C贸digo fuente para qiskit.quantum_info.analysis.z2_symmetries

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# (C) Copyright IBM 2022.
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"""Z2Symmetries for SparsePauliOp."""

from __future__ import annotations

import itertools
from collections.abc import Iterable
from copy import deepcopy
from typing import Union, cast

import numpy as np

from qiskit.exceptions import QiskitError
from ..operators import Pauli, SparsePauliOp


[documentos]class Z2Symmetries: r""" The $Z_2$ symmetry converter identifies symmetries from the problem hamiltonian and uses them to provide a tapered - more efficient - representation of operators as Paulis for this problem. For each identified symmetry, one qubit can be eliminated in the Pauli representation at the cost of having to test two symmetry sectors (for the two possible eigenvalues - tapering values - of the symmetry). In certain problems such as the finding of the main operator's ground state, one can a priori identify the symmetry sector of the solution and thus effectively reduce the computational overhead. The following attributes can be read and updated once the ``Z2Symmetries`` object has been constructed. Attributes: tapering_values (list[int] or None): Values determining the sector. tol (float): The tolerance threshold for ignoring real and complex parts of a coefficient. References: [1]: Bravyi, S., et al, "Tapering off qubits to simulate fermionic Hamiltonians" `arXiv:1701.08213 <https://arxiv.org/abs/1701.08213>`__ """ def __init__( self, symmetries: Iterable[Pauli], sq_paulis: Iterable[Pauli], sq_list: Iterable[int], tapering_values: Iterable[int] | None = None, *, tol: float = 1e-14, ): r""" Args: symmetries: Object representing the list of $Z_2$ symmetries. These correspond to the generators of the symmetry group $\langle \tau_1, \tau_2\dots \rangle>$. sq_paulis: Object representing the list of single-qubit Pauli $\sigma^x_{q(i)}$ anti-commuting with the symmetry $\tau_i$ and commuting with all the other symmetries $\tau_{j\neq i}$. These operators are used to construct the unitary Clifford operators. sq_list: The list of indices $q(i)$ of the single-qubit Pauli operators used to build the Clifford operators. tapering_values: List of eigenvalues determining the symmetry sector for each symmetry. tol: Tolerance threshold for ignoring real and complex parts of a coefficient. Raises: QiskitError: Invalid paulis. The lists of symmetries, single-qubit paulis support paulis and tapering values must be of equal length. This length is the number of applied symmetries and translates directly to the number of eliminated qubits. """ symmetries = list(symmetries) sq_paulis = list(sq_paulis) sq_list = list(sq_list) tapering_values = None if tapering_values is None else list(tapering_values) if len(symmetries) != len(sq_paulis): raise QiskitError( f"The number of Z2 symmetries, {len(symmetries)}, has to match the number \ of single-qubit pauli operators, {len(sq_paulis)}." ) if len(sq_paulis) != len(sq_list): raise QiskitError( f"The number of single-qubit pauli operators, {len(sq_paulis)}, has to match the length \ of the of single-qubit list, {len(sq_list)}." ) if tapering_values is not None: if len(sq_list) != len(tapering_values): raise QiskitError( f"The length of the single-qubit list, {len(sq_list)}, must match the length of the \ tapering values, {len(tapering_values)} ." ) self._symmetries = symmetries self._sq_paulis = sq_paulis self._sq_list = sq_list self.tapering_values = tapering_values self.tol = tol @property def symmetries(self) -> list[Pauli]: """Return symmetries.""" return self._symmetries @property def sq_paulis(self) -> list[Pauli]: """Return sq paulis.""" return self._sq_paulis @property def cliffords(self) -> list[SparsePauliOp]: """ Get clifford operators, built based on symmetries and single-qubit X. Returns: A list of unitaries used to diagonalize the Hamiltonian. """ cliffords = [ (SparsePauliOp(pauli_symm) + SparsePauliOp(sq_pauli)) / np.sqrt(2) for pauli_symm, sq_pauli in zip(self._symmetries, self._sq_paulis) ] return cliffords @property def sq_list(self) -> list[int]: """Return sq list.""" return self._sq_list @property def settings(self) -> dict: """Return operator settings.""" return { "symmetries": self._symmetries, "sq_paulis": self._sq_paulis, "sq_list": self._sq_list, "tapering_values": self.tapering_values, } def __str__(self): ret = ["Z2 symmetries:"] ret.append("Symmetries:") for symmetry in self._symmetries: ret.append(symmetry.to_label()) ret.append("Single-Qubit Pauli X:") for x in self._sq_paulis: ret.append(x.to_label()) ret.append("Cliffords:") for c in self.cliffords: ret.append(str(c)) ret.append("Qubit index:") ret.append(str(self._sq_list)) ret.append("Tapering values:") if self.tapering_values is None: possible_values = [ str(list(coeff)) for coeff in itertools.product([1, -1], repeat=len(self._sq_list)) ] possible_values = ", ".join(x for x in possible_values) ret.append(" - Possible values: " + possible_values) else: ret.append(str(self.tapering_values)) ret = "\n".join(ret) return ret
[documentos] def is_empty(self) -> bool: """ Check the z2_symmetries is empty or not. Returns: Empty or not. """ return len(self._symmetries) == 0 or len(self._sq_paulis) == 0 or len(self._sq_list) == 0
[documentos] @classmethod def find_z2_symmetries(cls, operator: SparsePauliOp) -> Z2Symmetries: """ Finds Z2 Pauli-type symmetries of a :class:`.SparsePauliOp`. Returns: A ``Z2Symmetries`` instance. """ pauli_symmetries = [] sq_paulis = [] sq_list = [] stacked_paulis = [] test_idx = { "X_or_I": [(0, 0), (1, 0)], "Y_or_I": [(0, 0), (1, 1)], "Z_or_I": [(0, 0), (0, 1)], } test_row = { "Z_or_I": [(1, 0), (1, 1)], "X_or_I": [(0, 1), (1, 1)], "Y_or_I": [(0, 1), (1, 0)], } pauli_bool = { "Z_or_I": [False, True], "X_or_I": [True, False], "Y_or_I": [True, True], } if _sparse_pauli_op_is_zero(operator): return cls([], [], [], None) for pauli in iter(operator): stacked_paulis.append( np.concatenate((pauli.paulis.x[0], pauli.paulis.z[0]), axis=0).astype(int) ) stacked_matrix = np.stack(stacked_paulis) symmetries = _kernel_f2(stacked_matrix) if not symmetries: return cls([], [], [], None) stacked_symmetries = np.stack(symmetries) symm_shape = stacked_symmetries.shape half_symm_shape = symm_shape[1] // 2 stacked_symm_del = [ np.delete(stacked_symmetries, row, axis=0) for row in range(symm_shape[0]) ] def _test_symmetry_row_col(row: int, col: int, idx_test: list, row_test: list) -> bool: """ Utility method that determines how to build the list of single-qubit Pauli X operators and the list of corresponding qubit indices from the stacked symmetries. This method is successively applied to Z type, X type and Y type symmetries (in this order) to build the letter at position (col) of the Pauli word corresponding to the symmetry at position (row). Args: row (int): Index of the symmetry for which the single-qubit Pauli X operator is being built. col (int): Index of the letter in the Pauli word corresponding to the single-qubit Pauli X operator. idx_test (list): List of possibilities for the stacked symmetries at all other rows than row. row_test (list): List of possibilities for the stacked symmetries at row. Returns: Whether or not this symmetry type should be used to build this letter of this single-qubit Pauli X operator. """ stacked_symm_idx_tests = np.array( [ ( stacked_symm_del[row][symm_idx, col], stacked_symm_del[row][symm_idx, col + half_symm_shape], ) in idx_test for symm_idx in range(symm_shape[0] - 1) ] ) stacked_symm_row_test = ( stacked_symmetries[row, col], stacked_symmetries[row, col + half_symm_shape], ) in row_test return bool(np.all(stacked_symm_idx_tests)) and stacked_symm_row_test for row in range(symm_shape[0]): pauli_symmetries.append( Pauli( ( stacked_symmetries[row, :half_symm_shape], stacked_symmetries[row, half_symm_shape:], ) ) ) # Try all cases for the symmetries other than row: Z or I, Y or I, X or I on col qubit. # One test will return true. # Build the single-qubit Pauli accordingly. # Build the index list accordingly. for col in range(half_symm_shape): for key in ("Z_or_I", "X_or_I", "Y_or_I"): current_test_result = _test_symmetry_row_col( row, col, test_idx[key], test_row[key] ) if current_test_result: sq_paulis.append( Pauli((np.zeros(half_symm_shape), np.zeros(half_symm_shape))) ) sq_paulis[row].z[col] = pauli_bool[key][0] sq_paulis[row].x[col] = pauli_bool[key][1] sq_list.append(col) break if current_test_result: # We break out of the loop over columns only when one valid test is identified. break return cls(pauli_symmetries, sq_paulis, sq_list, None)
[documentos] def convert_clifford(self, operator: SparsePauliOp) -> SparsePauliOp: """This method operates the first part of the tapering. It converts the operator by composing it with the clifford unitaries defined in the current symmetry. Args: operator: The to-be-tapered operator. Returns: ``SparsePauliOp`` corresponding to the converted operator. """ if not self.is_empty() and not _sparse_pauli_op_is_zero(operator): # If the operator is zero then we can skip the following. for clifford in self.cliffords: operator = cast(SparsePauliOp, clifford @ operator @ clifford) operator = operator.simplify(atol=0.0) return operator
[documentos] def taper_clifford(self, operator: SparsePauliOp) -> Union[SparsePauliOp, list[SparsePauliOp]]: """Operate the second part of the tapering. This function assumes that the input operators have already been transformed using :meth:`convert_clifford`. The redundant qubits due to the symmetries are dropped and replaced by their two possible eigenvalues. Args: operator: Partially tapered operator resulting from a call to :meth:`convert_clifford`. Returns: If tapering_values is None: [:class:`SparsePauliOp`]; otherwise, :class:`SparsePauliOp`. """ tapered_ops: Union[SparsePauliOp, list[SparsePauliOp]] if self.is_empty(): tapered_ops = operator else: # If the operator is zero we still need to taper the operator to reduce its size i.e. the # number of qubits so for example 0*"IIII" could taper to 0*"II" when symmetries remove # two qubits. if self.tapering_values is None: tapered_ops = [ self._taper(operator, list(coeff)) for coeff in itertools.product([1, -1], repeat=len(self._sq_list)) ] else: tapered_ops = self._taper(operator, self.tapering_values) return tapered_ops
[documentos] def taper(self, operator: SparsePauliOp) -> Union[SparsePauliOp, list[SparsePauliOp]]: """ Taper an operator based on the z2_symmetries info and sector defined by `tapering_values`. Returns operator if the symmetry object is empty. The tapering is a two-step algorithm which first converts the operator into a :class:`SparsePauliOp` with same eigenvalues but where some qubits are only acted upon with the Pauli operators I or X. The number M of these redundant qubits is equal to the number M of identified symmetries. The second step of the reduction consists in replacing these qubits with the possible eigenvalues of the corresponding Pauli X, giving 2^M new operators with M less qubits. If an eigenvalue sector was previously identified for the solution, then this reduces to 1 new operator with M less qubits. Args: operator: The to-be-tapered operator. Returns: If tapering_values is None: [:class:`SparsePauliOp`]; otherwise, :class:`SparsePauliOp`. """ converted_ops = self.convert_clifford(operator) tapered_ops = self.taper_clifford(converted_ops) return tapered_ops
def _taper(self, op: SparsePauliOp, curr_tapering_values: list[int]) -> SparsePauliOp: pauli_list = [] for pauli_term in iter(op): coeff_out = pauli_term.coeffs[0] for idx, qubit_idx in enumerate(self._sq_list): if pauli_term.paulis.z[0, qubit_idx] or pauli_term.paulis.x[0, qubit_idx]: coeff_out = curr_tapering_values[idx] * coeff_out z_temp = np.delete(pauli_term.paulis.z[0].copy(), np.asarray(self._sq_list)) x_temp = np.delete(pauli_term.paulis.x[0].copy(), np.asarray(self._sq_list)) pauli_list.append((Pauli((z_temp, x_temp)).to_label(), coeff_out)) spo = SparsePauliOp.from_list(pauli_list).simplify(atol=0.0) spo = spo.chop(self.tol) return spo def __eq__(self, other: Z2Symmetries) -> bool: """ Overload `==` operation to evaluate equality between Z2Symmetries. Args: other: The `Z2Symmetries` to compare to self. Returns: A bool equal to the equality of self and other. """ if not isinstance(other, Z2Symmetries): return False return ( self.symmetries == other.symmetries and self.sq_paulis == other.sq_paulis and self.sq_list == other.sq_list and self.tapering_values == other.tapering_values )
def _kernel_f2(matrix_in): """ Compute the kernel of a binary matrix on the binary finite field. Args: matrix_in (numpy.ndarray): Binary matrix. Returns: The list of kernel vectors. """ size = matrix_in.shape kernel = [] matrix_in_id = np.vstack((matrix_in, np.identity(size[1]))) matrix_in_id_ech = (_row_echelon_f2(matrix_in_id.transpose())).transpose() for col in range(size[1]): if np.array_equal( matrix_in_id_ech[0 : size[0], col], np.zeros(size[0]) ) and not np.array_equal(matrix_in_id_ech[size[0] :, col], np.zeros(size[1])): kernel.append(matrix_in_id_ech[size[0] :, col]) return kernel def _row_echelon_f2(matrix_in): """ Compute the row Echelon form of a binary matrix on the binary finite field. Args: matrix_in (numpy.ndarray): Binary matrix. Returns: Matrix_in in Echelon row form. """ size = matrix_in.shape for i in range(size[0]): pivot_index = 0 for j in range(size[1]): if matrix_in[i, j] == 1: pivot_index = j break for k in range(size[0]): if k != i and matrix_in[k, pivot_index] == 1: matrix_in[k, :] = np.mod(matrix_in[k, :] + matrix_in[i, :], 2) matrix_out_temp = deepcopy(matrix_in) indices = [] matrix_out = np.zeros(size) for i in range(size[0] - 1): if np.array_equal(matrix_out_temp[i, :], np.zeros(size[1])): indices.append(i) for row in np.sort(indices)[::-1]: matrix_out_temp = np.delete(matrix_out_temp, (row), axis=0) matrix_out[0 : size[0] - len(indices), :] = matrix_out_temp matrix_out = matrix_out.astype(int) return matrix_out def _sparse_pauli_op_is_zero(op: SparsePauliOp) -> bool: """Returns whether or not this operator represents a zero operation.""" op = op.simplify() return len(op.coeffs) == 1 and op.coeffs[0] == 0