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controlled_gate_decomposition.py
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controlled_gate_decomposition.py
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# Copyright 2020 The Cirq Developers
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# https://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
from typing import List, Tuple, TYPE_CHECKING
import numpy as np
from cirq import ops
from cirq.linalg import is_unitary, is_special_unitary, map_eigenvalues
from cirq.protocols import unitary
if TYPE_CHECKING:
import cirq
def _unitary_power(matrix: np.ndarray, power: float) -> np.ndarray:
return map_eigenvalues(matrix, lambda e: e**power)
def _is_identity(matrix):
"""Checks whether M is identity."""
return np.allclose(matrix, np.eye(matrix.shape[0]))
def _flatten(x):
return sum(x, [])
def _decompose_abc(matrix: np.ndarray) -> Tuple[np.ndarray, np.ndarray, np.ndarray, float]:
"""Decomposes 2x2 unitary matrix.
Returns 2x2 special unitary matrices A, B, C and phase delta, such that:
* ABC = I.
* AXBXC * exp(1j*delta) = matrix.
See [1], chapter 4.
"""
assert matrix.shape == (2, 2)
delta = np.angle(np.linalg.det(matrix)) * 0.5
alpha = np.angle(matrix[0, 0]) + np.angle(matrix[0, 1]) - 2 * delta
beta = np.angle(matrix[0, 0]) - np.angle(matrix[0, 1])
m00_abs = np.abs(matrix[0, 0])
if np.abs(m00_abs - 1.0) < 1e-9:
m00_abs = 1
theta = 2 * np.arccos(m00_abs)
a = unitary(ops.rz(-alpha)) @ unitary(ops.ry(-theta / 2))
b = unitary(ops.ry(theta / 2)) @ unitary(ops.rz((alpha + beta) / 2))
c = unitary(ops.rz((alpha - beta) / 2))
x = unitary(ops.X)
assert np.allclose(a @ b @ c, np.eye(2), atol=1e-2)
assert np.allclose((a @ x @ b @ x @ c) * np.exp(1j * delta), matrix, atol=1e-2)
return a, b, c, delta
def _decompose_single_ctrl(
matrix: np.ndarray, control: 'cirq.Qid', target: 'cirq.Qid'
) -> List['cirq.Operation']:
"""Decomposes controlled gate with one control.
See [1], chapter 5.1.
"""
a, b, c, delta = _decompose_abc(matrix)
result = [
ops.ZPowGate(exponent=delta / np.pi).on(control),
ops.MatrixGate(c).on(target),
ops.CNOT.on(control, target),
ops.MatrixGate(b).on(target),
ops.CNOT.on(control, target),
ops.MatrixGate(a).on(target),
]
# Remove no-ops.
result = [g for g in result if not _is_identity(unitary(g))]
return result
def _ccnot_congruent(c0: 'cirq.Qid', c1: 'cirq.Qid', target: 'cirq.Qid') -> List['cirq.Operation']:
"""Implements 3-qubit gate 'congruent' to CCNOT.
Returns sequence of operations which is equivalent to applying
CCNOT(c0, c1, target) and multiplying phase of |101> sate by -1.
See lemma 6.2 in [1]."""
return [
ops.ry(-np.pi / 4).on(target),
ops.CNOT(c1, target),
ops.ry(-np.pi / 4).on(target),
ops.CNOT(c0, target),
ops.ry(np.pi / 4).on(target),
ops.CNOT(c1, target),
ops.ry(np.pi / 4).on(target),
]
def decompose_multi_controlled_x(
controls: List['cirq.Qid'], target: 'cirq.Qid', free_qubits: List['cirq.Qid']
) -> List['cirq.Operation']:
"""Implements action of multi-controlled Pauli X gate.
Result is guaranteed to consist exclusively of 1-qubit, CNOT and CCNOT
gates.
If `free_qubits` has at least 1 element, result has lengts
O(len(controls)).
Args:
controls - control qubits.
targets - target qubits.
free_qubits - qubits which are neither controlled nor target. Can be
modified by algorithm, but will end up in their initial state.
"""
m = len(controls)
if m == 0:
return [ops.X.on(target)]
elif m == 1:
return [ops.CNOT.on(controls[0], target)]
elif m == 2:
return [ops.CCNOT.on(controls[0], controls[1], target)]
m = len(controls)
n = m + 1 + len(free_qubits)
if (n >= 2 * m - 1) and (m >= 3):
# See [1], Lemma 7.2.
gates1 = [
_ccnot_congruent(controls[m - 2 - i], free_qubits[m - 4 - i], free_qubits[m - 3 - i])
for i in range(m - 3)
]
gates2 = _ccnot_congruent(controls[0], controls[1], free_qubits[0])
gates3 = _flatten(gates1) + gates2 + _flatten(gates1[::-1])
first_ccnot = ops.CCNOT(controls[m - 1], free_qubits[m - 3], target)
return [first_ccnot, *gates3, first_ccnot, *gates3]
elif len(free_qubits) >= 1:
# See [1], Lemma 7.3.
m1 = n // 2
free1 = controls[m1:] + [target] + free_qubits[1:]
ctrl1 = controls[:m1]
part1 = decompose_multi_controlled_x(ctrl1, free_qubits[0], free1)
free2 = controls[:m1] + free_qubits[1:]
ctrl2 = controls[m1:] + [free_qubits[0]]
part2 = decompose_multi_controlled_x(ctrl2, target, free2)
return [*part1, *part2, *part1, *part2]
else:
# No free qubits - must use general algorithm.
# This will never happen if called from main algorithm and is added
# only for completeness.
return decompose_multi_controlled_rotation(unitary(ops.X), controls, target)
def _decompose_su(
matrix: np.ndarray, controls: List['cirq.Qid'], target: 'cirq.Qid'
) -> List['cirq.Operation']:
"""Decomposes controlled special unitary gate into elementary gates.
Result has O(len(controls)) operations.
See [1], lemma 7.9.
"""
assert matrix.shape == (2, 2)
assert is_special_unitary(matrix)
assert len(controls) >= 1
a, b, c, _ = _decompose_abc(matrix)
cnots = decompose_multi_controlled_x(controls[:-1], target, [controls[-1]])
return [
*_decompose_single_ctrl(c, controls[-1], target),
*cnots,
*_decompose_single_ctrl(b, controls[-1], target),
*cnots,
*_decompose_single_ctrl(a, controls[-1], target),
]
def _decompose_recursive(
matrix: np.ndarray,
power: float,
controls: List['cirq.Qid'],
target: 'cirq.Qid',
free_qubits: List['cirq.Qid'],
) -> List['cirq.Operation']:
"""Decomposes controlled unitary gate into elementary gates.
Result has O(len(controls)^2) operations.
See [1], lemma 7.5.
"""
if len(controls) == 1:
return _decompose_single_ctrl(_unitary_power(matrix, power), controls[0], target)
cnots = decompose_multi_controlled_x(controls[:-1], controls[-1], free_qubits + [target])
return [
*_decompose_single_ctrl(_unitary_power(matrix, 0.5 * power), controls[-1], target),
*cnots,
*_decompose_single_ctrl(_unitary_power(matrix, -0.5 * power), controls[-1], target),
*cnots,
*_decompose_recursive(
matrix, 0.5 * power, controls[:-1], target, [controls[-1]] + free_qubits
),
]
def decompose_multi_controlled_rotation(
matrix: np.ndarray, controls: List['cirq.Qid'], target: 'cirq.Qid'
) -> List['cirq.Operation']:
"""Implements action of multi-controlled unitary gate.
Returns a sequence of operations, which is equivalent to applying
single-qubit gate with matrix `matrix` on `target`, controlled by
`controls`.
Result is guaranteed to consist exclusively of 1-qubit, CNOT and CCNOT
gates.
If matrix is special unitary, result has length `O(len(controls))`.
Otherwise result has length `O(len(controls)**2)`.
References:
[1] Barenco, Bennett et al.
Elementary gates for quantum computation. 1995.
https://arxiv.org/pdf/quant-ph/9503016.pdf
Args:
matrix - 2x2 numpy unitary matrix (of real or complex dtype).
controls - control qubits.
targets - target qubits.
Returns:
A list of operations which, applied in a sequence, are equivalent to
applying `MatrixGate(matrix).on(target).controlled_by(*controls)`.
"""
assert is_unitary(matrix)
assert matrix.shape == (2, 2)
if len(controls) == 0:
return [ops.MatrixGate(matrix).on(target)]
elif len(controls) == 1:
return _decompose_single_ctrl(matrix, controls[0], target)
elif is_special_unitary(matrix):
return _decompose_su(matrix, controls, target)
else:
return _decompose_recursive(matrix, 1.0, controls, target, [])