##// END OF EJS Templates
upstream » ipython
RSS

Clone from https://github.com/ipython/ipython.git

Wrap os.path functions in method calls...
Wrap os.path functions in method calls Some functions from os.path are now references to C functions (e.g. isdir on Windows). This breaks the path module, because compiled functions do not get bound to an object instance. All os.path functions have been wrapped in method calls, out of general caution. Closes gh-737
Brian E. Granger - - Load All Authors

File last commit:

r4637:d919e2ec
r4833:fc05f375
Show More
quantum_computing.ipynb
424 lines | 76.1 KiB | text/plain | TextLexer
/ docs / examples / notebooks / quantum_computing.ipynb
History | Annotation | Raw |Copy content |Copy permalink

Symbolic Quantum Computing

In [1]:
%load_ext sympyprinting
In [2]:
from sympy import sqrt, symbols, Rational
from sympy import expand, Eq, Symbol, simplify, exp, sin
from sympy.physics.quantum import *
from sympy.physics.quantum.qubit import *
from sympy.physics.quantum.gate import *
from sympy.physics.quantum.grover import *
from sympy.physics.quantum.qft import QFT, IQFT, Fourier
from sympy.physics.quantum.circuitplot import circuit_plot

Qubits

The state of a set of qubits (Two state systems) is the quantum state that is of interest in Quantum Computing.

In [3]:
alpha, beta = symbols('alpha beta',real=True)
In [4]:
psi = alpha*Qubit('00') + beta*Qubit('11'); psi
Out[4]:
$$\alpha {\left|00\right\rangle } + \beta {\left|11\right\rangle }$$
In [5]:
Dagger(psi)
Out[5]:
$$\alpha {\left\langle 00\right|} + \beta {\left\langle 11\right|}$$
In [6]:
qapply(Dagger(Qubit('00'))*psi)
Out[6]:
$$\alpha$$

SymPy supports many different types of measurements.

In [7]:
for state, prob in measure_all(psi):
    display(state)
    display(prob)
$${\left|00\right\rangle }$$
$$\frac{\alpha^{2}}{\alpha^{2} + \beta^{2}}$$
$${\left|11\right\rangle }$$
$$\frac{\beta^{2}}{\alpha^{2} + \beta^{2}}$$

Qubits can be represented in the computational basis.

In [8]:
represent(psi)
Out[8]:
$$\left(\begin{smallmatrix}\alpha\\0\\0\\\beta\end{smallmatrix}\right)$$

Gates

Gate objects are the operators which act on a quantum state.

In [9]:
g = X(0); g
Out[9]:
$$X_{0}$$
In [10]:
represent(g, nqubits=2)
Out[10]:
$$\left(\begin{smallmatrix}0 & 1 & 0 & 0\\1 & 0 & 0 & 0\\0 & 0 & 0 & 1\\0 & 0 & 1 & 0\end{smallmatrix}\right)$$
In [11]:
c = H(0)*Qubit('00'); c
Out[11]:
$$H_{0} {\left|00\right\rangle }$$
In [12]:
qapply(c)
Out[12]:
$$\frac{1}{2} \sqrt{2} {\left|00\right\rangle } + \frac{1}{2} \sqrt{2} {\left|01\right\rangle }$$
In [13]:
for gate in [H,X,Y,Z,S,T]:
    for state in [Qubit('0'),Qubit('1')]:
        lhs = gate(0)*state
        rhs = qapply(lhs)
        display(Eq(lhs,rhs))
$$H_{0} {\left|0\right\rangle } = \frac{1}{2} \sqrt{2} {\left|0\right\rangle } + \frac{1}{2} \sqrt{2} {\left|1\right\rangle }$$
$$H_{0} {\left|1\right\rangle } = \frac{1}{2} \sqrt{2} {\left|0\right\rangle } - \frac{1}{2} \sqrt{2} {\left|1\right\rangle }$$
$$X_{0} {\left|0\right\rangle } = {\left|1\right\rangle }$$
$$X_{0} {\left|1\right\rangle } = {\left|0\right\rangle }$$
$$Y_{0} {\left|0\right\rangle } = \mathbf{\imath} {\left|1\right\rangle }$$
$$Y_{0} {\left|1\right\rangle } = - \mathbf{\imath} {\left|0\right\rangle }$$
$$Z_{0} {\left|0\right\rangle } = {\left|0\right\rangle }$$
$$Z_{0} {\left|1\right\rangle } = - {\left|1\right\rangle }$$
$$S_{0} {\left|0\right\rangle } = {\left|0\right\rangle }$$
$$S_{0} {\left|1\right\rangle } = \mathbf{\imath} {\left|1\right\rangle }$$
$$T_{0} {\left|0\right\rangle } = {\left|0\right\rangle }$$
$$T_{0} {\left|1\right\rangle } = e^{\frac{1}{4} \mathbf{\imath} \pi} {\left|1\right\rangle }$$

Symbolic gate rules and circuit simplification

In [14]:
for g1 in (Y,Z,H):
    for g2 in (Y,Z,H):
        e = Commutator(g1(0),g2(0))
        if g1 != g2:
            display(Eq(e,e.doit()))
$$\left[Y_{0},Z_{0}\right] = 2 \mathbf{\imath} X_{0}$$
$$- \left[H_{0},Y_{0}\right] = - \sqrt{2} \mathbf{\imath} \left(- X_{0} + Z_{0}\right)$$
$$- \left[Y_{0},Z_{0}\right] = - 2 \mathbf{\imath} X_{0}$$
$$- \left[H_{0},Z_{0}\right] = \sqrt{2} \mathbf{\imath} Y_{0}$$
$$\left[H_{0},Y_{0}\right] = \sqrt{2} \mathbf{\imath} \left(- X_{0} + Z_{0}\right)$$
$$\left[H_{0},Z_{0}\right] = - \sqrt{2} \mathbf{\imath} Y_{0}$$
In [15]:
c = H(0)*X(1)*H(0)**2*CNOT(0,1)*X(1)**3*X(0)*Z(1)**2; c
Out[15]:
$$H_{0} X_{1} \left(H_{0}\right)^{2} CNOT_{0,1} X_{1} X_{0}$$
In [16]:
circuit_plot(c, nqubits=2)
Out[16]:
<sympy.physics.quantum.circuitplot.CircuitPlot object at 0x48cd350>
No description has been provided for this image

This performs a commutator/anticommutator aware bubble sort algorithm to simplify a circuit.

In [17]:
gate_simp(c)
Out[17]:
$$H_{0} CNOT_{0,1} X_{0}$$
In [18]:
circuit_plot(gate_simp(c),nqubits=2)
Out[18]:
<sympy.physics.quantum.circuitplot.CircuitPlot object at 0x3da1990>
No description has been provided for this image