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# This file is part of QuTiP: Quantum Toolbox in Python.
#
# Copyright (c) 2011 and later, Paul D. Nation and Robert J. Johansson.
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from __future__ import division
import numpy as np
import scipy.sparse as sp
from qutip.qobj import Qobj
from qutip.operators import identity, qeye, sigmax
from qutip.tensor import tensor
from qutip.states import fock_dm
from itertools import product
from functools import partial, reduce
from operator import mul
__all__ = ['sigma_x', 'rx', 'ry', 'rz', 'sqrtnot', 'snot', 'phasegate', 'cphase', 'cnot',
'csign', 'berkeley', 'swapalpha', 'swap', 'iswap', 'sqrtswap',
'sqrtiswap', 'fredkin', 'toffoli', 'rotation', 'controlled_gate',
'globalphase', 'hadamard_transform', 'gate_sequence_product',
'gate_expand_1toN', 'gate_expand_2toN', 'gate_expand_3toN',
'qubit_clifford_group','controlled_s','controlled_s_inv','controlled_ry','controlled_f','controlled_h']
#
# Single Qubit Gates
#
def sigma_x(N=None, target=0):
"""Operator sigma X.
Returns
-------
result : qobj
Quantum object for operator describing the sigma X.
"""
if N is not None:
return gate_expand_1toN(sigma_x(), N, target)
else:
return Qobj([[0, 1],
[1, 0]])
def rx(phi, N=None, target=0):
"""Single-qubit rotation for operator sigmax with angle phi.
Returns
-------
result : qobj
Quantum object for operator describing the rotation.
"""
if N is not None:
return gate_expand_1toN(rx(phi), N, target)
else:
return Qobj([[np.cos(phi / 2), -1j * np.sin(phi / 2)],
[-1j * np.sin(phi / 2), np.cos(phi / 2)]])
def ry(phi, N=None, target=0):
"""Single-qubit rotation for operator sigmay with angle phi.
Returns
-------
result : qobj
Quantum object for operator describing the rotation.
"""
if N is not None:
return gate_expand_1toN(ry(phi), N, target)
else:
return Qobj([[np.cos(phi / 2), -np.sin(phi / 2)],
[np.sin(phi / 2), np.cos(phi / 2)]])
def rz(phi, N=None, target=0):
"""Single-qubit rotation for operator sigmaz with angle phi.
Returns
-------
result : qobj
Quantum object for operator describing the rotation.
"""
if N is not None:
return gate_expand_1toN(rz(phi), N, target)
else:
return Qobj([[np.exp(-1j * phi / 2), 0],
[0, np.exp(1j * phi / 2)]])
def sqrtnot(N=None, target=0):
"""Single-qubit square root NOT gate.
Returns
-------
result : qobj
Quantum object for operator describing the square root NOT gate.
"""
if N is not None:
return gate_expand_1toN(sqrtnot(), N, target)
else:
return Qobj([[0.5 + 0.5j, 0.5 - 0.5j],
[0.5 - 0.5j, 0.5 + 0.5j]])
def snot(N=None, target=0):
"""Quantum object representing the SNOT (Hadamard) gate.
Returns
-------
snot_gate : qobj
Quantum object representation of SNOT gate.
Examples
--------
>>> snot()
Quantum object: dims = [[2], [2]], \
shape = [2, 2], type = oper, isHerm = True
Qobj data =
[[ 0.70710678+0.j 0.70710678+0.j]
[ 0.70710678+0.j -0.70710678+0.j]]
"""
if N is not None:
return gate_expand_1toN(snot(), N, target)
else:
return 1 / np.sqrt(2.0) * Qobj([[1, 1],
[1, -1]])
def phasegate(theta, N=None, target=0):
"""
Returns quantum object representing the phase shift gate.
Parameters
----------
theta : float
Phase rotation angle.
Returns
-------
phase_gate : qobj
Quantum object representation of phase shift gate.
Examples
--------
>>> phasegate(pi/4)
Quantum object: dims = [[2], [2]], \
shape = [2, 2], type = oper, isHerm = False
Qobj data =
[[ 1.00000000+0.j 0.00000000+0.j ]
[ 0.00000000+0.j 0.70710678+0.70710678j]]
"""
if N is not None:
return gate_expand_1toN(phasegate(theta), N, target)
else:
return Qobj([[1, 0],
[0, np.exp(1.0j * theta)]],
dims=[[2], [2]])
#
# 2 Qubit Gates
#
def cphase(theta, N=2, control=0, target=1):
"""
Returns quantum object representing the controlled phase shift gate.
Parameters
----------
theta : float
Phase rotation angle.
N : integer
The number of qubits in the target space.
control : integer
The index of the control qubit.
target : integer
The index of the target qubit.
Returns
-------
U : qobj
Quantum object representation of controlled phase gate.
"""
if N < 1 or target < 0 or control < 0:
raise ValueError("Minimum value: N=1, control=0 and target=0")
if control >= N or target >= N:
raise ValueError("control and target need to be smaller than N")
U_list1 = [identity(2)] * N
U_list2 = [identity(2)] * N
U_list1[control] = fock_dm(2, 1)
U_list1[target] = phasegate(theta)
U_list2[control] = fock_dm(2, 0)
U = tensor(U_list1) + tensor(U_list2)
return U
def cnot(N=None, control=0, target=1):
"""
Quantum object representing the CNOT gate.
Returns
-------
cnot_gate : qobj
Quantum object representation of CNOT gate
Examples
--------
>>> cnot()
Quantum object: dims = [[2, 2], [2, 2]], \
shape = [4, 4], type = oper, isHerm = True
Qobj data =
[[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 1.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 1.+0.j]
[ 0.+0.j 0.+0.j 1.+0.j 0.+0.j]]
"""
if (control == 1 and target == 0) and N is None:
N = 2
if N is not None:
return gate_expand_2toN(cnot(), N, control, target)
else:
return Qobj([[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 0, 1],
[0, 0, 1, 0]],
dims=[[2, 2], [2, 2]])
def csign(N=None, control=0, target=1):
"""
Quantum object representing the CSIGN gate.
Returns
-------
csign_gate : qobj
Quantum object representation of CSIGN gate
Examples
--------
>>> csign()
Quantum object: dims = [[2, 2], [2, 2]], \
shape = [4, 4], type = oper, isHerm = True
Qobj data =
[[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 1.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 1.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j -1.+0.j]]
"""
if (control == 1 and target == 0) and N is None:
N = 2
if N is not None:
return gate_expand_2toN(csign(), N, control, target)
else:
return Qobj([[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, -1]],
dims=[[2, 2], [2, 2]])
def berkeley(N=None, targets=[0, 1]):
"""
Quantum object representing the Berkeley gate.
Returns
-------
berkeley_gate : qobj
Quantum object representation of Berkeley gate
Examples
--------
>>> berkeley()
Quantum object: dims = [[2, 2], [2, 2]], \
shape = [4, 4], type = oper, isHerm = True
Qobj data =
[[ cos(pi/8).+0.j 0.+0.j 0.+0.j 0.+sin(pi/8).j]
[ 0.+0.j cos(3pi/8).+0.j 0.+sin(3pi/8).j 0.+0.j]
[ 0.+0.j 0.+sin(3pi/8).j cos(3pi/8).+0.j 0.+0.j]
[ 0.+sin(pi/8).j 0.+0.j 0.+0.j cos(pi/8).+0.j]]
"""
if (targets[0] == 1 and targets[1] == 0) and N is None:
N = 2
if N is not None:
return gate_expand_2toN(cnot(), N, targets=targets)
else:
return Qobj([[np.cos(np.pi / 8), 0, 0, 1.0j * np.sin(np.pi / 8)],
[0, np.cos(3 * np.pi / 8), 1.0j *
np.sin(3 * np.pi / 8), 0],
[0, 1.0j * np.sin(3 * np.pi / 8),
np.cos(3 * np.pi / 8), 0],
[1.0j * np.sin(np.pi / 8), 0, 0, np.cos(np.pi / 8)]],
dims=[[2, 2], [2, 2]])
def swapalpha(alpha, N=None, targets=[0, 1]):
"""
Quantum object representing the SWAPalpha gate.
Returns
-------
swapalpha_gate : qobj
Quantum object representation of SWAPalpha gate
Examples
--------
>>> swapalpha(alpha)
Quantum object: dims = [[2, 2], [2, 2]], \
shape = [4, 4], type = oper, isHerm = True
Qobj data =
[[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.5*(1 + exp(j*pi*alpha) 0.5*(1 - exp(j*pi*alpha) 0.+0.j]
[ 0.+0.j 0.5*(1 - exp(j*pi*alpha) 0.5*(1 + exp(j*pi*alpha) 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 1.+0.j]]
"""
if (targets[0] == 1 and targets[1] == 0) and N is None:
N = 2
if N is not None:
return gate_expand_2toN(cnot(), N, targets=targets)
else:
return Qobj([[1, 0, 0, 0],
[0, 0.5 * (1 + np.exp(1.0j * np.pi * alpha)),
0.5 * (1 - np.exp(1.0j * np.pi * alpha)), 0],
[0, 0.5 * (1 - np.exp(1.0j * np.pi * alpha)),
0.5 * (1 + np.exp(1.0j * np.pi * alpha)), 0],
[0, 0, 0, 1]],
dims=[[2, 2], [2, 2]])
def swap(N=None, targets=[0, 1]):
"""Quantum object representing the SWAP gate.
Returns
-------
swap_gate : qobj
Quantum object representation of SWAP gate
Examples
--------
>>> swap()
Quantum object: dims = [[2, 2], [2, 2]], \
shape = [4, 4], type = oper, isHerm = True
Qobj data =
[[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 1.+0.j 0.+0.j]
[ 0.+0.j 1.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 1.+0.j]]
"""
if targets != [0, 1] and N is None:
N = 2
if N is not None:
return gate_expand_2toN(swap(), N, targets=targets)
else:
return Qobj([[1, 0, 0, 0],
[0, 0, 1, 0],
[0, 1, 0, 0],
[0, 0, 0, 1]],
dims=[[2, 2], [2, 2]])
def iswap(N=None, targets=[0, 1]):
"""Quantum object representing the iSWAP gate.
Returns
-------
iswap_gate : qobj
Quantum object representation of iSWAP gate
Examples
--------
>>> iswap()
Quantum object: dims = [[2, 2], [2, 2]], \
shape = [4, 4], type = oper, isHerm = False
Qobj data =
[[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+1.j 0.+0.j]
[ 0.+0.j 0.+1.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 1.+0.j]]
"""
if targets != [0, 1] and N is None:
N = 2
if N is not None:
return gate_expand_2toN(iswap(), N, targets=targets)
else:
return Qobj([[1, 0, 0, 0],
[0, 0, 1j, 0],
[0, 1j, 0, 0],
[0, 0, 0, 1]],
dims=[[2, 2], [2, 2]])
def sqrtswap(N=None, targets=[0, 1]):
"""Quantum object representing the square root SWAP gate.
Returns
-------
sqrtswap_gate : qobj
Quantum object representation of square root SWAP gate
"""
if targets != [0, 1] and N is None:
N = 2
if N is not None:
return gate_expand_2toN(sqrtswap(), N, targets=targets)
else:
return Qobj(np.array([[1, 0, 0, 0],
[0, 0.5 + 0.5j, 0.5 - 0.5j, 0],
[0, 0.5 - 0.5j, 0.5 + 0.5j, 0],
[0, 0, 0, 1]]),
dims=[[2, 2], [2, 2]])
def sqrtiswap(N=None, targets=[0, 1]):
"""Quantum object representing the square root iSWAP gate.
Returns
-------
sqrtiswap_gate : qobj
Quantum object representation of square root iSWAP gate
Examples
--------
>>> sqrtiswap()
Quantum object: dims = [[2, 2], [2, 2]], \
shape = [4, 4], type = oper, isHerm = False
Qobj data =
[[ 1.00000000+0.j 0.00000000+0.j \
0.00000000+0.j 0.00000000+0.j]
[ 0.00000000+0.j 0.70710678+0.j \
0.00000000-0.70710678j 0.00000000+0.j]
[ 0.00000000+0.j 0.00000000-0.70710678j\
0.70710678+0.j 0.00000000+0.j]
[ 0.00000000+0.j 0.00000000+0.j \
0.00000000+0.j 1.00000000+0.j]]
"""
if targets != [0, 1] and N is None:
N = 2
if N is not None:
return gate_expand_2toN(sqrtiswap(), N, targets=targets)
else:
return Qobj(np.array([[1, 0, 0, 0],
[0, 1 / np.sqrt(2), 1j / np.sqrt(2), 0],
[0, 1j / np.sqrt(2), 1 / np.sqrt(2), 0],
[0, 0, 0, 1]]), dims=[[2, 2], [2, 2]])
#
# 3 Qubit Gates
#
def fredkin(N=None, control=0, targets=[1, 2]):
"""Quantum object representing the Fredkin gate.
Returns
-------
fredkin_gate : qobj
Quantum object representation of Fredkin gate.
Examples
--------
>>> fredkin()
Quantum object: dims = [[2, 2, 2], [2, 2, 2]], \
shape = [8, 8], type = oper, isHerm = True
Qobj data =
[[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j]]
"""
if [control, targets[0], targets[1]] != [0, 1, 2] and N is None:
N = 3
if N is not None:
return gate_expand_3toN(fredkin(), N,
[control, targets[0]], targets[1])
else:
return Qobj([[1, 0, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 1]],
dims=[[2, 2, 2], [2, 2, 2]])
def toffoli(N=None, controls=[0, 1], target=2):
"""Quantum object representing the Toffoli gate.
Returns
-------
toff_gate : qobj
Quantum object representation of Toffoli gate.
Examples
--------
>>> toffoli()
Quantum object: dims = [[2, 2, 2], [2, 2, 2]], \
shape = [8, 8], type = oper, isHerm = True
Qobj data =
[[ 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j 0.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j]
[ 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 0.+0.j 1.+0.j 0.+0.j]]
"""
if [controls[0], controls[1], target] != [0, 1, 2] and N is None:
N = 3
if N is not None:
return gate_expand_3toN(toffoli(), N, controls, target)
else:
return Qobj([[1, 0, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 0, 1, 0]],
dims=[[2, 2, 2], [2, 2, 2]])
#
# Miscellaneous Gates
#
def rotation(op, phi, N=None, target=0):
"""Single-qubit rotation for operator op with angle phi.
Returns
-------
result : qobj
Quantum object for operator describing the rotation.
"""
if N is not None:
return gate_expand_1toN(rotation(op, phi), N, target)
else:
return (-1j * op * phi / 2).expm()
def controlled_gate(U, N=2, control=0, target=1, control_value=1):
"""
Create an N-qubit controlled gate from a single-qubit gate U with the given
control and target qubits.
Parameters
----------
U : Qobj
Arbitrary single-qubit gate.
N : integer
The number of qubits in the target space.
control : integer
The index of the first control qubit.
target : integer
The index of the target qubit.
control_value : integer (1)
The state of the control qubit that activates the gate U.
Returns
-------
result : qobj
Quantum object representing the controlled-U gate.
"""
if [N, control, target] == [2, 0, 1]:
return (tensor(fock_dm(2, control_value), U) +
tensor(fock_dm(2, 1 - control_value), identity(2)))
else:
U2 = controlled_gate(U, control_value=control_value)
return gate_expand_2toN(U2, N=N, control=control, target=target)
def controlled_s(N=2, control=0, target=1, control_value=1):
"""
Create an N-qubit controlled gate from phase gate with the given
control and target qubits.
Parameters
----------
N : integer
The number of qubits in the target space.
control : integer
The index of the first control qubit.
target : integer
The index of the target qubit.
control_value : integer (1)
The state of the control qubit that activates the gate U.
Returns
-------
result : qobj
Quantum object representing the controlled-s gate.
"""
U = Qobj([[1, 0],
[0, 1j]])
if [N, control, target] == [2, 0, 1]:
return (tensor(fock_dm(2, control_value), U) +
tensor(fock_dm(2, 1 - control_value), identity(2)))
else:
U2 = controlled_gate(U, control_value=control_value)
return gate_expand_2toN(U2, N=N, control=control, target=target)
def controlled_s_inv(N=2, control=0, target=1, control_value=1):
"""
Create an N-qubit controlled gate from inversed phase gate with the given
control and target qubits.
Parameters
----------
N : integer
The number of qubits in the target space.
control : integer
The index of the first control qubit.
target : integer
The index of the target qubit.
control_value : integer (1)
The state of the control qubit that activates the gate U.
Returns
-------
result : qobj
Quantum object representing the controlled-s gate.
"""
U = Qobj([[1, 0],
[0, -1j]])
if [N, control, target] == [2, 0, 1]:
return (tensor(fock_dm(2, control_value), U) +
tensor(fock_dm(2, 1 - control_value), identity(2)))
else:
U2 = controlled_gate(U, control_value=control_value)
return gate_expand_2toN(U2, N=N, control=control, target=target)
def controlled_ry(phi, N=2, control=0, target=1, control_value=1):
"""
Create an N-qubit controlled gate from y-rotation gate with the given
control and target qubits.
Parameters
----------
N : integer
The number of qubits in the target space.
control : integer
The index of the first control qubit.
target : integer
The index of the target qubit.
control_value : integer (1)
The state of the control qubit that activates the gate U.
Returns
-------
result : qobj
Quantum object representing the controlled-s gate.
"""
U = ry(phi)
if [N, control, target] == [2, 0, 1]:
return (tensor(fock_dm(2, control_value), U) +
tensor(fock_dm(2, 1 - control_value), identity(2)))
else:
U2 = controlled_gate(U, control_value=control_value)
return gate_expand_2toN(U2, N=N, control=control, target=target)
def controlled_f(co, N=2, control=0, target=1, control_value=1):
"""
Create an N-qubit controlled gate from f gate with the given
control and target qubits.
Parameters
----------
N : integer
The number of qubits in the target space.
control : integer
The index of the first control qubit.
target : integer
The index of the target qubit.
control_value : integer (1)
The state of the control qubit that activates the gate U.
Returns
-------
result : qobj
Quantum object representing the controlled-s gate.
"""
U0 = Qobj([[1.0065, 0.2425],
[0.2425, 0.9935]])
U = (co * 1j * np.pi * U0).expm()
if [N, control, target] == [2, 0, 1]:
return (tensor(fock_dm(2, control_value), U) +
tensor(fock_dm(2, 1 - control_value), identity(2)))
else:
U2 = controlled_gate(U, control_value=control_value)
return gate_expand_2toN(U2, N=N, control=control, target=target)
def controlled_h(N=2, control=0, target=1, control_value=1):
"""
Create an N-qubit controlled gate from hadamard gate with the given
control and target qubits.
Parameters
----------
N : integer
The number of qubits in the target space.
control : integer
The index of the first control qubit.
target : integer
The index of the target qubit.
control_value : integer (1)
The state of the control qubit that activates the gate U.
Returns
-------
result : qobj
Quantum object representing the controlled-s gate.
"""
U = 1 / np.sqrt(2.0) * Qobj([[1, 1],
[1, -1]])
if [N, control, target] == [2, 0, 1]:
return (tensor(fock_dm(2, control_value), U) +
tensor(fock_dm(2, 1 - control_value), identity(2)))
else:
U2 = controlled_gate(U, control_value=control_value)
return gate_expand_2toN(U2, N=N, control=control, target=target)
def globalphase(theta, N=1):
"""
Returns quantum object representing the global phase shift gate.
Parameters
----------
theta : float
Phase rotation angle.
Returns
-------
phase_gate : qobj
Quantum object representation of global phase shift gate.
Examples
--------
>>> phasegate(pi/4)
Quantum object: dims = [[2], [2]], \
shape = [2, 2], type = oper, isHerm = False
Qobj data =
[[ 0.70710678+0.70710678j 0.00000000+0.j]
[ 0.00000000+0.j 0.70710678+0.70710678j]]
"""
data = (np.exp(1.0j * theta) * sp.eye(2 ** N, 2 ** N,
dtype=complex, format="csr"))
return Qobj(data, dims=[[2] * N, [2] * N])
#
# Operation on Gates
#
def _hamming_distance(x, bits=32):
"""
Calculate the bit-wise Hamming distance of x from 0: That is, the number
1s in the integer x.
"""
tot = 0
while x:
tot += 1
x &= x - 1
return tot
def hadamard_transform(N=1):
"""Quantum object representing the N-qubit Hadamard gate.
Returns
-------
q : qobj
Quantum object representation of the N-qubit Hadamard gate.
"""
data = 2 ** (-N / 2) * np.array([[(-1) ** _hamming_distance(i & j)
for i in range(2 ** N)]
for j in range(2 ** N)])
return Qobj(data, dims=[[2] * N, [2] * N])
def gate_sequence_product(U_list, left_to_right=True):
"""
Calculate the overall unitary matrix for a given list of unitary operations
Parameters
----------
U_list : list
List of gates implementing the quantum circuit.
left_to_right : Boolean
Check if multiplication is to be done from left to right.
Returns
-------
U_overall : qobj
Overall unitary matrix of a given quantum circuit.
"""
U_overall = 1
for U in U_list:
if left_to_right:
U_overall = U * U_overall
else:
U_overall = U_overall * U
return U_overall
def _powers(op, N):
"""
Generator that yields powers of an operator `op`,
through to `N`.
"""
acc = qeye(op.dims[0])
yield acc
for _ in range(N - 1):
acc *= op
yield acc
def qubit_clifford_group(N=None, target=0):
"""
Generates the Clifford group on a single qubit,
using the presentation of the group given by Ross and Selinger
(http://www.mathstat.dal.ca/~selinger/newsynth/).
Parameters
-----------
N : int or None
Number of qubits on which each operator is to be defined
(default: 1).
target : int
Index of the target qubit on which the single-qubit
Clifford operators are to act.
Yields
------
op : Qobj
Clifford operators, represented as Qobj instances.
"""
# The Ross-Selinger presentation of the single-qubit Clifford
# group expresses each element in the form C_{ijk} = E^i X^j S^k
# for gates E, X and S, and for i in range(3), j in range(2) and
# k in range(4).
#
# We start by defining these gates. E is defined in terms of H,
# \omega and S, so we define \omega and H first.
w = np.exp(1j * 2 * np.pi / 8)
H = snot()
X = sigmax()
S = phasegate(np.pi / 2)
E = H * (S ** 3) * w ** 3
for op in map(partial(reduce, mul), product(_powers(E, 3),
_powers(X, 2), _powers(S, 4))):
# partial(reduce, mul) returns a function that takes products
# of its argument, by analogy to sum. Note that by analogy,
# sum can be written as partial(reduce, add).
# product(...) yields the Cartesian product of its arguments.
# Here, each element is a tuple (E**i, X**j, S**k) such that
# partial(reduce, mul) acting on the tuple yields E**i * X**j * S**k.
# Finally, we optionally expand the gate.
if N is not None:
yield gate_expand_1toN(op, N, target)
else:
yield op
#
# Gate Expand
#
def gate_expand_1toN(U, N, target):
"""
Create a Qobj representing a one-qubit gate that act on a system with N
qubits.
Parameters
----------
U : Qobj
The one-qubit gate
N : integer
The number of qubits in the target space.
target : integer
The index of the target qubit.
Returns
-------
gate : qobj
Quantum object representation of N-qubit gate.
"""
if N < 1:
raise ValueError("integer N must be larger or equal to 1")
if target >= N:
raise ValueError("target must be integer < integer N")
return tensor([identity(2)] * (target) + [U] +
[identity(2)] * (N - target - 1))
def gate_expand_2toN(U, N, control=None, target=None, targets=None):
"""
Create a Qobj representing a two-qubit gate that act on a system with N
qubits.
Parameters
----------
U : Qobj
The two-qubit gate
N : integer
The number of qubits in the target space.
control : integer
The index of the control qubit.
target : integer
The index of the target qubit.
targets : list
List of target qubits.
Returns
-------
gate : qobj
Quantum object representation of N-qubit gate.
"""
if targets is not None:
control, target = targets
if control is None or target is None:
raise ValueError("Specify value of control and target")
if N < 2:
raise ValueError("integer N must be larger or equal to 2")
if control >= N or target >= N:
raise ValueError("control and not target must be integer < integer N")
if control == target:
raise ValueError("target and not control cannot be equal")
p = list(range(N))
if target == 0 and control == 1:
p[control], p[target] = p[target], p[control]
elif target == 0:
p[1], p[target] = p[target], p[1]
p[1], p[control] = p[control], p[1]
else:
p[1], p[target] = p[target], p[1]
p[0], p[control] = p[control], p[0]
return tensor([U] + [identity(2)] * (N - 2)).permute(p)
def gate_expand_3toN(U, N, controls=[0, 1], target=2):
"""
Create a Qobj representing a three-qubit gate that act on a system with N
qubits.
Parameters
----------
U : Qobj
The three-qubit gate
N : integer
The number of qubits in the target space.
controls : list
The list of the control qubits.
target : integer
The index of the target qubit.
Returns
-------
gate : qobj
Quantum object representation of N-qubit gate.
"""
if N < 3:
raise ValueError("integer N must be larger or equal to 3")
if controls[0] >= N or controls[1] >= N or target >= N:
raise ValueError("control and not target is None."
" Must be integer < integer N")
if (controls[0] == target or
controls[1] == target or
controls[0] == controls[1]):
raise ValueError("controls[0], controls[1], and target"
" cannot be equal")
p = list(range(N))
p1 = list(range(N))
p2 = list(range(N))
if controls[0] <= 2 and controls[1] <= 2 and target <= 2:
p[controls[0]] = 0
p[controls[1]] = 1
p[target] = 2
#
# N > 3 cases
#
elif controls[0] == 0 and controls[1] == 1:
p[2], p[target] = p[target], p[2]
elif controls[0] == 0 and target == 2:
p[1], p[controls[1]] = p[controls[1]], p[1]
elif controls[1] == 1 and target == 2:
p[0], p[controls[0]] = p[controls[0]], p[0]
elif controls[0] == 1 and controls[1] == 0:
p[controls[1]], p[controls[0]] = p[controls[0]], p[controls[1]]
p2[2], p2[target] = p2[target], p2[2]
p = [p2[p[k]] for k in range(N)]
elif controls[0] == 2 and target == 0:
p[target], p[controls[0]] = p[controls[0]], p[target]
p1[1], p1[controls[1]] = p1[controls[1]], p1[1]
p = [p1[p[k]] for k in range(N)]
elif controls[1] == 2 and target == 1:
p[target], p[controls[1]] = p[controls[1]], p[target]
p1[0], p1[controls[0]] = p1[controls[0]], p1[0]
p = [p1[p[k]] for k in range(N)]
elif controls[0] == 1 and controls[1] == 2:
# controls[0] -> controls[1] -> target -> outside
p[0], p[1] = p[1], p[0]
p[0], p[2] = p[2], p[0]
p[0], p[target] = p[target], p[0]
elif controls[0] == 2 and target == 1:
# controls[0] -> target -> controls[1] -> outside
p[0], p[2] = p[2], p[0]
p[0], p[1] = p[1], p[0]
p[0], p[controls[1]] = p[controls[1]], p[0]
elif controls[1] == 0 and controls[0] == 2:
# controls[1] -> controls[0] -> target -> outside
p[1], p[0] = p[0], p[1]
p[1], p[2] = p[2], p[1]
p[1], p[target] = p[target], p[1]
elif controls[1] == 2 and target == 0:
# controls[1] -> target -> controls[0] -> outside
p[1], p[2] = p[2], p[1]
p[1], p[0] = p[0], p[1]
p[1], p[controls[0]] = p[controls[0]], p[1]
elif target == 1 and controls[1] == 0:
# target -> controls[1] -> controls[0] -> outside
p[2], p[1] = p[1], p[2]
p[2], p[0] = p[0], p[2]
p[2], p[controls[0]] = p[controls[0]], p[2]
elif target == 0 and controls[0] == 1:
# target -> controls[0] -> controls[1] -> outside
p[2], p[0] = p[0], p[2]
p[2], p[1] = p[1], p[2]
p[2], p[controls[1]] = p[controls[1]], p[2]
elif controls[0] == 0 and controls[1] == 2:
# controls[0] -> self, controls[1] -> target -> outside
p[1], p[2] = p[2], p[1]
p[1], p[target] = p[target], p[1]
elif controls[1] == 1 and controls[0] == 2:
# controls[1] -> self, controls[0] -> target -> outside
p[0], p[2] = p[2], p[0]
p[0], p[target] = p[target], p[0]
elif target == 2 and controls[0] == 1:
# target -> self, controls[0] -> controls[1] -> outside
p[0], p[1] = p[1], p[0]
p[0], p[controls[1]] = p[controls[1]], p[0]
#
# N > 4 cases
#
elif controls[0] == 1 and controls[1] > 2 and target > 2:
# controls[0] -> controls[1] -> outside, target -> outside
p[0], p[1] = p[1], p[0]
p[0], p[controls[1]] = p[controls[1]], p[0]
p[2], p[target] = p[target], p[2]
elif controls[0] == 2 and controls[1] > 2 and target > 2:
# controls[0] -> target -> outside, controls[1] -> outside
p[0], p[2] = p[2], p[0]
p[0], p[target] = p[target], p[0]
p[1], p[controls[1]] = p[controls[1]], p[1]
elif controls[1] == 2 and controls[0] > 2 and target > 2:
# controls[1] -> target -> outside, controls[0] -> outside
p[1], p[2] = p[2], p[1]
p[1], p[target] = p[target], p[1]
p[0], p[controls[0]] = p[controls[0]], p[0]
else:
p[0], p[controls[0]] = p[controls[0]], p[0]
p1[1], p1[controls[1]] = p1[controls[1]], p1[1]
p2[2], p2[target] = p2[target], p2[2]
p = [p[p1[p2[k]]] for k in range(N)]
return tensor([U] + [identity(2)] * (N - 3)).permute(p)
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