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Source code for qiskit.quantum_info.operators.symplectic.pauli_list

# This code is part of Qiskit.
#
# (C) Copyright IBM 2017, 2022
#
# 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.
"""
Optimized list of Pauli operators
"""

from collections import defaultdict

import numpy as np
import retworkx as rx

from qiskit.exceptions import QiskitError
from qiskit.quantum_info.operators.custom_iterator import CustomIterator
from qiskit.quantum_info.operators.mixins import GroupMixin, LinearMixin
from qiskit.quantum_info.operators.symplectic.base_pauli import BasePauli
from qiskit.quantum_info.operators.symplectic.pauli import Pauli
from qiskit.quantum_info.operators.symplectic.pauli_table import PauliTable
from qiskit.quantum_info.operators.symplectic.stabilizer_table import StabilizerTable


[docs]class PauliList(BasePauli, LinearMixin, GroupMixin): r"""List of N-qubit Pauli operators. This class is an efficient representation of a list of :class:`Pauli` operators. It supports 1D numpy array indexing returning a :class:`Pauli` for integer indexes or a :class:`PauliList` for slice or list indices. **Initialization** A PauliList object can be initialized in several ways. ``PauliList(list[str])`` where strings are same representation with :class:`~qiskit.quantum_info.Pauli`. ``PauliList(Pauli) and PauliList(list[Pauli])`` where Pauli is :class:`~qiskit.quantum_info.Pauli`. ``PauliList.from_symplectic(z, x, phase)`` where ``z`` and ``x`` are 2 dimensional boolean ``numpy.ndarrays`` and ``phase`` is an integer in ``[0, 1, 2, 3]``. For example, .. jupyter-execute:: import numpy as np from qiskit.quantum_info import Pauli, PauliList # 1. init from list[str] pauli_list = PauliList(["II", "+ZI", "-iYY"]) print("1. ", pauli_list) pauli1 = Pauli("iXI") pauli2 = Pauli("iZZ") # 2. init from Pauli print("2. ", PauliList(pauli1)) # 3. init from list[Pauli] print("3. ", PauliList([pauli1, pauli2])) # 4. init from np.ndarray z = np.array([[True, True], [False, False]]) x = np.array([[False, True], [True, False]]) phase = np.array([0, 1]) pauli_list = PauliList.from_symplectic(z, x, phase) print("4. ", pauli_list) **Data Access** The individual Paulis can be accessed and updated using the ``[]`` operator which accepts integer, lists, or slices for selecting subsets of PauliList. If integer is given, it returns Pauli not PauliList. .. jupyter-execute:: pauli_list = PauliList(["XX", "ZZ", "IZ"]) print("Integer: ", repr(pauli_list[1])) print("List: ", repr(pauli_list[[0, 2]])) print("Slice: ", repr(pauli_list[0:2])) **Iteration** Rows in the Pauli table can be iterated over like a list. Iteration can also be done using the label or matrix representation of each row using the :meth:`label_iter` and :meth:`matrix_iter` methods. """ # Set the max number of qubits * paulis before string truncation __truncate__ = 2000 def __init__(self, data): """Initialize the PauliList. Args: data (Pauli or list): input data for Paulis. If input is a list each item in the list must be a Pauli object or Pauli str. Raises: QiskitError: if input array is invalid shape. Additional Information: The input array is not copied so multiple Pauli tables can share the same underlying array. """ if isinstance(data, BasePauli): base_z, base_x, base_phase = data._z, data._x, data._phase elif isinstance(data, StabilizerTable): # Conversion from legacy StabilizerTable base_z, base_x, base_phase = self._from_array(data.Z, data.X, 2 * data.phase) elif isinstance(data, PauliTable): # Conversion from legacy PauliTable base_z, base_x, base_phase = self._from_array(data.Z, data.X) else: # Conversion as iterable of Paulis base_z, base_x, base_phase = self._from_paulis(data) # Initialize BasePauli super().__init__(base_z, base_x, base_phase) # --------------------------------------------------------------------- # Representation conversions # --------------------------------------------------------------------- @property def settings(self): """Return settings.""" return {"data": self.to_labels()} def __array__(self, dtype=None): """Convert to numpy array""" # pylint: disable=unused-argument shape = (len(self),) + 2 * (2**self.num_qubits,) ret = np.zeros(shape, dtype=complex) for i, mat in enumerate(self.matrix_iter()): ret[i] = mat return ret @staticmethod def _from_paulis(data): """Construct a PauliList from a list of Pauli data. Args: data (iterable): list of Pauli data. Returns: PauliList: the constructed PauliList. Raises: QiskitError: If the input list is empty or contains invalid Pauli strings. """ if not isinstance(data, (list, tuple, set, np.ndarray)): data = [data] num_paulis = len(data) if num_paulis == 0: raise QiskitError("Input Pauli list is empty.") paulis = [] for i in data: if not isinstance(i, Pauli): paulis.append(Pauli(i)) else: paulis.append(i) num_qubits = paulis[0].num_qubits base_z = np.zeros((num_paulis, num_qubits), dtype=bool) base_x = np.zeros((num_paulis, num_qubits), dtype=bool) base_phase = np.zeros(num_paulis, dtype=int) for i, pauli in enumerate(paulis): base_z[i] = pauli._z base_x[i] = pauli._x base_phase[i] = pauli._phase return base_z, base_x, base_phase def __repr__(self): """Display representation.""" return self._truncated_str(True) def __str__(self): """Print representation.""" return self._truncated_str(False) def _truncated_str(self, show_class): stop = self._num_paulis if self.__truncate__ and self.num_qubits > 0: max_paulis = self.__truncate__ // self.num_qubits if self._num_paulis > max_paulis: stop = max_paulis labels = [str(self[i]) for i in range(stop)] prefix = "PauliList(" if show_class else "" tail = ")" if show_class else "" if stop != self._num_paulis: suffix = ", ...]" + tail else: suffix = "]" + tail list_str = np.array2string( np.array(labels), threshold=stop + 1, separator=", ", prefix=prefix, suffix=suffix ) return prefix + list_str[:-1] + suffix def __eq__(self, other): """Entrywise comparison of Pauli equality.""" if not isinstance(other, PauliList): other = PauliList(other) if not isinstance(other, BasePauli): return False return self._eq(other)
[docs] def equiv(self, other): """Entrywise comparison of Pauli equivalence up to global phase. Args: other (PauliList or Pauli): a comparison object. Returns: np.ndarray: An array of True or False for entrywise equivalence of the current table. """ if not isinstance(other, PauliList): other = PauliList(other) return np.all(self.z == other.z, axis=1) & np.all(self.x == other.x, axis=1)
# --------------------------------------------------------------------- # Direct array access # --------------------------------------------------------------------- @property def phase(self): """Return the phase exponent of the PauliList.""" # Convert internal ZX-phase convention to group phase convention return np.mod(self._phase - self._count_y(dtype=self._phase.dtype), 4) @phase.setter def phase(self, value): # Convert group phase convetion to internal ZX-phase convention self._phase[:] = np.mod(value + self._count_y(dtype=self._phase.dtype), 4) @property def x(self): """The x array for the symplectic representation.""" return self._x @x.setter def x(self, val): self._x[:] = val @property def z(self): """The z array for the symplectic representation.""" return self._z @z.setter def z(self, val): self._z[:] = val # --------------------------------------------------------------------- # Size Properties # --------------------------------------------------------------------- @property def shape(self): """The full shape of the :meth:`array`""" return self._num_paulis, self.num_qubits @property def size(self): """The number of Pauli rows in the table.""" return self._num_paulis def __len__(self): """Return the number of Pauli rows in the table.""" return self._num_paulis # --------------------------------------------------------------------- # Pauli Array methods # --------------------------------------------------------------------- def __getitem__(self, index): """Return a view of the PauliList.""" # Returns a view of specified rows of the PauliList # This supports all slicing operations the underlying array supports. if isinstance(index, tuple): if len(index) == 1: index = index[0] elif len(index) > 2: raise IndexError(f"Invalid PauliList index {index}") # Row-only indexing if isinstance(index, (int, np.integer)): # Single Pauli return Pauli( BasePauli( self._z[np.newaxis, index], self._x[np.newaxis, index], self._phase[np.newaxis, index], ) ) elif isinstance(index, (slice, list, np.ndarray)): # Sub-Table view return PauliList(BasePauli(self._z[index], self._x[index], self._phase[index])) # Row and Qubit indexing return PauliList((self._z[index], self._x[index], 0)) def __setitem__(self, index, value): """Update PauliList.""" if isinstance(index, tuple): if len(index) == 1: index = index[0] elif len(index) > 2: raise IndexError(f"Invalid PauliList index {index}") # Modify specified rows of the PauliList if not isinstance(value, PauliList): value = PauliList(value) self._z[index] = value._z self._x[index] = value._x if not isinstance(index, tuple): # Row-only indexing self._phase[index] = value._phase else: # Row and Qubit indexing self._phase[index[0]] += value._phase self._phase %= 4
[docs] def delete(self, ind, qubit=False): """Return a copy with Pauli rows deleted from table. When deleting qubits the qubit index is the same as the column index of the underlying :attr:`X` and :attr:`Z` arrays. Args: ind (int or list): index(es) to delete. qubit (bool): if True delete qubit columns, otherwise delete Pauli rows (Default: False). Returns: PauliList: the resulting table with the entries removed. Raises: QiskitError: if ind is out of bounds for the array size or number of qubits. """ if isinstance(ind, int): ind = [ind] # Row deletion if not qubit: if max(ind) >= len(self): raise QiskitError( "Indices {} are not all less than the size" " of the PauliList ({})".format(ind, len(self)) ) z = np.delete(self._z, ind, axis=0) x = np.delete(self._x, ind, axis=0) phase = np.delete(self._phase, ind) return PauliList(BasePauli(z, x, phase)) # Column (qubit) deletion if max(ind) >= self.num_qubits: raise QiskitError( "Indices {} are not all less than the number of" " qubits in the PauliList ({})".format(ind, self.num_qubits) ) z = np.delete(self._z, ind, axis=1) x = np.delete(self._x, ind, axis=1) # Use self.phase, not self._phase as deleting qubits can change the # ZX phase convention return PauliList.from_symplectic(z, x, self.phase)
[docs] def insert(self, ind, value, qubit=False): """Insert Pauli's into the table. When inserting qubits the qubit index is the same as the column index of the underlying :attr:`X` and :attr:`Z` arrays. Args: ind (int): index to insert at. value (PauliList): values to insert. qubit (bool): if True delete qubit columns, otherwise delete Pauli rows (Default: False). Returns: PauliList: the resulting table with the entries inserted. Raises: QiskitError: if the insertion index is invalid. """ if not isinstance(ind, int): raise QiskitError("Insert index must be an integer.") if not isinstance(value, PauliList): value = PauliList(value) # Row insertion size = self._num_paulis if not qubit: if ind > size: raise QiskitError( "Index {} is larger than the number of rows in the" " PauliList ({}).".format(ind, size) ) base_z = np.insert(self._z, ind, value._z, axis=0) base_x = np.insert(self._x, ind, value._x, axis=0) base_phase = np.insert(self._phase, ind, value._phase) return PauliList(BasePauli(base_z, base_x, base_phase)) # Column insertion if ind > self.num_qubits: raise QiskitError( "Index {} is greater than number of qubits" " in the PauliList ({})".format(ind, self.num_qubits) ) if len(value) == 1: # Pad blocks to correct size value_x = np.vstack(size * [value.x]) value_z = np.vstack(size * [value.z]) value_phase = np.vstack(size * [value.phase]) elif len(value) == size: # Blocks are already correct size value_x = value.x value_z = value.z value_phase = value.phase else: # Blocks are incorrect size raise QiskitError( "Input PauliList must have a single row, or" " the same number of rows as the Pauli Table" " ({}).".format(size) ) # Build new array by blocks z = np.hstack([self.z[:, :ind], value_z, self.z[:, ind:]]) x = np.hstack([self.x[:, :ind], value_x, self.x[:, ind:]]) phase = self.phase + value_phase return PauliList.from_symplectic(z, x, phase)
[docs] def argsort(self, weight=False, phase=False): """Return indices for sorting the rows of the table. The default sort method is lexicographic sorting by qubit number. By using the `weight` kwarg the output can additionally be sorted by the number of non-identity terms in the Pauli, where the set of all Pauli's of a given weight are still ordered lexicographically. Args: weight (bool): Optionally sort by weight if True (Default: False). phase (bool): Optionally sort by phase before weight or order (Default: False). Returns: array: the indices for sorting the table. """ # Get order of each Pauli using # I => 0, X => 1, Y => 2, Z => 3 x = self.x z = self.z order = 1 * (x & ~z) + 2 * (x & z) + 3 * (~x & z) phases = self.phase # Optionally get the weight of Pauli # This is the number of non identity terms if weight: weights = np.sum(x | z, axis=1) # To preserve ordering between successive sorts we # are use the 'stable' sort method indices = np.arange(self._num_paulis) # Initial sort by phases sort_inds = phases.argsort(kind="stable") indices = indices[sort_inds] order = order[sort_inds] if phase: phases = phases[sort_inds] if weight: weights = weights[sort_inds] # Sort by order for i in range(self.num_qubits): sort_inds = order[:, i].argsort(kind="stable") order = order[sort_inds] indices = indices[sort_inds] if weight: weights = weights[sort_inds] if phase: phases = phases[sort_inds] # If using weights we implement a sort by total number # of non-identity Paulis if weight: sort_inds = weights.argsort(kind="stable") indices = indices[sort_inds] phases = phases[sort_inds] # If sorting by phase we perform a final sort by the phase value # of each pauli if phase: indices = indices[phases.argsort(kind="stable")] return indices
[docs] def sort(self, weight=False, phase=False): """Sort the rows of the table. The default sort method is lexicographic sorting by qubit number. By using the `weight` kwarg the output can additionally be sorted by the number of non-identity terms in the Pauli, where the set of all Pauli's of a given weight are still ordered lexicographically. **Example** Consider sorting all a random ordering of all 2-qubit Paulis .. jupyter-execute:: from numpy.random import shuffle from qiskit.quantum_info.operators import PauliList # 2-qubit labels labels = ['II', 'IX', 'IY', 'IZ', 'XI', 'XX', 'XY', 'XZ', 'YI', 'YX', 'YY', 'YZ', 'ZI', 'ZX', 'ZY', 'ZZ'] # Shuffle Labels shuffle(labels) pt = PauliList(labels) print('Initial Ordering') print(pt) # Lexicographic Ordering srt = pt.sort() print('Lexicographically sorted') print(srt) # Weight Ordering srt = pt.sort(weight=True) print('Weight sorted') print(srt) Args: weight (bool): optionally sort by weight if True (Default: False). phase (bool): Optionally sort by phase before weight or order (Default: False). Returns: PauliList: a sorted copy of the original table. """ return self[self.argsort(weight=weight, phase=phase)]
[docs] def unique(self, return_index=False, return_counts=False): """Return unique Paulis from the table. **Example** .. jupyter-execute:: from qiskit.quantum_info.operators import PauliList pt = PauliList(['X', 'Y', '-X', 'I', 'I', 'Z', 'X', 'iZ']) unique = pt.unique() print(unique) Args: return_index (bool): If True, also return the indices that result in the unique array. (Default: False) return_counts (bool): If True, also return the number of times each unique item appears in the table. Returns: PauliList: unique the table of the unique rows. unique_indices: np.ndarray, optional The indices of the first occurrences of the unique values in the original array. Only provided if ``return_index`` is True. unique_counts: np.array, optional The number of times each of the unique values comes up in the original array. Only provided if ``return_counts`` is True. """ # Check if we need to stack the phase array if np.any(self._phase != self._phase[0]): # Create a single array of Pauli's and phases for calling np.unique on # so that we treat different phased Pauli's as unique array = np.hstack([self._z, self._x, self.phase.reshape((self.phase.shape[0], 1))]) else: # All Pauli's have the same phase so we only need to sort the array array = np.hstack([self._z, self._x]) # Get indexes of unique entries if return_counts: _, index, counts = np.unique(array, return_index=True, return_counts=True, axis=0) else: _, index = np.unique(array, return_index=True, axis=0) # Sort the index so we return unique rows in the original array order sort_inds = index.argsort() index = index[sort_inds] unique = PauliList(BasePauli(self._z[index], self._x[index], self._phase[index])) # Concatinate return tuples ret = (unique,) if return_index: ret += (index,) if return_counts: ret += (counts[sort_inds],) if len(ret) == 1: return ret[0] return ret
# --------------------------------------------------------------------- # BaseOperator methods # ---------------------------------------------------------------------
[docs] def tensor(self, other): """Return the tensor product with each Pauli in the list. Args: other (PauliList): another PauliList. Returns: PauliList: the list of tensor product Paulis. Raises: QiskitError: if other cannot be converted to a PauliList, does not have either 1 or the same number of Paulis as the current list. """ if not isinstance(other, PauliList): other = PauliList(other) return PauliList(super().tensor(other))
[docs] def expand(self, other): """Return the expand product of each Pauli in the list. Args: other (PauliList): another PauliList. Returns: PauliList: the list of tensor product Paulis. Raises: QiskitError: if other cannot be converted to a PauliList, does not have either 1 or the same number of Paulis as the current list. """ if not isinstance(other, PauliList): other = PauliList(other) if len(other) not in [1, len(self)]: raise QiskitError( "Incompatible PauliLists. Other list must " "have either 1 or the same number of Paulis." ) return PauliList(super().expand(other))
[docs] def compose(self, other, qargs=None, front=False, inplace=False): """Return the composition self∘other for each Pauli in the list. Args: other (PauliList): another PauliList. qargs (None or list): qubits to apply dot product on (Default: None). front (bool): If True use `dot` composition method [default: False]. inplace (bool): If True update in-place (default: False). Returns: PauliList: the list of composed Paulis. Raises: QiskitError: if other cannot be converted to a PauliList, does not have either 1 or the same number of Paulis as the current list, or has the wrong number of qubits for the specified qargs. """ if qargs is None: qargs = getattr(other, "qargs", None) if not isinstance(other, PauliList): other = PauliList(other) if len(other) not in [1, len(self)]: raise QiskitError( "Incompatible PauliLists. Other list must " "have either 1 or the same number of Paulis." ) return PauliList(super().compose(other, qargs=qargs, front=front, inplace=inplace))
# pylint: disable=arguments-differ
[docs] def dot(self, other, qargs=None, inplace=False): """Return the composition other∘self for each Pauli in the list. Args: other (PauliList): another PauliList. qargs (None or list): qubits to apply dot product on (Default: None). inplace (bool): If True update in-place (default: False). Returns: PauliList: the list of composed Paulis. Raises: QiskitError: if other cannot be converted to a PauliList, does not have either 1 or the same number of Paulis as the current list, or has the wrong number of qubits for the specified qargs. """ return self.compose(other, qargs=qargs, front=True, inplace=inplace)
def _add(self, other, qargs=None): """Append two PauliLists. If ``qargs`` are specified the other operator will be added assuming it is identity on all other subsystems. Args: other (PauliList): another table. qargs (None or list): optional subsystems to add on (Default: None) Returns: PauliList: the concatenated list self + other. """ if qargs is None: qargs = getattr(other, "qargs", None) if not isinstance(other, PauliList): other = PauliList(other) self._op_shape._validate_add(other._op_shape, qargs) base_phase = np.hstack((self._phase, other._phase)) if qargs is None or (sorted(qargs) == qargs and len(qargs) == self.num_qubits): base_z = np.vstack([self._z, other._z]) base_x = np.vstack([self._x, other._x]) else: # Pad other with identity and then add padded = BasePauli( np.zeros((other.size, self.num_qubits), dtype=bool), np.zeros((other.size, self.num_qubits), dtype=bool), np.zeros(other.size, dtype=int), ) padded = padded.compose(other, qargs=qargs, inplace=True) base_z = np.vstack([self._z, padded._z]) base_x = np.vstack([self._x, padded._x]) return PauliList(BasePauli(base_z, base_x, base_phase)) def _multiply(self, other): """Multiply each Pauli in the list by a phase. Args: other (complex or array): a complex number in [1, -1j, -1, 1j] Returns: PauliList: the list of Paulis other * self. Raises: QiskitError: if the phase is not in the set [1, -1j, -1, 1j]. """ return PauliList(super()._multiply(other))
[docs] def conjugate(self): """Return the conjugate of each Pauli in the list.""" return PauliList(super().conjugate())
[docs] def transpose(self): """Return the transpose of each Pauli in the list.""" return PauliList(super().transpose())
[docs] def adjoint(self): """Return the adjoint of each Pauli in the list.""" return PauliList(super().adjoint())
[docs] def inverse(self): """Return the inverse of each Pauli in the list.""" return PauliList(super().adjoint())
# --------------------------------------------------------------------- # Utility methods # ---------------------------------------------------------------------
[docs] def commutes(self, other, qargs=None): """Return True for each Pauli that commutes with other. Args: other (PauliList): another PauliList operator. qargs (list): qubits to apply dot product on (default: None). Returns: bool: True if Pauli's commute, False if they anti-commute. """ if qargs is None: qargs = getattr(other, "qargs", None) if not isinstance(other, BasePauli): other = PauliList(other) return super().commutes(other, qargs=qargs)
[docs] def anticommutes(self, other, qargs=None): """Return True if other Pauli that anticommutes with other. Args: other (PauliList): another PauliList operator. qargs (list): qubits to apply dot product on (default: None). Returns: bool: True if Pauli's anticommute, False if they commute. """ return np.logical_not(self.commutes(other, qargs=qargs))
[docs] def commutes_with_all(self, other): """Return indexes of rows that commute other. If other is a multi-row Pauli list the returned vector indexes rows of the current PauliList that commute with *all* Pauli's in other. If no rows satisfy the condition the returned array will be empty. Args: other (PauliList): a single Pauli or multi-row PauliList. Returns: array: index array of the commuting rows. """ return self._commutes_with_all(other)
[docs] def anticommutes_with_all(self, other): """Return indexes of rows that commute other. If other is a multi-row Pauli list the returned vector indexes rows of the current PauliList that anti-commute with *all* Pauli's in other. If no rows satisfy the condition the returned array will be empty. Args: other (PauliList): a single Pauli or multi-row PauliList. Returns: array: index array of the anti-commuting rows. """ return self._commutes_with_all(other, anti=True)
def _commutes_with_all(self, other, anti=False): """Return row indexes that commute with all rows in another PauliList. Args: other (PauliList): a PauliList. anti (bool): if True return rows that anti-commute, otherwise return rows that commute (Default: False). Returns: array: index array of commuting or anti-commuting row. """ if not isinstance(other, PauliList): other = PauliList(other) comms = self.commutes(other[0]) (inds,) = np.where(comms == int(not anti)) for pauli in other[1:]: comms = self[inds].commutes(pauli) (new_inds,) = np.where(comms == int(not anti)) if new_inds.size == 0: # No commuting rows return new_inds inds = inds[new_inds] return inds
[docs] def evolve(self, other, qargs=None, frame="h"): r"""Evolve the Pauli by a Clifford. This returns the Pauli :math:`P^\prime = C.P.C^\dagger`. By choosing the parameter frame='s', this function returns the Schrödinger evolution of the Pauli :math:`P^\prime = C.P.C^\dagger`. This option yields a faster calculation. Args: other (Pauli or Clifford or QuantumCircuit): The Clifford operator to evolve by. qargs (list): a list of qubits to apply the Clifford to. frame (string): 'h' for Heisenberg or 's' for Schrödinger framework. Returns: Pauli: the Pauli :math:`C.P.C^\dagger`. Raises: QiskitError: if the Clifford number of qubits and qargs don't match. """ from qiskit.circuit import Instruction, QuantumCircuit from qiskit.quantum_info.operators.symplectic.clifford import Clifford if qargs is None: qargs = getattr(other, "qargs", None) if not isinstance(other, (BasePauli, Instruction, QuantumCircuit, Clifford)): # Convert to a PauliList other = PauliList(other) return PauliList(super().evolve(other, qargs=qargs, frame=frame))
[docs] def to_labels(self, array=False): r"""Convert a PauliList to a list Pauli string labels. For large PauliLists converting using the ``array=True`` kwarg will be more efficient since it allocates memory for the full Numpy array of labels in advance. .. list-table:: Pauli Representations :header-rows: 1 * - Label - Symplectic - Matrix * - ``"I"`` - :math:`[0, 0]` - :math:`\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}` * - ``"X"`` - :math:`[1, 0]` - :math:`\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}` * - ``"Y"`` - :math:`[1, 1]` - :math:`\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}` * - ``"Z"`` - :math:`[0, 1]` - :math:`\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}` Args: array (bool): return a Numpy array if True, otherwise return a list (Default: False). Returns: list or array: The rows of the PauliList in label form. """ if (self.phase == 1).any(): prefix_len = 2 elif (self.phase > 0).any(): prefix_len = 1 else: prefix_len = 0 str_len = self.num_qubits + prefix_len ret = np.zeros(self.size, dtype=f"<U{str_len}") iterator = self.label_iter() for i in range(self.size): ret[i] = next(iterator) if array: return ret return ret.tolist()
[docs] def to_matrix(self, sparse=False, array=False): r"""Convert to a list or array of Pauli matrices. For large PauliLists converting using the ``array=True`` kwarg will be more efficient since it allocates memory a full rank-3 Numpy array of matrices in advance. .. list-table:: Pauli Representations :header-rows: 1 * - Label - Symplectic - Matrix * - ``"I"`` - :math:`[0, 0]` - :math:`\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}` * - ``"X"`` - :math:`[1, 0]` - :math:`\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}` * - ``"Y"`` - :math:`[1, 1]` - :math:`\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}` * - ``"Z"`` - :math:`[0, 1]` - :math:`\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}` Args: sparse (bool): if True return sparse CSR matrices, otherwise return dense Numpy arrays (Default: False). array (bool): return as rank-3 numpy array if True, otherwise return a list of Numpy arrays (Default: False). Returns: list: A list of dense Pauli matrices if `array=False` and `sparse=False`. list: A list of sparse Pauli matrices if `array=False` and `sparse=True`. array: A dense rank-3 array of Pauli matrices if `array=True`. """ if not array: # We return a list of Numpy array matrices return list(self.matrix_iter(sparse=sparse)) # For efficiency we also allow returning a single rank-3 # array where first index is the Pauli row, and second two # indices are the matrix indices dim = 2**self.num_qubits ret = np.zeros((self.size, dim, dim), dtype=complex) iterator = self.matrix_iter(sparse=sparse) for i in range(self.size): ret[i] = next(iterator) return ret
# --------------------------------------------------------------------- # Custom Iterators # ---------------------------------------------------------------------
[docs] def label_iter(self): """Return a label representation iterator. This is a lazy iterator that converts each row into the string label only as it is used. To convert the entire table to labels use the :meth:`to_labels` method. Returns: LabelIterator: label iterator object for the PauliList. """ class LabelIterator(CustomIterator): """Label representation iteration and item access.""" def __repr__(self): return f"<PauliList_label_iterator at {hex(id(self))}>" def __getitem__(self, key): return self.obj._to_label(self.obj._z[key], self.obj._x[key], self.obj._phase[key]) return LabelIterator(self)
[docs] def matrix_iter(self, sparse=False): """Return a matrix representation iterator. This is a lazy iterator that converts each row into the Pauli matrix representation only as it is used. To convert the entire table to matrices use the :meth:`to_matrix` method. Args: sparse (bool): optionally return sparse CSR matrices if True, otherwise return Numpy array matrices (Default: False) Returns: MatrixIterator: matrix iterator object for the PauliList. """ class MatrixIterator(CustomIterator): """Matrix representation iteration and item access.""" def __repr__(self): return f"<PauliList_matrix_iterator at {hex(id(self))}>" def __getitem__(self, key): return self.obj._to_matrix( self.obj._z[key], self.obj._x[key], self.obj._phase[key], sparse=sparse ) return MatrixIterator(self)
# --------------------------------------------------------------------- # Class methods # ---------------------------------------------------------------------
[docs] @classmethod def from_symplectic(cls, z, x, phase=0): """Construct a PauliList from a symplectic data. Args: z (np.ndarray): 2D boolean Numpy array. x (np.ndarray): 2D boolean Numpy array. phase (np.ndarray or None): Optional, 1D integer array from Z_4. Returns: PauliList: the constructed PauliList. """ base_z, base_x, base_phase = cls._from_array(z, x, phase) return cls(BasePauli(base_z, base_x, base_phase))
def _noncommutation_graph(self, qubit_wise): """Create an edge list representing the non-commutation graph (Pauli Graph). An edge (i, j) is present if i and j are not commutable. Args: qubit_wise (bool): whether the commutation rule is applied to the whole operator, or on a per-qubit basis. Returns: List[Tuple(int,int)]: A list of pairs of indices of the PauliList that are not commutable. """ # convert a Pauli operator into int vector where {I: 0, X: 2, Y: 3, Z: 1} mat1 = np.array( [op.z + 2 * op.x for op in self], dtype=np.int8, ) mat2 = mat1[:, None] # This is 0 (false-y) iff one of the operators is the identity and/or both operators are the # same. In other cases, it is non-zero (truth-y). qubit_anticommutation_mat = (mat1 * mat2) * (mat1 - mat2) # 'adjacency_mat[i, j]' is True iff Paulis 'i' and 'j' do not commute in the given strategy. if qubit_wise: adjacency_mat = np.logical_or.reduce(qubit_anticommutation_mat, axis=2) else: # Don't commute if there's an odd number of element-wise anti-commutations. adjacency_mat = np.logical_xor.reduce(qubit_anticommutation_mat, axis=2) # Convert into list where tuple elements are non-commuting operators. We only want to # results from one triangle to avoid symmetric duplications. return list(zip(*np.where(np.triu(adjacency_mat, k=1)))) def _create_graph(self, qubit_wise): """Transform measurement operator grouping problem into graph coloring problem Args: qubit_wise (bool): whether the commutation rule is applied to the whole operator, or on a per-qubit basis. Returns: retworkx.PyGraph: A class of undirected graphs """ edges = self._noncommutation_graph(qubit_wise) graph = rx.PyGraph() graph.add_nodes_from(range(self.size)) graph.add_edges_from_no_data(edges) return graph
[docs] def group_qubit_wise_commuting(self): """Partition a PauliList into sets of mutually qubit-wise commuting Pauli strings. Returns: List[PauliList]: List of PauliLists where each PauliList contains commutable Pauli operators. """ return self.group_commuting(qubit_wise=True)
[docs] def group_commuting(self, qubit_wise=False): """Partition a PauliList into sets of commuting Pauli strings. Args: qubit_wise (bool): whether the commutation rule is applied to the whole operator, or on a per-qubit basis. For example: .. code-block:: python >>> from qiskit.quantum_info import PauliList >>> op = PauliList(["XX", "YY", "IZ", "ZZ"]) >>> op.group_commuting() [PauliList(['XX', 'YY']), PauliList(['IZ', 'ZZ'])] >>> op.group_commuting(qubit_wise=True) [PauliList(['XX']), PauliList(['YY']), PauliList(['IZ', 'ZZ'])] Returns: List[PauliList]: List of PauliLists where each PauliList contains commuting Pauli operators. """ graph = self._create_graph(qubit_wise) # Keys in coloring_dict are nodes, values are colors coloring_dict = rx.graph_greedy_color(graph) groups = defaultdict(list) for idx, color in coloring_dict.items(): groups[color].append(idx) return [self[group] for group in groups.values()]