Symbolic Quantum Computing¶
Imports¶
In [1]:
%matplotlib inline
In [2]:
from IPython.display import display
In [3]:
from sympy import init_printing
init_printing(use_latex=True)
In [4]:
from sympy import sqrt, symbols, Rational
from sympy import expand, Eq, Symbol, simplify, exp, sin, srepr
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¶
In [5]:
alpha, beta = symbols('alpha beta', real=True)
In [6]:
psi = alpha*Qubit('00') + beta*Qubit('11'); psi
Out[6]:
$$\alpha {\left|00\right\rangle } + \beta {\left|11\right\rangle }$$
In [7]:
Dagger(psi)
Out[7]:
$$\alpha {\left\langle 00\right|} + \beta {\left\langle 11\right|}$$
In [8]:
qapply(Dagger(Qubit('00'))*psi)
Out[8]:
$$\alpha$$
In [9]:
for state, prob in measure_all(psi):
display(state)
display(prob)
$${\left|00\right\rangle }$$
$$\frac{\alpha^{2} \overline{\frac{1}{\sqrt{\alpha^{2} + \beta^{2}}}}}{\sqrt{\alpha^{2} + \beta^{2}}}$$
$${\left|11\right\rangle }$$
$$\frac{\beta^{2} \overline{\frac{1}{\sqrt{\alpha^{2} + \beta^{2}}}}}{\sqrt{\alpha^{2} + \beta^{2}}}$$
Qubits can be represented in the computational basis.
In [10]:
represent(psi)
Out[10]:
$$\left[\begin{matrix}\alpha\\0\\0\\\beta\end{matrix}\right]$$
Gates¶
Gate objects are the operators which act on a quantum state.
In [11]:
g = X(0)
g
Out[11]:
$$X_{0}$$
In [12]:
represent(g, nqubits=2)
Out[12]:
$$\left[\begin{matrix}0 & 1 & 0 & 0\\1 & 0 & 0 & 0\\0 & 0 & 0 & 1\\0 & 0 & 1 & 0\end{matrix}\right]$$
In [13]:
c = H(0)*Qubit('00')
c
Out[13]:
$$H_{0} {\left|00\right\rangle }$$
In [14]:
qapply(c)
Out[14]:
$$\frac{\sqrt{2}}{2} {\left|00\right\rangle } + \frac{\sqrt{2}}{2} {\left|01\right\rangle }$$
In [15]:
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))
$$H_{0} {\left|0\right\rangle } = \frac{\sqrt{2}}{2} {\left|0\right\rangle } + \frac{\sqrt{2}}{2} {\left|1\right\rangle }$$
$$H_{0} {\left|1\right\rangle } = \frac{\sqrt{2}}{2} {\left|0\right\rangle } - \frac{\sqrt{2}}{2} {\left|1\right\rangle }$$
$$X_{0} {\left|0\right\rangle } = {\left|1\right\rangle }$$
$$X_{0} {\left|1\right\rangle } = {\left|0\right\rangle }$$
$$Y_{0} {\left|0\right\rangle } = i {\left|1\right\rangle }$$
$$Y_{0} {\left|1\right\rangle } = - i {\left|0\right\rangle }$$
$$Z_{0} {\left|0\right\rangle } = {\left|0\right\rangle }$$
$$Z_{0} {\left|1\right\rangle } = - {\left|1\right\rangle }$$
$$S_{0} {\left|0\right\rangle } = {\left|0\right\rangle }$$
$$S_{0} {\left|1\right\rangle } = i {\left|1\right\rangle }$$
$$T_{0} {\left|0\right\rangle } = {\left|0\right\rangle }$$
$$T_{0} {\left|1\right\rangle } = e^{\frac{i \pi}{4}} {\left|1\right\rangle }$$
Symbolic gate rules and circuit simplification
In [16]:
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()))
$$\left[Y_{0},Z_{0}\right] = 2 i X_{0}$$
$$- \left[H_{0},Y_{0}\right] = - \sqrt{2} i \left(- X_{0} + Z_{0}\right)$$
$$- \left[Y_{0},Z_{0}\right] = - 2 i X_{0}$$
$$- \left[H_{0},Z_{0}\right] = \sqrt{2} i Y_{0}$$
$$\left[H_{0},Y_{0}\right] = \sqrt{2} i \left(- X_{0} + Z_{0}\right)$$
$$\left[H_{0},Z_{0}\right] = - \sqrt{2} i Y_{0}$$
In [17]:
c = H(0)*X(1)*H(0)**2*CNOT(0,1)*X(1)**3*X(0)*Z(1)**2
c
Out[17]:
$$H_{0} X_{1} CNOT_{0,1} X_{1} X_{0}$$
In [19]:
circuit_plot(c, nqubits=2);
This performs a commutator/anticommutator aware bubble sort algorithm to simplify a circuit:
In [20]:
gate_simp(c)
Out[20]:
$$H_{0} CNOT_{0,1} X_{0}$$
In [21]:
circuit_plot(gate_simp(c),nqubits=2)
Out[21]:
<sympy.physics.quantum.circuitplot.CircuitPlot at 0x110567940>
