGR5SHV4VGVGNVRrcy81ejJqT1Mz
			1022	CkgyTldtU0tUdWs0SWZXNUJ1UnVUL09OeThwWUtZVnUrM3EwbTcvOFVUWTduaUMyQjdmSURvdy9n
			1023	YlJYYktUT3BzNFVDZDNQbG5iWmsKdVNXUi95TWl1dXlGYzRSaVovdHBjODZ6ZTBXL0VKYXVhRS8w
			1024	RzJLeHlYdUhLbU5XQzBVS3QxVW8xSktjdkpuNEczNVhVT2RtWEo2Ngp2eHlmUzkxdkVyTitXYWJj
			1025	c3FUZTNXS1dwUmtScG1rY2o2L1l6cUEyZnRjSTg3ZXR4L0tuQ2JmbmpXSVZibDBiVE40UVh5VDZa
			1026	RVh5Cjk4YUNPa2RFaEYva0FwWDFaVzJrRlc0bVBpVDhrRkh0Zlo5MVNmNHlZVEsrcDMyUWY3VXdI
			1027	Mmx1TTlIUlg0dGZwKzZ2RmN2K1cxTnAKczRRU2pvdHRrTmJYSGwvQnI0UXpQaTZVc1dvN2cxSzQx
			1028	dEZjMGVsTUp4YUtiWTJpLzExNFUwNlpHMFNmWG1IeThkUnlNV1k3a3ZKMwpKZmNyTXMrVjlXVmY5
			1029	dUZHOFNWdHhXdnhRUEs3WDM3RTlaZk0vZkVtQmhvUG1iaDE4bnJoK0orZlNoc1Q1NlJsZnVpOVhi
			1030	UXpLQTdqCnViaTdoN0tIeEo1YkVYbTdBdWVKOTc0L0orOXU3VEZMVzdJL1paNHI3Y3QrYmZ3V0hm
			1031	QVNSMXk3TytTM3VNZkVpR3F1V0tGYWpJdWoKcXoycHRMSE1mUlhLMmhra3ZTZ2I0V1A5cUlkeWVj
			1032	cEcrTU4xak5sUWYvRzdTNXo5dFZoazhpcjRNcE9qclg2MDAxQ04wcjRjZG9WTApSMmRuNHF1cCs3
			1033	UEVJWHoydzhtNjIybW96clQxNVdyeDZjbzFlSHBOZFI2amZXNTNnVGlIblozSjcyYzc1NHYzMldl
			1034	NFBrOGFac3JHCnJMYi8rRm1ndlFPOVZ6dEFtQ29qWXZQNEFQNWNVNTFWMnpsQisweHhqeVAwbjFW
			1035	NllMckdyS0dob2FHaG9hR2hZWnI0RDB0Y2p3bmcKNUcraEFBQUFBRWxGVGtTdVFtQ0MK
			1036	</png>
			1037	<latex>$$- \left[H_{0},Z_{0}\right] = \sqrt{2} \mathbf{\imath} Y_{0}$$</latex>
			1038	</output>
			1039	<output>
			1040	<output_type>display_data</output_type>
			1041	<text>⎽⎽⎽
			1042	⎡H ,Y ⎤ = ╲╱ 2 ⋅ⅈ⋅⎛- X + Z ⎞
			1043	⎣ 0 0⎦ ⎝ 0 0⎠</text>
			1044	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQU5RQUFBQWhDQVlBQUFDaGdZV3lBQUFBQkhOQ1NWUUlD
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			1097	T0FyN3RVTWhtZEdadFNrRkI3b0s5RXN4NXFMYUtONmRRK2cvY29pTjFWdGhBaFdxRWJneVZPUnRj
			1098	VApKaDZQeCtQeGVEd2VqOGZqU2VEL004TWJMbnFHbE9jQUFBQUFTVVZPUks1Q1lJST0K
			1099	</png>
			1100	<latex>$$\left[H_{0},Y_{0}\right] = \sqrt{2} \mathbf{\imath} \left(- X_{0} + Z_{0}\right)$$</latex>
			1101	</output>
			1102	<output>
			1103	<output_type>display_data</output_type>
			1104	<text>⎽⎽⎽
			1105	⎡H ,Z ⎤ = -╲╱ 2 ⋅ⅈ⋅Y
			1106	⎣ 0 0⎦ 0</text>
			1107	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUp3QUFBQWhDQVlBQUFBczdNTG1BQUFBQkhOQ1NWUUlD
			1108	QWdJZkFoa2lBQUFCbjFKUkVGVQplSnp0MjN1c0hVVWR3UEZQTDdjOHFyYTBVbmtaZ1ZEcXF5aEdq
			1109	TEUyWUVtTUZJMEJxUkFTRVMyUGlrcFVsSUF4alJwUmpGR2pGa090CjhvcnhoWllLUm9wSzhCU0pK
			1110	QkNKQ1BpMlVNcWpGTEVnV0pXMDFEOSt1NTY5ZTNiUDJUMzMzSHQ3eW42VHplNloyZm45Wm5kK00v
			1111	UDcKemM2aG9hR2hvYUZoVjJVR0RzVE9BUndUenVoa0tHbVlVTTdIYlpnMjFSVnAyUDJaZ1o5T2RT
			1112	VWFuanQ4Q0NkTmRTVWFuaHRNeDQyRwpiQ29kbWVvS05QVE42Zml1U1hMMko0cnpzUXFmbXVKNkRJ
			1113	cFI4VHlyY09vVTEyV1FqT0JuaGpEb3kxZjQ3Zmd3L3BOTG40L1ZtS2s5Ckt2NFgzOFMzc0RRcDl6
			1114	enRJZjVmU2RvZDQ2amZEL0RTek8rdGlWN0NZWjZKSCtIaWt2STdoTEVkanNXSnZOMkJrM0U5dHZl
			1115	NGJ6L1IKQnEvRUgvRUwzRnhUMTNIaS9jNUlmdS9FRnB3ZzNtL0tFYmhHdS8yMzRTMTRxcHZ3Vmcv
			1116	bG4wd1VubG1TZnpPZXhTdDZ5S25Dd1ltdQpXN0VJTDh6a3pjR2ZzQUV2cWlCckFTNGRRSjEyRmRa
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			1120	c25Zbmc3dGVjWWZQc2hmK2lVL24waGVJTnJxcQpUOTIzSitWZmxVdmZHMnZFck5TVk9sSHFrWml0
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			1143	bmZtTG11MUE3ZERPN3hFdVZaWlJ0TDhyYm8KakliMk50YUIzSnFrYmN1a1Bab2NkYWlpYTFqcHg5
			1144	aUlUdC9xbzF4WmUyL0RiM3VVcmRRT0U3WGo5MDdoaEdlWlkreTN0c04wOW9pSgowdFV3OGZUVkRx
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			1146	eVhDNDJBR2JwCnRjVm1FTHBHeGZPczBSamNaTkNyelR1WWg2TlUzOWpZaTMzRmpvWHBBNUpYVjlj
			1147	MDhUeEhxYkJYcTJFZ1RHYWJOelEwTkRRME5EUTAKVEFML0ExWVBpV1VGdktGZkFBQUFBRWxGVGtT
			1148	dVFtQ0MK
			1149	</png>
			1150	<latex>$$\left[H_{0},Z_{0}\right] = - \sqrt{2} \mathbf{\imath} Y_{0}$$</latex>
			1151	</output>
			1152	</outputs>
			1153	</codecell>
			1154	<codecell>
			1155	<input>c = H(0)X(1)H(0)*2CNOT(0,1)X(1)3X(0)Z(1)*2; c
			1156	</input>
			1157	<language>python</language>
			1158	<prompt_number>15</prompt_number>
			1159	<collapsed>0</collapsed>
			1160	<outputs>
			1161	<output>
			1162	<output_type>pyout</output_type>
			1163	<text>2
			1164	H ⋅X ⋅⎛H ⎞ ⋅CNOT ⋅X ⋅X
			1165	0 1 ⎝ 0⎠ 0,1 1 0</text>
			1166	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQU53QUFBQWdDQVlBQUFCNUNoYmpBQUFBQkhOQ1NWUUlD
			1167	QWdJZkFoa2lBQUFDSkJKUkVGVQplSnp0bkhtUUY4VVZ4ei91SW5JWVFCYU1vR0tpQzBKaWhJWFNF
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			1220	</png>
			1221	<latex>$$H_{0} X_{1} \left(H_{0}\right)^{2} CNOT_{0,1} X_{1} X_{0}$$</latex>
			1222	<prompt_number>15</prompt_number>
			1223	</output>
			1224	</outputs>
			1225	</codecell>
			1226	<codecell>
			1227	<input>circuit_plot(c, nqubits=2)</input>
			1228	<language>python</language>
			1229	<prompt_number>16</prompt_number>
			1230	<collapsed>0</collapsed>
			1231	<outputs>
			1232	<output>
			1233	<output_type>pyout</output_type>
			1234	<text>&lt;sympy.physics.quantum.circuitplot.CircuitPlot object at 0x48cd350&gt;</text>
			1235	<prompt_number>16</prompt_number>
			1236	</output>
			1237	<output>
			1238	<output_type>display_data</output_type>
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			1325	V094R1JnbGp1UkVRS1lya1RFU21JNVU1RQpwQ0NXT3hHUmdsanVSRVFLWXJrVEVTbUk1VTVFcEND
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			1327	L0FWNklkZEJoanV6V0FBQUFBRWxGVGtTdVFtQ0MK
			1328	</png>
			1329	</output>
			1330	</outputs>
			1331	</codecell>
			1332	<markdowncell>
			1333	<source>This performs a commutator/anticommutator aware bubble sort algorithm to simplify a circuit.</source>
			1334	</markdowncell>
			1335	<codecell>
			1336	<input>gate_simp(c)
			1337	</input>
			1338	<language>python</language>
			1339	<prompt_number>17</prompt_number>
			1340	<collapsed>0</collapsed>
			1341	<outputs>
			1342	<output>
			1343	<output_type>pyout</output_type>
			1344	<text>H ⋅CNOT ⋅X
			1345	0 0,1 0</text>
			1346	<png>aVZCT1J3MEtHZ29BQUFBTlNVaEVVZ0FBQUh3QUFBQWNDQVlBQUFDajZ0dmtBQUFBQkhOQ1NWUUlD
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			1373	CittMEUvMWh6Y0o1RFgwLytDWHNFdFNRRHRFYUwvaGI1dS9sZzFEMmJUbjc3TWtCN0ZNTGlxb0Uz
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			1383	PT0K
			1384	</png>
			1385	<latex>$$H_{0} CNOT_{0,1} X_{0}$$</latex>
			1386	<prompt_number>17</prompt_number>
			1387	</output>
			1388	</outputs>
			1389	</codecell>
			1390	<codecell>
			1391	<input>circuit_plot(gate_simp(c),nqubits=2)</input>
			1392	<language>python</language>
			1393	<prompt_number>18</prompt_number>
			1394	<collapsed>0</collapsed>
			1395	<outputs>
			1396	<output>
			1397	<output_type>pyout</output_type>
			1398	<text>&lt;sympy.physics.quantum.circuitplot.CircuitPlot object at 0x3da1990&gt;</text>
			1399	<prompt_number>18</prompt_number>
			1400	</output>
			1401	<output>
			1402	<output_type>display_data</output_type>
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			1463	UlU1RXJrSmdnZz09Cg==
			1464	</png>
			1465	</output>
			1466	</outputs>
			1467	</codecell>
			1468	</cells>
			1469	</worksheet>
			1470	</worksheets>
			1471	</notebook>

docs/examples/notebooks/smooth_dos.ipynb

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	1	{"cells":[{"code":"%notebook save dos.ipynb","cell_type":"code","prompt_number":5},{"code":"from sympy import \nimport numpy as np\nimport math","cell_type":"code","prompt_number":47},{"cell_type":"text","text":"<h2>Strutinsky Energy Averaging Method</h2>"},{"cell_type":"text","text":"Define a callable class for computing the smooth part of the density of states $\\tilde{g}_m(E)$ with curvature\ncorrection of order $2M$."},{"code":"class SmoothDOS(object):\n\n def __init__(self, energies):\n self.energies = energies\n\n def gaussian_smoothing_func(self, M, x):\n return (mpmath.laguerre(M,0.5,x2)exp(-x2)/sqrt(pi)).evalf()\n\n def __call__(self, e, M=3, gamma=1.0):\n return sum(self.gaussian_smoothing_func(M, (e-ei)/gamma)/gamma for ei in self.energies)","cell_type":"code","prompt_number":67},{"code":"def avgf(M, x):\n return (mpmath.laguerre(M,0.5,x2)exp(-x2)/sqrt(pi)).evalf()","cell_type":"code","prompt_number":14},{"code":"def smooth_dos(gamma, M, e, energies):\n return sum(avgf(M, (e-ei)/gamma)/gamma for ei in energies)","cell_type":"code","prompt_number":2},{"cell_type":"text","text":"<h2>1D Simple Harmonic Oscillator DOS</h2>"},{"code":"def sho_smooth_dos(e):\n \"\"\"Compute the exact smooth DOS.\"\"\"\n return 1","cell_type":"code","prompt_number":50},{"code":"def sho_spectrum(nmax):\n \"\"\"Compute the first nmax energies of the 1D SHO.\"\"\"\n return [n+0.5 for n in range(nmax)]","cell_type":"code","prompt_number":51},{"code":"sho_evalues = np.linspace(0.0,20,100)","cell_type":"code","prompt_number":63},{"code":"sho_dos = SmoothDOS(sho_spectrum(30))","cell_type":"code","prompt_number":68},{"code":"sho_exact_dos = [sho_smooth_dos(e) for e in evalues]","cell_type":"code","prompt_number":69},{"code":"sho_approx_dos = [sho_dos(e,M=3,gamma=10.0) for e in evalues]","cell_type":"code","prompt_number":72},{"code":"plot(evalues, sho_exact, label=\"exact\")\nplot(evalues, sho_approx, label=\"approx\")\ntitle(\"Smooth part of the DOS for the 1D SHO\")\nxlabel(\"Energy\"); ylabel(\"$g(E)$\")\nlegend(loc=4)","cell_type":"code","prompt_number":73},{"cell_type":"text","text":"Note how the exact smoothed DOS and the Strutinsky approximation agree once the energy is away from\n0. In general, these are edge effects and are also present at the right limit if you don't use enough energy\nlevels. Here we have used 30, so the right side doesn't show these artifacts."},{"cell_type":"text","text":"<h2>1D Particle in a BOX (PIAB) DOS</h2>"},{"code":"def piab_smooth_dos(e):\n \"\"\"Compute the exact smooth DOS.\"\"\"\n return 1.0/(2.0math.sqrt(emath.pi2/2))","cell_type":"code","prompt_number":76},{"code":"def piab_spectrum(nmax):\n \"\"\"Compute the first nmax energies of the 1D PIAB.\"\"\"\n return [0.5math.pi*2(n+1)**2 for n in range(nmax)]","cell_type":"code","prompt_number":77},{"code":"piab_evalues = linspace(1.0,1000.0,100)","cell_type":"code","prompt_number":78},{"code":"piab_dos = SmoothDOS(piab_spectrum(20))","cell_type":"code","prompt_number":79},{"code":"piab_exact_dos = [piab_smooth_dos(e) for e in piab_evalues]","cell_type":"code","prompt_number":80},{"code":"piab_approx_dos = [piab_dos(e, M=4, gamma=150.0) for e in piab_evalues]","cell_type":"code","prompt_number":81},{"code":"plot(piab_evalues, piab_exact, label=\"exact\")\nplot(piab_evalues, piab_approx, label=\"approx\")\ntitle(\"Smooth part of the DOS for the 1D PIAB\")\nxlabel(\"Energy\"); ylabel(\"$g(E)$\")\nlegend(loc=1)","cell_type":"code","prompt_number":85},{"code":"","cell_type":"code","prompt_number":24}]} No newline at end of file	1	<?xml version="1.0" encoding="utf-8"?>
			2	<notebook>
			3	<name>dos</name>
			4	<nbformat>2</nbformat>
			5	<worksheets>
			6	<worksheet>
			7	<cells>
			8	<codecell>
			9	<input>from sympy import *
			10	import numpy as np
			11	import math</input>
			12	<language>python</language>
			13	<prompt_number>1</prompt_number>
			14	<collapsed>1</collapsed>
			15	<outputs />
			16	</codecell>
			17	<markdowncell>
			18	<source><h2>Strutinsky Energy Averaging Method</h2></source>
			19	</markdowncell>
			20	<markdowncell>
			21	<source>Define a callable class for computing the smooth part of the density of states $\tilde{g}_m(E)$ with curvature
			22	correction of order $2M$.</source>
			23	</markdowncell>
			24	<codecell>
			25	<input>class SmoothDOS(object):
			26
			27	def __init__(self, energies):
			28	self.energies = energies
			29
			30	def gaussian_smoothing_func(self, M, x):
			31	return (mpmath.laguerre(M,0.5,x*2)exp(-x**2)/sqrt(pi)).evalf()
			32
			33	def __call__(self, e, M=3, gamma=1.0):
			34	return sum(self.gaussian_smoothing_func(M, (e-ei)/gamma)/gamma for ei in self.energies)</input>
			35	<language>python</language>
			36	<prompt_number>2</prompt_number>
			37	<collapsed>1</collapsed>
			38	<outputs />
			39	</codecell>
			40	<codecell>
			41	<input>def avgf(M, x):
			42	return (mpmath.laguerre(M,0.5,x*2)exp(-x**2)/sqrt(pi)).evalf()</input>
			43	<language>python</language>
			44	<prompt_number>3</prompt_number>
			45	<collapsed>1</collapsed>
			46	<outputs />
			47	</codecell>
			48	<codecell>
			49	<input>def smooth_dos(gamma, M, e, energies):
			50	return sum(avgf(M, (e-ei)/gamma)/gamma for ei in energies)</input>
			51	<language>python</language>
			52	<prompt_number>4</prompt_number>
			53	<collapsed>1</collapsed>
			54	<outputs />
			55	</codecell>
			56	<markdowncell>
			57	<source><h2>1D Simple Harmonic Oscillator DOS</h2></source>
			58	</markdowncell>
			59	<codecell>
			60	<input>def sho_smooth_dos(e):
			61	"""Compute the exact smooth DOS."""
			62	return 1</input>
			63	<language>python</language>
			64	<prompt_number>5</prompt_number>
			65	<collapsed>1</collapsed>
			66	<outputs />
			67	</codecell>
			68	<codecell>
			69	<input>def sho_spectrum(nmax):
			70	"""Compute the first nmax energies of the 1D SHO."""
			71	return [n+0.5 for n in range(nmax)]</input>
			72	<language>python</language>
			73	<prompt_number>6</prompt_number>
			74	<collapsed>1</collapsed>
			75	<outputs />
			76	</codecell>
			77	<codecell>
			78	<input>sho_evalues = np.linspace(0.0,20,100)</input>
			79	<language>python</language>
			80	<prompt_number>7</prompt_number>
			81	<collapsed>1</collapsed>
			82	<outputs />
			83	</codecell>
			84	<codecell>
			85	<input>sho_dos = SmoothDOS(sho_spectrum(30))</input>
			86	<language>python</language>
			87	<prompt_number>8</prompt_number>
			88	<collapsed>1</collapsed>
			89	<outputs />
			90	</codecell>
			91	<codecell>
			92	<input>sho_exact_dos = [sho_smooth_dos(e) for e in sho_evalues]</input>
			93	<language>python</language>
			94	<prompt_number>10</prompt_number>
			95	<collapsed>0</collapsed>
			96	<outputs />
			97	</codecell>
			98	<codecell>
			99	<input>sho_approx_dos = [sho_dos(e,M=3,gamma=10.0) for e in sho_evalues]</input>
			100	<language>python</language>
			101	<prompt_number>23</prompt_number>
			102	<collapsed>1</collapsed>
			103	<outputs />
			104	</codecell>
			105	<codecell>
			106	<input>plot(sho_evalues, sho_exact_dos, label="exact")
			107	plot(sho_evalues, sho_approx_dos, label="approx")
			108	title("Smooth part of the DOS for the 1D SHO")
			109	xlabel("Energy"); ylabel("$g(E)$")
			110	legend(loc=4)</input>
			111	<language>python</language>
			112	<prompt_number>24</prompt_number>
			113	<collapsed>0</collapsed>
			114	<outputs>
			115	<output>
			116	<output_type>pyout</output_type>
			117	<text>&lt;matplotlib.legend.Legend at 0x7f1790f77990&gt;</text>
			118	<prompt_number>24</prompt_number>
			119	</output>
			120	<output>
			121	<output_type>display_data</output_type>
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			491	</png>
			492	</output>
			493	</outputs>
			494	</codecell>
			495	<markdowncell>
			496	<source>Note how the exact smoothed DOS and the Strutinsky approximation agree once the energy is away from
			497	0. In general, these are edge effects and are also present at the right limit if you don't use enough energy
			498	levels. Here we have used 30, so the right side doesn't show these artifacts.</source>
			499	</markdowncell>
			500	<markdowncell>
			501	<source><h2>1D Particle in a BOX (PIAB) DOS</h2></source>
			502	</markdowncell>
			503	<codecell>
			504	<input>def piab_smooth_dos(e):
			505	"""Compute the exact smooth DOS."""
			506	return 1.0/(2.0math.sqrt(emath.pi**2/2))</input>
			507	<language>python</language>
			508	<prompt_number>16</prompt_number>
			509	<collapsed>1</collapsed>
			510	<outputs />
			511	</codecell>
			512	<codecell>
			513	<input>def piab_spectrum(nmax):
			514	"""Compute the first nmax energies of the 1D PIAB."""
			515	return [0.5math.pi2(n+1)**2 for n in range(nmax)]</input>
			516	<language>python</language>
			517	<prompt_number>17</prompt_number>
			518	<collapsed>1</collapsed>
			519	<outputs />
			520	</codecell>
			521	<codecell>
			522	<input>piab_evalues = linspace(1.0,1000.0,100)</input>
			523	<language>python</language>
			524	<prompt_number>18</prompt_number>
			525	<collapsed>1</collapsed>
			526	<outputs />
			527	</codecell>
			528	<codecell>
			529	<input>piab_dos = SmoothDOS(piab_spectrum(20))</input>
			530	<language>python</language>
			531	<prompt_number>19</prompt_number>
			532	<collapsed>1</collapsed>
			533	<outputs />
			534	</codecell>
			535	<codecell>
			536	<input>piab_exact_dos = [piab_smooth_dos(e) for e in piab_evalues]</input>
			537	<language>python</language>
			538	<prompt_number>20</prompt_number>
			539	<collapsed>1</collapsed>
			540	<outputs />
			541	</codecell>
			542	<codecell>
			543	<input>piab_approx_dos = [piab_dos(e, M=4, gamma=150.0) for e in piab_evalues]</input>
			544	<language>python</language>
			545	<prompt_number>21</prompt_number>
			546	<collapsed>1</collapsed>
			547	<outputs />
			548	</codecell>
			549	<codecell>
			550	<input>plot(piab_evalues, piab_exact_dos, label="exact")
			551	plot(piab_evalues, piab_approx_dos, label="approx")
			552	title("Smooth part of the DOS for the 1D PIAB")
			553	xlabel("Energy"); ylabel("$g(E)$")
			554	legend(loc=1)</input>
			555	<language>python</language>
			556	<prompt_number>22</prompt_number>
			557	<collapsed>0</collapsed>
			558	<outputs>
			559	<output>
			560	<output_type>pyout</output_type>
			561	<text>&lt;matplotlib.legend.Legend at 0x6107410&gt;</text>
			562	<prompt_number>22</prompt_number>
			563	</output>
			564	<output>
			565	<output_type>display_data</output_type>
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			919	</png>
			920	</output>
			921	</outputs>
			922	</codecell>
			923	<codecell>
			924	<input />
			925	<language>python</language>
			926	<collapsed>1</collapsed>
			927	<outputs />
			928	</codecell>
			929	</cells>
			930	</worksheet>
			931	</worksheets>
			932	</notebook>

docs/examples/notebooks/sympy.ipynb

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