# This code is part of Qiskit. # # (C) Copyright IBM 2017, 2019. # # This code is licensed under the Apache License, Version 2.0. You may # obtain a copy of this license in the LICENSE.txt file in the root directory # of this source tree or at http://www.apache.org/licenses/LICENSE-2.0. # # Any modifications or derivative works of this code must retain this # copyright notice, and modified files need to carry a notice indicating # that they have been altered from the originals. """ Implementation of the GraySynth algorithm for synthesizing CNOT-Phase circuits with efficient CNOT cost, and the Patel-Hayes-Markov algorithm for optimal synthesis of linear (CNOT-only) reversible circuits. """ from __future__ import annotations import copy import numpy as np from qiskit.circuit import QuantumCircuit from qiskit.exceptions import QiskitError from qiskit.synthesis.linear import synth_cnot_count_full_pmh def synth_cnot_phase_aam( cnots: list[list[int]], angles: list[str], section_size: int = 2 ) -> QuantumCircuit: r"""This function is an implementation of the `GraySynth` algorithm of Amy, Azimadeh and Mosca. GraySynth is a heuristic algorithm from [1] for synthesizing small parity networks. It is inspired by Gray codes. Given a set of binary strings :math:`S` (called ``cnots`` bellow), the algorithm synthesizes a parity network for :math:`S` by repeatedly choosing an index :math:`i` to expand and then effectively recursing on the co-factors :math:`S_0` and :math:`S_1`, consisting of the strings :math:`y \in S`, with :math:`y_i = 0` or :math:`1` respectively. As a subset :math:`S` is recursively expanded, ``cx`` gates are applied so that a designated target bit contains the (partial) parity :math:`\chi_y(x)` where :math:`y_i = 1` if and only if :math:`y'_i = 1` for all :math:`y' \in S`. If :math:`S` contains a single element :math:`\{y'\}`, then :math:`y = y'`, and the target bit contains the value :math:`\chi_{y'}(x)` as desired. Notably, rather than uncomputing this sequence of ``cx`` (CNOT) gates when a subset :math:`S` is finished being synthesized, the algorithm maintains the invariant that the remaining parities to be computed are expressed over the current state of bits. This allows the algorithm to avoid the 'backtracking' inherent in uncomputing-based methods. The algorithm is described in detail in section 4 of [1]. Args: cnots: A matrix whose columns are the parities to be synthesized e.g.:: [[0, 1, 1, 1, 1, 1], [1, 0, 0, 1, 1, 1], [1, 0, 0, 1, 0, 0], [0, 0, 1, 0, 1, 0]] corresponds to:: x1^x2 + x0 + x0^x3 + x0^x1^x2 + x0^x1^x3 + x0^x1 angles: A list containing all the phase-shift gates which are to be applied, in the same order as in ``cnots``. A number is interpreted as the angle of :math:`p(angle)`, otherwise the elements have to be ``'t'``, ``'tdg'``, ``'s'``, ``'sdg'`` or ``'z'``. section_size: The size of every section in the Patel–Markov–Hayes algorithm. ``section_size`` must be a factor of the number of qubits. Returns: The decomposed quantum circuit. Raises: QiskitError: when dimensions of ``cnots`` and ``angles`` don't align. References: 1. Matthew Amy, Parsiad Azimzadeh, and Michele Mosca. *On the controlled-NOT complexity of controlled-NOT–phase circuits.*, Quantum Science and Technology 4.1 (2018): 015002. `arXiv:1712.01859 `_ """ num_qubits = len(cnots) # Create a quantum circuit on num_qubits qcir = QuantumCircuit(num_qubits) if len(cnots[0]) != len(angles): raise QiskitError('Size of "cnots" and "angles" do not match.') range_list = list(range(num_qubits)) epsilon = num_qubits sta = [] cnots_copy = np.transpose(np.array(copy.deepcopy(cnots))) # This matrix keeps track of the state in the algorithm state = np.eye(num_qubits).astype("int") # Check if some phase-shift gates can be applied, before adding any C-NOT gates for qubit in range(num_qubits): index = 0 for icnots in cnots_copy: if np.array_equal(icnots, state[qubit]): if angles[index] == "t": qcir.t(qubit) elif angles[index] == "tdg": qcir.tdg(qubit) elif angles[index] == "s": qcir.s(qubit) elif angles[index] == "sdg": qcir.sdg(qubit) elif angles[index] == "z": qcir.z(qubit) else: qcir.p(angles[index] % np.pi, qubit) del angles[index] cnots_copy = np.delete(cnots_copy, index, axis=0) if index == len(cnots_copy): break index -= 1 index += 1 # Implementation of the pseudo-code (Algorithm 1) in the aforementioned paper sta.append([cnots, range_list, epsilon]) while sta: [cnots, ilist, qubit] = sta.pop() if cnots == []: continue if 0 <= qubit < num_qubits: condition = True while condition: condition = False for j in range(num_qubits): if (j != qubit) and (sum(cnots[j]) == len(cnots[j])): condition = True qcir.cx(j, qubit) state[qubit] ^= state[j] index = 0 for icnots in cnots_copy: if np.array_equal(icnots, state[qubit]): if angles[index] == "t": qcir.t(qubit) elif angles[index] == "tdg": qcir.tdg(qubit) elif angles[index] == "s": qcir.s(qubit) elif angles[index] == "sdg": qcir.sdg(qubit) elif angles[index] == "z": qcir.z(qubit) else: qcir.p(angles[index] % np.pi, qubit) del angles[index] cnots_copy = np.delete(cnots_copy, index, axis=0) if index == len(cnots_copy): break index -= 1 index += 1 for x in _remove_duplicates(sta + [[cnots, ilist, qubit]]): [cnotsp, _, _] = x if cnotsp == []: continue for ttt in range(len(cnotsp[j])): cnotsp[j][ttt] ^= cnotsp[qubit][ttt] if ilist == []: continue # See line 18 in pseudo-code of Algorithm 1 in the aforementioned paper # this choice of j maximizes the size of the largest subset (S_0 or S_1) # and the larger a subset, the closer it gets to the ideal in the # Gray code of one CNOT per string. j = ilist[np.argmax([[max(row.count(0), row.count(1)) for row in cnots][k] for k in ilist])] cnots0 = [] cnots1 = [] for y in list(map(list, zip(*cnots))): if y[j] == 0: cnots0.append(y) elif y[j] == 1: cnots1.append(y) cnots0 = list(map(list, zip(*cnots0))) cnots1 = list(map(list, zip(*cnots1))) if qubit == epsilon: sta.append([cnots1, list(set(ilist).difference([j])), j]) else: sta.append([cnots1, list(set(ilist).difference([j])), qubit]) sta.append([cnots0, list(set(ilist).difference([j])), qubit]) qcir &= synth_cnot_count_full_pmh(state, section_size).inverse() return qcir def _remove_duplicates(lists): """ Remove duplicates in list Args: lists (list): a list which may contain duplicate elements. Returns: list: a list which contains only unique elements. """ unique_list = [] for element in lists: if element not in unique_list: unique_list.append(element) return unique_list