854 lines
27 KiB
Python
854 lines
27 KiB
Python
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from collections import deque
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from sympy.core.random import randint
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from sympy.external import import_module
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from sympy.core.basic import Basic
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from sympy.core.mul import Mul
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from sympy.core.numbers import Number, equal_valued
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from sympy.core.power import Pow
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from sympy.core.singleton import S
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from sympy.physics.quantum.represent import represent
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from sympy.physics.quantum.dagger import Dagger
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__all__ = [
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# Public interfaces
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'generate_gate_rules',
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'generate_equivalent_ids',
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'GateIdentity',
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'bfs_identity_search',
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'random_identity_search',
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# "Private" functions
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'is_scalar_sparse_matrix',
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'is_scalar_nonsparse_matrix',
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'is_degenerate',
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'is_reducible',
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]
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np = import_module('numpy')
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scipy = import_module('scipy', import_kwargs={'fromlist': ['sparse']})
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def is_scalar_sparse_matrix(circuit, nqubits, identity_only, eps=1e-11):
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"""Checks if a given scipy.sparse matrix is a scalar matrix.
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A scalar matrix is such that B = bI, where B is the scalar
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matrix, b is some scalar multiple, and I is the identity
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matrix. A scalar matrix would have only the element b along
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it's main diagonal and zeroes elsewhere.
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Parameters
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==========
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circuit : Gate tuple
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Sequence of quantum gates representing a quantum circuit
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nqubits : int
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Number of qubits in the circuit
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identity_only : bool
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Check for only identity matrices
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eps : number
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The tolerance value for zeroing out elements in the matrix.
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Values in the range [-eps, +eps] will be changed to a zero.
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"""
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if not np or not scipy:
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pass
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matrix = represent(Mul(*circuit), nqubits=nqubits,
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format='scipy.sparse')
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# In some cases, represent returns a 1D scalar value in place
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# of a multi-dimensional scalar matrix
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if (isinstance(matrix, int)):
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return matrix == 1 if identity_only else True
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# If represent returns a matrix, check if the matrix is diagonal
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# and if every item along the diagonal is the same
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else:
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# Due to floating pointing operations, must zero out
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# elements that are "very" small in the dense matrix
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# See parameter for default value.
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# Get the ndarray version of the dense matrix
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dense_matrix = matrix.todense().getA()
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# Since complex values can't be compared, must split
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# the matrix into real and imaginary components
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# Find the real values in between -eps and eps
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bool_real = np.logical_and(dense_matrix.real > -eps,
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dense_matrix.real < eps)
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# Find the imaginary values between -eps and eps
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bool_imag = np.logical_and(dense_matrix.imag > -eps,
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dense_matrix.imag < eps)
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# Replaces values between -eps and eps with 0
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corrected_real = np.where(bool_real, 0.0, dense_matrix.real)
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corrected_imag = np.where(bool_imag, 0.0, dense_matrix.imag)
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# Convert the matrix with real values into imaginary values
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corrected_imag = corrected_imag * complex(1j)
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# Recombine the real and imaginary components
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corrected_dense = corrected_real + corrected_imag
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# Check if it's diagonal
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row_indices = corrected_dense.nonzero()[0]
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col_indices = corrected_dense.nonzero()[1]
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# Check if the rows indices and columns indices are the same
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# If they match, then matrix only contains elements along diagonal
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bool_indices = row_indices == col_indices
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is_diagonal = bool_indices.all()
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first_element = corrected_dense[0][0]
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# If the first element is a zero, then can't rescale matrix
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# and definitely not diagonal
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if (first_element == 0.0 + 0.0j):
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return False
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# The dimensions of the dense matrix should still
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# be 2^nqubits if there are elements all along the
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# the main diagonal
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trace_of_corrected = (corrected_dense/first_element).trace()
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expected_trace = pow(2, nqubits)
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has_correct_trace = trace_of_corrected == expected_trace
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# If only looking for identity matrices
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# first element must be a 1
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real_is_one = abs(first_element.real - 1.0) < eps
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imag_is_zero = abs(first_element.imag) < eps
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is_one = real_is_one and imag_is_zero
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is_identity = is_one if identity_only else True
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return bool(is_diagonal and has_correct_trace and is_identity)
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def is_scalar_nonsparse_matrix(circuit, nqubits, identity_only, eps=None):
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"""Checks if a given circuit, in matrix form, is equivalent to
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a scalar value.
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Parameters
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==========
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circuit : Gate tuple
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Sequence of quantum gates representing a quantum circuit
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nqubits : int
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Number of qubits in the circuit
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identity_only : bool
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Check for only identity matrices
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eps : number
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This argument is ignored. It is just for signature compatibility with
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is_scalar_sparse_matrix.
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Note: Used in situations when is_scalar_sparse_matrix has bugs
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"""
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matrix = represent(Mul(*circuit), nqubits=nqubits)
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# In some cases, represent returns a 1D scalar value in place
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# of a multi-dimensional scalar matrix
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if (isinstance(matrix, Number)):
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return matrix == 1 if identity_only else True
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# If represent returns a matrix, check if the matrix is diagonal
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# and if every item along the diagonal is the same
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else:
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# Added up the diagonal elements
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matrix_trace = matrix.trace()
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# Divide the trace by the first element in the matrix
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# if matrix is not required to be the identity matrix
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adjusted_matrix_trace = (matrix_trace/matrix[0]
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if not identity_only
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else matrix_trace)
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is_identity = equal_valued(matrix[0], 1) if identity_only else True
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has_correct_trace = adjusted_matrix_trace == pow(2, nqubits)
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# The matrix is scalar if it's diagonal and the adjusted trace
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# value is equal to 2^nqubits
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return bool(
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matrix.is_diagonal() and has_correct_trace and is_identity)
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if np and scipy:
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is_scalar_matrix = is_scalar_sparse_matrix
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else:
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is_scalar_matrix = is_scalar_nonsparse_matrix
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def _get_min_qubits(a_gate):
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if isinstance(a_gate, Pow):
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return a_gate.base.min_qubits
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else:
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return a_gate.min_qubits
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def ll_op(left, right):
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"""Perform a LL operation.
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A LL operation multiplies both left and right circuits
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with the dagger of the left circuit's leftmost gate, and
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the dagger is multiplied on the left side of both circuits.
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If a LL is possible, it returns the new gate rule as a
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2-tuple (LHS, RHS), where LHS is the left circuit and
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and RHS is the right circuit of the new rule.
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If a LL is not possible, None is returned.
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Parameters
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==========
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left : Gate tuple
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The left circuit of a gate rule expression.
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right : Gate tuple
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The right circuit of a gate rule expression.
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Examples
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========
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Generate a new gate rule using a LL operation:
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>>> from sympy.physics.quantum.identitysearch import ll_op
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>>> from sympy.physics.quantum.gate import X, Y, Z
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>>> x = X(0); y = Y(0); z = Z(0)
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>>> ll_op((x, y, z), ())
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((Y(0), Z(0)), (X(0),))
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>>> ll_op((y, z), (x,))
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((Z(0),), (Y(0), X(0)))
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"""
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if (len(left) > 0):
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ll_gate = left[0]
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ll_gate_is_unitary = is_scalar_matrix(
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(Dagger(ll_gate), ll_gate), _get_min_qubits(ll_gate), True)
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if (len(left) > 0 and ll_gate_is_unitary):
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# Get the new left side w/o the leftmost gate
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new_left = left[1:len(left)]
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# Add the leftmost gate to the left position on the right side
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new_right = (Dagger(ll_gate),) + right
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# Return the new gate rule
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return (new_left, new_right)
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return None
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def lr_op(left, right):
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"""Perform a LR operation.
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A LR operation multiplies both left and right circuits
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with the dagger of the left circuit's rightmost gate, and
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the dagger is multiplied on the right side of both circuits.
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If a LR is possible, it returns the new gate rule as a
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2-tuple (LHS, RHS), where LHS is the left circuit and
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and RHS is the right circuit of the new rule.
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If a LR is not possible, None is returned.
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Parameters
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==========
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left : Gate tuple
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The left circuit of a gate rule expression.
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right : Gate tuple
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The right circuit of a gate rule expression.
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Examples
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========
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Generate a new gate rule using a LR operation:
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>>> from sympy.physics.quantum.identitysearch import lr_op
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>>> from sympy.physics.quantum.gate import X, Y, Z
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>>> x = X(0); y = Y(0); z = Z(0)
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>>> lr_op((x, y, z), ())
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((X(0), Y(0)), (Z(0),))
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>>> lr_op((x, y), (z,))
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((X(0),), (Z(0), Y(0)))
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"""
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if (len(left) > 0):
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lr_gate = left[len(left) - 1]
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lr_gate_is_unitary = is_scalar_matrix(
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(Dagger(lr_gate), lr_gate), _get_min_qubits(lr_gate), True)
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if (len(left) > 0 and lr_gate_is_unitary):
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# Get the new left side w/o the rightmost gate
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new_left = left[0:len(left) - 1]
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# Add the rightmost gate to the right position on the right side
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new_right = right + (Dagger(lr_gate),)
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# Return the new gate rule
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return (new_left, new_right)
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return None
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def rl_op(left, right):
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"""Perform a RL operation.
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A RL operation multiplies both left and right circuits
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with the dagger of the right circuit's leftmost gate, and
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the dagger is multiplied on the left side of both circuits.
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If a RL is possible, it returns the new gate rule as a
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2-tuple (LHS, RHS), where LHS is the left circuit and
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and RHS is the right circuit of the new rule.
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If a RL is not possible, None is returned.
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Parameters
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==========
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left : Gate tuple
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The left circuit of a gate rule expression.
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right : Gate tuple
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The right circuit of a gate rule expression.
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Examples
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========
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Generate a new gate rule using a RL operation:
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>>> from sympy.physics.quantum.identitysearch import rl_op
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>>> from sympy.physics.quantum.gate import X, Y, Z
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>>> x = X(0); y = Y(0); z = Z(0)
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>>> rl_op((x,), (y, z))
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((Y(0), X(0)), (Z(0),))
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>>> rl_op((x, y), (z,))
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((Z(0), X(0), Y(0)), ())
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"""
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if (len(right) > 0):
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rl_gate = right[0]
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rl_gate_is_unitary = is_scalar_matrix(
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(Dagger(rl_gate), rl_gate), _get_min_qubits(rl_gate), True)
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if (len(right) > 0 and rl_gate_is_unitary):
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# Get the new right side w/o the leftmost gate
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new_right = right[1:len(right)]
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# Add the leftmost gate to the left position on the left side
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new_left = (Dagger(rl_gate),) + left
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# Return the new gate rule
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return (new_left, new_right)
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return None
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def rr_op(left, right):
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"""Perform a RR operation.
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A RR operation multiplies both left and right circuits
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with the dagger of the right circuit's rightmost gate, and
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the dagger is multiplied on the right side of both circuits.
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If a RR is possible, it returns the new gate rule as a
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2-tuple (LHS, RHS), where LHS is the left circuit and
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and RHS is the right circuit of the new rule.
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If a RR is not possible, None is returned.
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Parameters
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==========
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left : Gate tuple
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The left circuit of a gate rule expression.
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right : Gate tuple
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The right circuit of a gate rule expression.
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Examples
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========
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Generate a new gate rule using a RR operation:
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>>> from sympy.physics.quantum.identitysearch import rr_op
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>>> from sympy.physics.quantum.gate import X, Y, Z
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>>> x = X(0); y = Y(0); z = Z(0)
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>>> rr_op((x, y), (z,))
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((X(0), Y(0), Z(0)), ())
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>>> rr_op((x,), (y, z))
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((X(0), Z(0)), (Y(0),))
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"""
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if (len(right) > 0):
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rr_gate = right[len(right) - 1]
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rr_gate_is_unitary = is_scalar_matrix(
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(Dagger(rr_gate), rr_gate), _get_min_qubits(rr_gate), True)
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if (len(right) > 0 and rr_gate_is_unitary):
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# Get the new right side w/o the rightmost gate
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new_right = right[0:len(right) - 1]
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# Add the rightmost gate to the right position on the right side
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|
new_left = left + (Dagger(rr_gate),)
|
||
|
|
# Return the new gate rule
|
||
|
|
return (new_left, new_right)
|
||
|
|
|
||
|
|
return None
|
||
|
|
|
||
|
|
|
||
|
|
def generate_gate_rules(gate_seq, return_as_muls=False):
|
||
|
|
"""Returns a set of gate rules. Each gate rules is represented
|
||
|
|
as a 2-tuple of tuples or Muls. An empty tuple represents an arbitrary
|
||
|
|
scalar value.
|
||
|
|
|
||
|
|
This function uses the four operations (LL, LR, RL, RR)
|
||
|
|
to generate the gate rules.
|
||
|
|
|
||
|
|
A gate rule is an expression such as ABC = D or AB = CD, where
|
||
|
|
A, B, C, and D are gates. Each value on either side of the
|
||
|
|
equal sign represents a circuit. The four operations allow
|
||
|
|
one to find a set of equivalent circuits from a gate identity.
|
||
|
|
The letters denoting the operation tell the user what
|
||
|
|
activities to perform on each expression. The first letter
|
||
|
|
indicates which side of the equal sign to focus on. The
|
||
|
|
second letter indicates which gate to focus on given the
|
||
|
|
side. Once this information is determined, the inverse
|
||
|
|
of the gate is multiplied on both circuits to create a new
|
||
|
|
gate rule.
|
||
|
|
|
||
|
|
For example, given the identity, ABCD = 1, a LL operation
|
||
|
|
means look at the left value and multiply both left sides by the
|
||
|
|
inverse of the leftmost gate A. If A is Hermitian, the inverse
|
||
|
|
of A is still A. The resulting new rule is BCD = A.
|
||
|
|
|
||
|
|
The following is a summary of the four operations. Assume
|
||
|
|
that in the examples, all gates are Hermitian.
|
||
|
|
|
||
|
|
LL : left circuit, left multiply
|
||
|
|
ABCD = E -> AABCD = AE -> BCD = AE
|
||
|
|
LR : left circuit, right multiply
|
||
|
|
ABCD = E -> ABCDD = ED -> ABC = ED
|
||
|
|
RL : right circuit, left multiply
|
||
|
|
ABC = ED -> EABC = EED -> EABC = D
|
||
|
|
RR : right circuit, right multiply
|
||
|
|
AB = CD -> ABD = CDD -> ABD = C
|
||
|
|
|
||
|
|
The number of gate rules generated is n*(n+1), where n
|
||
|
|
is the number of gates in the sequence (unproven).
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
gate_seq : Gate tuple, Mul, or Number
|
||
|
|
A variable length tuple or Mul of Gates whose product is equal to
|
||
|
|
a scalar matrix
|
||
|
|
return_as_muls : bool
|
||
|
|
True to return a set of Muls; False to return a set of tuples
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
Find the gate rules of the current circuit using tuples:
|
||
|
|
|
||
|
|
>>> from sympy.physics.quantum.identitysearch import generate_gate_rules
|
||
|
|
>>> from sympy.physics.quantum.gate import X, Y, Z
|
||
|
|
>>> x = X(0); y = Y(0); z = Z(0)
|
||
|
|
>>> generate_gate_rules((x, x))
|
||
|
|
{((X(0),), (X(0),)), ((X(0), X(0)), ())}
|
||
|
|
|
||
|
|
>>> generate_gate_rules((x, y, z))
|
||
|
|
{((), (X(0), Z(0), Y(0))), ((), (Y(0), X(0), Z(0))),
|
||
|
|
((), (Z(0), Y(0), X(0))), ((X(0),), (Z(0), Y(0))),
|
||
|
|
((Y(0),), (X(0), Z(0))), ((Z(0),), (Y(0), X(0))),
|
||
|
|
((X(0), Y(0)), (Z(0),)), ((Y(0), Z(0)), (X(0),)),
|
||
|
|
((Z(0), X(0)), (Y(0),)), ((X(0), Y(0), Z(0)), ()),
|
||
|
|
((Y(0), Z(0), X(0)), ()), ((Z(0), X(0), Y(0)), ())}
|
||
|
|
|
||
|
|
Find the gate rules of the current circuit using Muls:
|
||
|
|
|
||
|
|
>>> generate_gate_rules(x*x, return_as_muls=True)
|
||
|
|
{(1, 1)}
|
||
|
|
|
||
|
|
>>> generate_gate_rules(x*y*z, return_as_muls=True)
|
||
|
|
{(1, X(0)*Z(0)*Y(0)), (1, Y(0)*X(0)*Z(0)),
|
||
|
|
(1, Z(0)*Y(0)*X(0)), (X(0)*Y(0), Z(0)),
|
||
|
|
(Y(0)*Z(0), X(0)), (Z(0)*X(0), Y(0)),
|
||
|
|
(X(0)*Y(0)*Z(0), 1), (Y(0)*Z(0)*X(0), 1),
|
||
|
|
(Z(0)*X(0)*Y(0), 1), (X(0), Z(0)*Y(0)),
|
||
|
|
(Y(0), X(0)*Z(0)), (Z(0), Y(0)*X(0))}
|
||
|
|
"""
|
||
|
|
|
||
|
|
if isinstance(gate_seq, Number):
|
||
|
|
if return_as_muls:
|
||
|
|
return {(S.One, S.One)}
|
||
|
|
else:
|
||
|
|
return {((), ())}
|
||
|
|
|
||
|
|
elif isinstance(gate_seq, Mul):
|
||
|
|
gate_seq = gate_seq.args
|
||
|
|
|
||
|
|
# Each item in queue is a 3-tuple:
|
||
|
|
# i) first item is the left side of an equality
|
||
|
|
# ii) second item is the right side of an equality
|
||
|
|
# iii) third item is the number of operations performed
|
||
|
|
# The argument, gate_seq, will start on the left side, and
|
||
|
|
# the right side will be empty, implying the presence of an
|
||
|
|
# identity.
|
||
|
|
queue = deque()
|
||
|
|
# A set of gate rules
|
||
|
|
rules = set()
|
||
|
|
# Maximum number of operations to perform
|
||
|
|
max_ops = len(gate_seq)
|
||
|
|
|
||
|
|
def process_new_rule(new_rule, ops):
|
||
|
|
if new_rule is not None:
|
||
|
|
new_left, new_right = new_rule
|
||
|
|
|
||
|
|
if new_rule not in rules and (new_right, new_left) not in rules:
|
||
|
|
rules.add(new_rule)
|
||
|
|
# If haven't reached the max limit on operations
|
||
|
|
if ops + 1 < max_ops:
|
||
|
|
queue.append(new_rule + (ops + 1,))
|
||
|
|
|
||
|
|
queue.append((gate_seq, (), 0))
|
||
|
|
rules.add((gate_seq, ()))
|
||
|
|
|
||
|
|
while len(queue) > 0:
|
||
|
|
left, right, ops = queue.popleft()
|
||
|
|
|
||
|
|
# Do a LL
|
||
|
|
new_rule = ll_op(left, right)
|
||
|
|
process_new_rule(new_rule, ops)
|
||
|
|
# Do a LR
|
||
|
|
new_rule = lr_op(left, right)
|
||
|
|
process_new_rule(new_rule, ops)
|
||
|
|
# Do a RL
|
||
|
|
new_rule = rl_op(left, right)
|
||
|
|
process_new_rule(new_rule, ops)
|
||
|
|
# Do a RR
|
||
|
|
new_rule = rr_op(left, right)
|
||
|
|
process_new_rule(new_rule, ops)
|
||
|
|
|
||
|
|
if return_as_muls:
|
||
|
|
# Convert each rule as tuples into a rule as muls
|
||
|
|
mul_rules = set()
|
||
|
|
for rule in rules:
|
||
|
|
left, right = rule
|
||
|
|
mul_rules.add((Mul(*left), Mul(*right)))
|
||
|
|
|
||
|
|
rules = mul_rules
|
||
|
|
|
||
|
|
return rules
|
||
|
|
|
||
|
|
|
||
|
|
def generate_equivalent_ids(gate_seq, return_as_muls=False):
|
||
|
|
"""Returns a set of equivalent gate identities.
|
||
|
|
|
||
|
|
A gate identity is a quantum circuit such that the product
|
||
|
|
of the gates in the circuit is equal to a scalar value.
|
||
|
|
For example, XYZ = i, where X, Y, Z are the Pauli gates and
|
||
|
|
i is the imaginary value, is considered a gate identity.
|
||
|
|
|
||
|
|
This function uses the four operations (LL, LR, RL, RR)
|
||
|
|
to generate the gate rules and, subsequently, to locate equivalent
|
||
|
|
gate identities.
|
||
|
|
|
||
|
|
Note that all equivalent identities are reachable in n operations
|
||
|
|
from the starting gate identity, where n is the number of gates
|
||
|
|
in the sequence.
|
||
|
|
|
||
|
|
The max number of gate identities is 2n, where n is the number
|
||
|
|
of gates in the sequence (unproven).
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
gate_seq : Gate tuple, Mul, or Number
|
||
|
|
A variable length tuple or Mul of Gates whose product is equal to
|
||
|
|
a scalar matrix.
|
||
|
|
return_as_muls: bool
|
||
|
|
True to return as Muls; False to return as tuples
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
Find equivalent gate identities from the current circuit with tuples:
|
||
|
|
|
||
|
|
>>> from sympy.physics.quantum.identitysearch import generate_equivalent_ids
|
||
|
|
>>> from sympy.physics.quantum.gate import X, Y, Z
|
||
|
|
>>> x = X(0); y = Y(0); z = Z(0)
|
||
|
|
>>> generate_equivalent_ids((x, x))
|
||
|
|
{(X(0), X(0))}
|
||
|
|
|
||
|
|
>>> generate_equivalent_ids((x, y, z))
|
||
|
|
{(X(0), Y(0), Z(0)), (X(0), Z(0), Y(0)), (Y(0), X(0), Z(0)),
|
||
|
|
(Y(0), Z(0), X(0)), (Z(0), X(0), Y(0)), (Z(0), Y(0), X(0))}
|
||
|
|
|
||
|
|
Find equivalent gate identities from the current circuit with Muls:
|
||
|
|
|
||
|
|
>>> generate_equivalent_ids(x*x, return_as_muls=True)
|
||
|
|
{1}
|
||
|
|
|
||
|
|
>>> generate_equivalent_ids(x*y*z, return_as_muls=True)
|
||
|
|
{X(0)*Y(0)*Z(0), X(0)*Z(0)*Y(0), Y(0)*X(0)*Z(0),
|
||
|
|
Y(0)*Z(0)*X(0), Z(0)*X(0)*Y(0), Z(0)*Y(0)*X(0)}
|
||
|
|
"""
|
||
|
|
|
||
|
|
if isinstance(gate_seq, Number):
|
||
|
|
return {S.One}
|
||
|
|
elif isinstance(gate_seq, Mul):
|
||
|
|
gate_seq = gate_seq.args
|
||
|
|
|
||
|
|
# Filter through the gate rules and keep the rules
|
||
|
|
# with an empty tuple either on the left or right side
|
||
|
|
|
||
|
|
# A set of equivalent gate identities
|
||
|
|
eq_ids = set()
|
||
|
|
|
||
|
|
gate_rules = generate_gate_rules(gate_seq)
|
||
|
|
for rule in gate_rules:
|
||
|
|
l, r = rule
|
||
|
|
if l == ():
|
||
|
|
eq_ids.add(r)
|
||
|
|
elif r == ():
|
||
|
|
eq_ids.add(l)
|
||
|
|
|
||
|
|
if return_as_muls:
|
||
|
|
convert_to_mul = lambda id_seq: Mul(*id_seq)
|
||
|
|
eq_ids = set(map(convert_to_mul, eq_ids))
|
||
|
|
|
||
|
|
return eq_ids
|
||
|
|
|
||
|
|
|
||
|
|
class GateIdentity(Basic):
|
||
|
|
"""Wrapper class for circuits that reduce to a scalar value.
|
||
|
|
|
||
|
|
A gate identity is a quantum circuit such that the product
|
||
|
|
of the gates in the circuit is equal to a scalar value.
|
||
|
|
For example, XYZ = i, where X, Y, Z are the Pauli gates and
|
||
|
|
i is the imaginary value, is considered a gate identity.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
args : Gate tuple
|
||
|
|
A variable length tuple of Gates that form an identity.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
Create a GateIdentity and look at its attributes:
|
||
|
|
|
||
|
|
>>> from sympy.physics.quantum.identitysearch import GateIdentity
|
||
|
|
>>> from sympy.physics.quantum.gate import X, Y, Z
|
||
|
|
>>> x = X(0); y = Y(0); z = Z(0)
|
||
|
|
>>> an_identity = GateIdentity(x, y, z)
|
||
|
|
>>> an_identity.circuit
|
||
|
|
X(0)*Y(0)*Z(0)
|
||
|
|
|
||
|
|
>>> an_identity.equivalent_ids
|
||
|
|
{(X(0), Y(0), Z(0)), (X(0), Z(0), Y(0)), (Y(0), X(0), Z(0)),
|
||
|
|
(Y(0), Z(0), X(0)), (Z(0), X(0), Y(0)), (Z(0), Y(0), X(0))}
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __new__(cls, *args):
|
||
|
|
# args should be a tuple - a variable length argument list
|
||
|
|
obj = Basic.__new__(cls, *args)
|
||
|
|
obj._circuit = Mul(*args)
|
||
|
|
obj._rules = generate_gate_rules(args)
|
||
|
|
obj._eq_ids = generate_equivalent_ids(args)
|
||
|
|
|
||
|
|
return obj
|
||
|
|
|
||
|
|
@property
|
||
|
|
def circuit(self):
|
||
|
|
return self._circuit
|
||
|
|
|
||
|
|
@property
|
||
|
|
def gate_rules(self):
|
||
|
|
return self._rules
|
||
|
|
|
||
|
|
@property
|
||
|
|
def equivalent_ids(self):
|
||
|
|
return self._eq_ids
|
||
|
|
|
||
|
|
@property
|
||
|
|
def sequence(self):
|
||
|
|
return self.args
|
||
|
|
|
||
|
|
def __str__(self):
|
||
|
|
"""Returns the string of gates in a tuple."""
|
||
|
|
return str(self.circuit)
|
||
|
|
|
||
|
|
|
||
|
|
def is_degenerate(identity_set, gate_identity):
|
||
|
|
"""Checks if a gate identity is a permutation of another identity.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
identity_set : set
|
||
|
|
A Python set with GateIdentity objects.
|
||
|
|
gate_identity : GateIdentity
|
||
|
|
The GateIdentity to check for existence in the set.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
Check if the identity is a permutation of another identity:
|
||
|
|
|
||
|
|
>>> from sympy.physics.quantum.identitysearch import (
|
||
|
|
... GateIdentity, is_degenerate)
|
||
|
|
>>> from sympy.physics.quantum.gate import X, Y, Z
|
||
|
|
>>> x = X(0); y = Y(0); z = Z(0)
|
||
|
|
>>> an_identity = GateIdentity(x, y, z)
|
||
|
|
>>> id_set = {an_identity}
|
||
|
|
>>> another_id = (y, z, x)
|
||
|
|
>>> is_degenerate(id_set, another_id)
|
||
|
|
True
|
||
|
|
|
||
|
|
>>> another_id = (x, x)
|
||
|
|
>>> is_degenerate(id_set, another_id)
|
||
|
|
False
|
||
|
|
"""
|
||
|
|
|
||
|
|
# For now, just iteratively go through the set and check if the current
|
||
|
|
# gate_identity is a permutation of an identity in the set
|
||
|
|
for an_id in identity_set:
|
||
|
|
if (gate_identity in an_id.equivalent_ids):
|
||
|
|
return True
|
||
|
|
return False
|
||
|
|
|
||
|
|
|
||
|
|
def is_reducible(circuit, nqubits, begin, end):
|
||
|
|
"""Determines if a circuit is reducible by checking
|
||
|
|
if its subcircuits are scalar values.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
circuit : Gate tuple
|
||
|
|
A tuple of Gates representing a circuit. The circuit to check
|
||
|
|
if a gate identity is contained in a subcircuit.
|
||
|
|
nqubits : int
|
||
|
|
The number of qubits the circuit operates on.
|
||
|
|
begin : int
|
||
|
|
The leftmost gate in the circuit to include in a subcircuit.
|
||
|
|
end : int
|
||
|
|
The rightmost gate in the circuit to include in a subcircuit.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
Check if the circuit can be reduced:
|
||
|
|
|
||
|
|
>>> from sympy.physics.quantum.identitysearch import is_reducible
|
||
|
|
>>> from sympy.physics.quantum.gate import X, Y, Z
|
||
|
|
>>> x = X(0); y = Y(0); z = Z(0)
|
||
|
|
>>> is_reducible((x, y, z), 1, 0, 3)
|
||
|
|
True
|
||
|
|
|
||
|
|
Check if an interval in the circuit can be reduced:
|
||
|
|
|
||
|
|
>>> is_reducible((x, y, z), 1, 1, 3)
|
||
|
|
False
|
||
|
|
|
||
|
|
>>> is_reducible((x, y, y), 1, 1, 3)
|
||
|
|
True
|
||
|
|
"""
|
||
|
|
|
||
|
|
current_circuit = ()
|
||
|
|
# Start from the gate at "end" and go down to almost the gate at "begin"
|
||
|
|
for ndx in reversed(range(begin, end)):
|
||
|
|
next_gate = circuit[ndx]
|
||
|
|
current_circuit = (next_gate,) + current_circuit
|
||
|
|
|
||
|
|
# If a circuit as a matrix is equivalent to a scalar value
|
||
|
|
if (is_scalar_matrix(current_circuit, nqubits, False)):
|
||
|
|
return True
|
||
|
|
|
||
|
|
return False
|
||
|
|
|
||
|
|
|
||
|
|
def bfs_identity_search(gate_list, nqubits, max_depth=None,
|
||
|
|
identity_only=False):
|
||
|
|
"""Constructs a set of gate identities from the list of possible gates.
|
||
|
|
|
||
|
|
Performs a breadth first search over the space of gate identities.
|
||
|
|
This allows the finding of the shortest gate identities first.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
gate_list : list, Gate
|
||
|
|
A list of Gates from which to search for gate identities.
|
||
|
|
nqubits : int
|
||
|
|
The number of qubits the quantum circuit operates on.
|
||
|
|
max_depth : int
|
||
|
|
The longest quantum circuit to construct from gate_list.
|
||
|
|
identity_only : bool
|
||
|
|
True to search for gate identities that reduce to identity;
|
||
|
|
False to search for gate identities that reduce to a scalar.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
Find a list of gate identities:
|
||
|
|
|
||
|
|
>>> from sympy.physics.quantum.identitysearch import bfs_identity_search
|
||
|
|
>>> from sympy.physics.quantum.gate import X, Y, Z
|
||
|
|
>>> x = X(0); y = Y(0); z = Z(0)
|
||
|
|
>>> bfs_identity_search([x], 1, max_depth=2)
|
||
|
|
{GateIdentity(X(0), X(0))}
|
||
|
|
|
||
|
|
>>> bfs_identity_search([x, y, z], 1)
|
||
|
|
{GateIdentity(X(0), X(0)), GateIdentity(Y(0), Y(0)),
|
||
|
|
GateIdentity(Z(0), Z(0)), GateIdentity(X(0), Y(0), Z(0))}
|
||
|
|
|
||
|
|
Find a list of identities that only equal to 1:
|
||
|
|
|
||
|
|
>>> bfs_identity_search([x, y, z], 1, identity_only=True)
|
||
|
|
{GateIdentity(X(0), X(0)), GateIdentity(Y(0), Y(0)),
|
||
|
|
GateIdentity(Z(0), Z(0))}
|
||
|
|
"""
|
||
|
|
|
||
|
|
if max_depth is None or max_depth <= 0:
|
||
|
|
max_depth = len(gate_list)
|
||
|
|
|
||
|
|
id_only = identity_only
|
||
|
|
|
||
|
|
# Start with an empty sequence (implicitly contains an IdentityGate)
|
||
|
|
queue = deque([()])
|
||
|
|
|
||
|
|
# Create an empty set of gate identities
|
||
|
|
ids = set()
|
||
|
|
|
||
|
|
# Begin searching for gate identities in given space.
|
||
|
|
while (len(queue) > 0):
|
||
|
|
current_circuit = queue.popleft()
|
||
|
|
|
||
|
|
for next_gate in gate_list:
|
||
|
|
new_circuit = current_circuit + (next_gate,)
|
||
|
|
|
||
|
|
# Determines if a (strict) subcircuit is a scalar matrix
|
||
|
|
circuit_reducible = is_reducible(new_circuit, nqubits,
|
||
|
|
1, len(new_circuit))
|
||
|
|
|
||
|
|
# In many cases when the matrix is a scalar value,
|
||
|
|
# the evaluated matrix will actually be an integer
|
||
|
|
if (is_scalar_matrix(new_circuit, nqubits, id_only) and
|
||
|
|
not is_degenerate(ids, new_circuit) and
|
||
|
|
not circuit_reducible):
|
||
|
|
ids.add(GateIdentity(*new_circuit))
|
||
|
|
|
||
|
|
elif (len(new_circuit) < max_depth and
|
||
|
|
not circuit_reducible):
|
||
|
|
queue.append(new_circuit)
|
||
|
|
|
||
|
|
return ids
|
||
|
|
|
||
|
|
|
||
|
|
def random_identity_search(gate_list, numgates, nqubits):
|
||
|
|
"""Randomly selects numgates from gate_list and checks if it is
|
||
|
|
a gate identity.
|
||
|
|
|
||
|
|
If the circuit is a gate identity, the circuit is returned;
|
||
|
|
Otherwise, None is returned.
|
||
|
|
"""
|
||
|
|
|
||
|
|
gate_size = len(gate_list)
|
||
|
|
circuit = ()
|
||
|
|
|
||
|
|
for i in range(numgates):
|
||
|
|
next_gate = gate_list[randint(0, gate_size - 1)]
|
||
|
|
circuit = circuit + (next_gate,)
|
||
|
|
|
||
|
|
is_scalar = is_scalar_matrix(circuit, nqubits, False)
|
||
|
|
|
||
|
|
return circuit if is_scalar else None
|