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This code is part of Qiskit.","#","# (C) Copyright IBM 2017, 2023","#","# 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","\"\"\"","# pylint: disable=invalid-name","","from __future__ import annotations","import copy","from typing import Literal, TYPE_CHECKING","","import numpy as np","","from qiskit.circuit import QuantumCircuit","from qiskit.circuit.barrier import Barrier","from qiskit.circuit.delay import Delay","from qiskit.exceptions import QiskitError","from qiskit.quantum_info.operators.base_operator import BaseOperator","from qiskit.quantum_info.operators.mixins import AdjointMixin, MultiplyMixin","","if TYPE_CHECKING:"," from qiskit.quantum_info.operators.symplectic.clifford import Clifford","","","# utility for _to_matrix","_PARITY = np.array([-1 if bin(i).count(\"1\") % 2 else 1 for i in range(256)], dtype=complex)","","","class BasePauli(BaseOperator, AdjointMixin, MultiplyMixin):"," r\"\"\"Symplectic representation of a list of N-qubit Paulis.",""," Base class for Pauli and PauliList."," \"\"\"",""," def __init__(self, z: np.ndarray, x: np.ndarray, phase: np.ndarray):"," \"\"\"Initialize the BasePauli.",""," This is an array of M N-qubit Paulis defined as"," P = (-i)^phase Z^z X^x.",""," Args:"," z (np.ndarray): input z matrix."," x (np.ndarray): input x matrix."," phase (np.ndarray): input phase vector."," \"\"\""," self._z = z"," self._x = x"," self._phase = phase"," self._num_paulis, num_qubits = self._z.shape"," super().__init__(num_qubits=num_qubits)",""," def copy(self):"," \"\"\"Make a deep copy of current operator.\"\"\""," # Deepcopy has terrible performance on objects with Numpy arrays"," # attributes so we make a shallow copy and then manually copy the"," # Numpy arrays to efficiently mimic a deepcopy"," ret = copy.copy(self)"," ret._z = self._z.copy()"," ret._x = self._x.copy()"," ret._phase = self._phase.copy()"," return ret",""," # ---------------------------------------------------------------------"," # BaseOperator methods"," # ---------------------------------------------------------------------",""," def tensor(self, other):"," return self._tensor(self, other)",""," def expand(self, other):"," return self._tensor(other, self)",""," @classmethod"," def _tensor(cls, a, b):"," x1 = cls._stack(a._x, b._num_paulis, False)"," x2 = cls._stack(b._x, a._num_paulis)"," z1 = cls._stack(a._z, b._num_paulis, False)"," z2 = cls._stack(b._z, a._num_paulis)"," phase1 = ("," np.vstack(b._num_paulis * [a._phase])"," .transpose(1, 0)"," .reshape(a._num_paulis * b._num_paulis)"," )"," phase2 = cls._stack(b._phase, a._num_paulis)"," z = np.hstack([z2, z1])"," x = np.hstack([x2, x1])"," phase = np.mod(phase1 + phase2, 4)"," return BasePauli(z, x, phase)",""," def compose(self, other, qargs: list | None = None, front: bool = False, inplace=False):"," \"\"\"Return the composition of Paulis.",""," Args:"," a ({cls}): an operator object."," b ({cls}): an operator object."," qargs (list or None): Optional, qubits to apply dot product"," on (default: None)."," inplace (bool): If True update in-place (default: False).",""," Returns:"," {cls}: The operator a.compose(b)",""," Raises:"," QiskitError: if number of qubits of other does not match qargs."," \"\"\".format("," cls=type(self).__name__"," )"," # Validation"," if qargs is None and other.num_qubits != self.num_qubits:"," raise QiskitError(f\"other {type(self).__name__} must be on the same number of qubits.\")",""," if qargs and other.num_qubits != len(qargs):"," raise QiskitError("," f\"Number of qubits of the other {type(self).__name__} does not match qargs.\""," )",""," if other._num_paulis not in [1, self._num_paulis]:"," raise QiskitError("," \"Incompatible BasePaulis. Second list must \""," \"either have 1 or the same number of Paulis.\""," )",""," # Compute phase shift"," if qargs is not None:"," x1, z1 = self._x[:, qargs], self._z[:, qargs]"," else:"," x1, z1 = self._x, self._z"," x2, z2 = other._x, other._z",""," # Get phase shift"," phase = self._phase + other._phase"," if front:"," phase += 2 * _count_y(x1, z2, dtype=phase.dtype)"," else:"," phase += 2 * _count_y(x2, z1, dtype=phase.dtype)",""," # Update Pauli"," x = np.logical_xor(x1, x2)"," z = np.logical_xor(z1, z2)",""," if qargs is None:"," if not inplace:"," return BasePauli(z, x, phase)"," # Inplace update"," self._x = x"," self._z = z"," self._phase = phase"," return self",""," # Qargs update"," ret = self if inplace else self.copy()"," ret._x[:, qargs] = x"," ret._z[:, qargs] = z"," ret._phase = np.mod(phase, 4)"," return ret",""," def _multiply(self, other):"," \"\"\"Return the {cls} other * self.",""," Args:"," other (complex): a complex number in ``[1, -1j, -1, 1j]``.",""," Returns:"," {cls}: the {cls} other * self.",""," Raises:"," QiskitError: if the phase is not in the set ``[1, -1j, -1, 1j]``."," \"\"\".format("," cls=type(self).__name__"," )"," if isinstance(other, (np.ndarray, list, tuple)):"," phase = np.array([self._phase_from_complex(phase) for phase in other])"," else:"," phase = self._phase_from_complex(other)"," return BasePauli(self._z, self._x, np.mod(self._phase + phase, 4))",""," def conjugate(self):"," \"\"\"Return the conjugate of each Pauli in the list.\"\"\""," complex_phase = np.mod(self._phase, 2)"," if np.all(complex_phase == 0):"," return self"," return BasePauli(self._z, self._x, np.mod(self._phase + 2 * complex_phase, 4))",""," def transpose(self):"," \"\"\"Return the transpose of each Pauli in the list.\"\"\""," # Transpose sets Y -> -Y. This has effect on changing the phase"," parity_y = self._count_y(dtype=self._phase.dtype) % 2"," if np.all(parity_y == 0):"," return self"," return BasePauli(self._z, self._x, np.mod(self._phase + 2 * parity_y, 4))",""," def commutes(self, other: BasePauli, qargs: list | None = None) -> np.ndarray:"," \"\"\"Return ``True`` if Pauli commutes with ``other``.",""," Args:"," other (BasePauli): another BasePauli operator."," qargs (list): qubits to apply dot product on (default: ``None``).",""," Returns:"," np.array: Boolean array of ``True`` if Paulis commute, ``False`` if"," they anti-commute.",""," Raises:"," QiskitError: if number of qubits of ``other`` does not match ``qargs``."," \"\"\""," if qargs is not None and len(qargs) != other.num_qubits:"," raise QiskitError("," \"Number of qubits of other Pauli does not match number of \""," \"qargs ({} != {}).\".format(other.num_qubits, len(qargs))"," )"," if qargs is None and self.num_qubits != other.num_qubits:"," raise QiskitError("," \"Number of qubits of other Pauli does not match the current \""," \"Pauli ({} != {}).\".format(other.num_qubits, self.num_qubits)"," )"," if qargs is not None:"," inds = list(qargs)"," x1, z1 = self._x[:, inds], self._z[:, inds]"," else:"," x1, z1 = self._x, self._z"," a_dot_b = np.mod(_count_y(x1, other._z), 2)"," b_dot_a = np.mod(_count_y(other._x, z1), 2)"," return a_dot_b == b_dot_a",""," def evolve("," self,"," other: BasePauli | QuantumCircuit | Clifford,"," qargs: list | None = None,"," frame: Literal[\"h\", \"s\"] = \"h\","," ) -> BasePauli:"," r\"\"\"Performs either Heisenberg (default) or Schrödinger picture"," evolution of the Pauli by a Clifford and returns the evolved Pauli.",""," Schrödinger picture evolution can be chosen by passing parameter ``frame='s'``."," This option yields a faster calculation.",""," Heisenberg picture evolves the Pauli as :math:`P^\\prime = C^\\dagger.P.C`.",""," Schrödinger picture evolves the Pauli as :math:`P^\\prime = C.P.C^\\dagger`.",""," Args:"," other (BasePauli or QuantumCircuit): The Clifford circuit 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:"," BasePauli: the Pauli :math:`C^\\dagger.P.C` (Heisenberg picture)"," or the Pauli :math:`C.P.C^\\dagger` (Schrödinger picture).",""," Raises:"," QiskitError: if the Clifford number of qubits and ``qargs`` don't match."," \"\"\""," # Check dimension"," if qargs is not None and len(qargs) != other.num_qubits:"," raise QiskitError("," \"Incorrect number of qubits for Clifford circuit ({} != {}).\".format("," other.num_qubits, len(qargs)"," )"," )"," if qargs is None and self.num_qubits != other.num_qubits:"," raise QiskitError("," \"Incorrect number of qubits for Clifford circuit ({} != {}).\".format("," other.num_qubits, self.num_qubits"," )"," )",""," # Evolve via Pauli"," if isinstance(other, BasePauli):"," if frame == \"s\":"," ret = self.compose(other, qargs=qargs)"," ret = ret.compose(other.adjoint(), front=True, qargs=qargs)"," else:"," ret = self.compose(other.adjoint(), qargs=qargs)"," ret = ret.compose(other, front=True, qargs=qargs)"," return ret",""," # pylint: disable=cyclic-import"," from qiskit.quantum_info.operators.symplectic.clifford import Clifford",""," # Convert Clifford to quantum circuits"," if isinstance(other, Clifford):"," return self._evolve_clifford(other, qargs=qargs, frame=frame)",""," # Otherwise evolve by the inverse circuit to compute C^dg.P.C"," if frame == \"s\":"," return self.copy()._append_circuit(other, qargs=qargs)"," return self.copy()._append_circuit(other.inverse(), qargs=qargs)",""," def _evolve_clifford(self, other, qargs=None, frame=\"h\"):"," \"\"\"Heisenberg picture evolution of a Pauli by a Clifford.\"\"\"",""," if frame == \"s\":"," adj = other"," else:"," adj = other.adjoint()",""," if qargs is None:"," qargs_ = slice(None)"," else:"," qargs_ = list(qargs)",""," # pylint: disable=cyclic-import"," from qiskit.quantum_info.operators.symplectic.pauli_list import PauliList",""," num_paulis = self._x.shape[0]",""," ret = self.copy()"," ret._x[:, qargs_] = False"," ret._z[:, qargs_] = False",""," idx = np.concatenate((self._x[:, qargs_], self._z[:, qargs_]), axis=1)"," for idx_, row in zip("," idx.T,"," PauliList.from_symplectic(z=adj.z, x=adj.x, phase=2 * adj.phase),"," ):"," # most of the logic below is to properly index if self is a PauliList (2D),"," # while not trying to index if the object is just a Pauli (1D)."," if idx_.any():"," if np.sum(idx_) == num_paulis:"," ret.compose(row, qargs=qargs, inplace=True)"," else:"," ret[idx_] = ret[idx_].compose(row, qargs=qargs)",""," return ret",""," def _eq(self, other):"," \"\"\"Entrywise comparison of Pauli equality.\"\"\""," return ("," self.num_qubits == other.num_qubits"," and np.all(np.mod(self._phase, 4) == np.mod(other._phase, 4))"," and np.all(self._z == other._z)"," and np.all(self._x == other._x)"," )",""," # ---------------------------------------------------------------------"," # Helper Methods"," # ---------------------------------------------------------------------",""," def __imul__(self, other):"," return self.compose(other, front=True, inplace=True)",""," def __neg__(self):"," ret = copy.copy(self)"," ret._phase = np.mod(self._phase + 2, 4)"," return ret",""," def _count_y(self, dtype=None):"," \"\"\"Count the number of I Paulis\"\"\""," return _count_y(self._x, self._z, dtype=dtype)",""," @staticmethod"," def _stack(array, size, vertical=True):"," \"\"\"Stack array.\"\"\""," if size == 1:"," return array"," if vertical:"," return np.vstack(size * [array]).reshape((size * len(array),) + array.shape[1:])"," return np.hstack(size * [array]).reshape((size * len(array),) + array.shape[1:])",""," @staticmethod"," def _phase_from_complex(coeff):"," \"\"\"Return the phase from a label\"\"\""," if np.isclose(coeff, 1):"," return 0"," if np.isclose(coeff, -1j):"," return 1"," if np.isclose(coeff, -1):"," return 2"," if np.isclose(coeff, 1j):"," return 3"," raise QiskitError(\"Pauli can only be multiplied by 1, -1j, -1, 1j.\")",""," @staticmethod"," def _from_array(z, x, phase=0):"," \"\"\"Convert array data to BasePauli data.\"\"\""," if isinstance(z, np.ndarray) and z.dtype == bool:"," base_z = z"," else:"," base_z = np.asarray(z, dtype=bool)"," if base_z.ndim == 1:"," base_z = base_z.reshape((1, base_z.size))"," elif base_z.ndim != 2:"," raise QiskitError(\"Invalid Pauli z vector shape.\")",""," if isinstance(x, np.ndarray) and x.dtype == bool:"," base_x = x"," else:"," base_x = np.asarray(x, dtype=bool)"," if base_x.ndim == 1:"," base_x = base_x.reshape((1, base_x.size))"," elif base_x.ndim != 2:"," raise QiskitError(\"Invalid Pauli x vector shape.\")",""," if base_z.shape != base_x.shape:"," raise QiskitError(\"z and x vectors are different size.\")",""," # Convert group phase convention to internal ZX-phase conversion."," dtype = getattr(phase, \"dtype\", None)"," base_phase = np.mod(_count_y(base_x, base_z, dtype=dtype) + phase, 4)"," return base_z, base_x, base_phase",""," @staticmethod"," def _to_matrix(z, x, phase=0, group_phase=False, sparse=False):"," \"\"\"Return the matrix from symplectic representation.",""," The Pauli is defined as :math:`P = (-i)^{phase + z.x} * Z^z.x^x`"," where ``array = [x, z]``.",""," Args:"," z (array): The symplectic representation z vector."," x (array): The symplectic representation x vector."," phase (int): Pauli phase."," group_phase (bool): Optional. If ``True`` use group-phase convention"," instead of BasePauli ZX-phase convention."," (default: ``False``)."," sparse (bool): Optional. Of ``True`` return a sparse CSR matrix,"," otherwise return a dense Numpy array"," (default: ``False``).",""," Returns:"," array: if ``sparse=False``."," csr_matrix: if ``sparse=True``."," \"\"\""," num_qubits = z.size",""," # Convert to zx_phase"," if group_phase:"," phase += np.sum(x & z)"," phase %= 4",""," dim = 2**num_qubits"," twos_array = 1 << np.arange(num_qubits, dtype=np.uint)"," x_indices = np.asarray(x).dot(twos_array)"," z_indices = np.asarray(z).dot(twos_array)",""," indptr = np.arange(dim + 1, dtype=np.uint)"," indices = indptr ^ x_indices"," if phase:"," coeff = (-1j) ** phase"," else:"," coeff = 1",""," # Compute parities of `z_indices & indptr`, i.e.,"," # np.array([(-1) ** bin(i).count(\"1\") for i in z_indices & indptr])"," vec_u64 = z_indices & indptr"," mat_u8 = np.zeros((vec_u64.size, 8), dtype=np.uint8)"," for i in range(8):"," mat_u8[:, i] = vec_u64 & 255"," vec_u64 >>= 8"," if np.all(vec_u64 == 0):"," break"," parity = _PARITY[np.bitwise_xor.reduce(mat_u8, axis=1)]",""," data = coeff * parity"," if sparse:"," # Return sparse matrix"," from scipy.sparse import csr_matrix",""," return csr_matrix((data, indices, indptr), shape=(dim, dim), dtype=complex)",""," # Build dense matrix using csr format"," mat = np.zeros((dim, dim), dtype=complex)"," mat[range(dim), indices[:dim]] = data[:dim]"," return mat",""," @staticmethod"," def _to_label(z, x, phase, group_phase=False, full_group=True, return_phase=False):"," \"\"\"Return the label string for a Pauli.",""," Args:"," z (array): The symplectic representation z vector."," x (array): The symplectic representation x vector."," phase (int): Pauli phase."," group_phase (bool): Optional. If ``True`` use group-phase convention"," instead of BasePauli ZX-phase convention."," (default: ``False``)."," full_group (bool): If True return the Pauli label from the full Pauli group"," including complex coefficient from [1, -1, 1j, -1j]. If"," ``False`` return the unsigned Pauli label with coefficient 1"," (default: ``True``)."," return_phase (bool): If ``True`` return the adjusted phase for the coefficient"," of the returned Pauli label. This can be used even if"," ``full_group=False``.",""," Returns:"," str: the Pauli label from the full Pauli group (if ``full_group=True``) or"," from the unsigned Pauli group (if ``full_group=False``)."," tuple[str, int]: if ``return_phase=True`` returns a tuple of the Pauli"," label (from either the full or unsigned Pauli group) and"," the phase ``q`` for the coefficient :math:`(-i)^(q + x.z)`"," for the label from the full Pauli group."," \"\"\""," num_qubits = z.size"," phase = int(phase)"," coeff_labels = {0: \"\", 1: \"-i\", 2: \"-\", 3: \"i\"}"," label = \"\""," for i in range(num_qubits):"," if not z[num_qubits - 1 - i]:"," if not x[num_qubits - 1 - i]:"," label += \"I\""," else:"," label += \"X\""," elif not x[num_qubits - 1 - i]:"," label += \"Z\""," else:"," label += \"Y\""," if not group_phase:"," phase -= 1"," phase %= 4"," if phase and full_group:"," label = coeff_labels[phase] + label"," if return_phase:"," return label, phase"," return label",""," def _append_circuit(self, circuit, qargs=None):"," \"\"\"Update BasePauli inplace by applying a Clifford circuit.",""," Args:"," circuit (QuantumCircuit or Instruction): the gate or composite gate to apply."," qargs (list or None): The qubits to apply gate to.",""," Returns:"," BasePauli: the updated Pauli.",""," Raises:"," QiskitError: if input gate cannot be decomposed into Clifford gates."," \"\"\""," if isinstance(circuit, (Barrier, Delay)):"," return self",""," if qargs is None:"," qargs = list(range(self.num_qubits))",""," if isinstance(circuit, QuantumCircuit):"," gate = circuit.to_instruction()"," else:"," gate = circuit",""," # Basis Clifford Gates"," basis_1q = {"," \"i\": _evolve_i,"," \"id\": _evolve_i,"," \"iden\": _evolve_i,"," \"x\": _evolve_x,"," \"y\": _evolve_y,"," \"z\": _evolve_z,"," \"h\": _evolve_h,"," \"s\": _evolve_s,"," \"sdg\": _evolve_sdg,"," \"sinv\": _evolve_sdg,"," }"," basis_2q = {\"cx\": _evolve_cx, \"cz\": _evolve_cz, \"cy\": _evolve_cy, \"swap\": _evolve_swap}",""," # Non-Clifford gates"," non_clifford = [\"t\", \"tdg\", \"ccx\", \"ccz\"]",""," if isinstance(gate, str):"," # Check if gate is a valid Clifford basis gate string"," if gate not in basis_1q and gate not in basis_2q:"," raise QiskitError(f\"Invalid Clifford gate name string {gate}\")"," name = gate"," else:"," # Assume gate is an Instruction"," name = gate.name",""," # Apply gate if it is a Clifford basis gate"," if name in non_clifford:"," raise QiskitError(f\"Cannot update Pauli with non-Clifford gate {name}\")"," if name in basis_1q:"," if len(qargs) != 1:"," raise QiskitError(\"Invalid qubits for 1-qubit gate.\")"," return basis_1q[name](self, qargs[0])"," if name in basis_2q:"," if len(qargs) != 2:"," raise QiskitError(\"Invalid qubits for 2-qubit gate.\")"," return basis_2q[name](self, qargs[0], qargs[1])",""," # If not a Clifford basis gate we try to unroll the gate and"," # raise an exception if unrolling reaches a non-Clifford gate."," if gate.definition is None:"," raise QiskitError(f\"Cannot apply Instruction: {gate.name}\")"," if not isinstance(gate.definition, QuantumCircuit):"," raise QiskitError("," \"{} instruction definition is {}; expected QuantumCircuit\".format("," gate.name, type(gate.definition)"," )"," )",""," flat_instr = gate.definition"," bit_indices = {"," bit: index"," for bits in [flat_instr.qubits, flat_instr.clbits]"," for index, bit in enumerate(bits)"," }",""," for instruction in flat_instr:"," if instruction.clbits:"," raise QiskitError("," f\"Cannot apply Instruction with classical bits: {instruction.operation.name}\""," )"," # Get the integer position of the flat register"," new_qubits = [qargs[bit_indices[tup]] for tup in instruction.qubits]"," self._append_circuit(instruction.operation, new_qubits)",""," # Since the individual gate evolution functions don't take mod"," # of phase we update it at the end"," self._phase %= 4"," return self","","","# 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_evolve_i(base_pauli, qubit):"," \"\"\"Update P -> P\"\"\""," return base_pauli","","","def _evolve_x(base_pauli, qubit):"," \"\"\"Update P -> X.P.X\"\"\""," base_pauli._phase += 2 * base_pauli._z[:, qubit].T.astype(base_pauli._phase.dtype)"," return base_pauli","","","def _evolve_y(base_pauli, qubit):"," \"\"\"Update P -> Y.P.Y\"\"\""," xp = base_pauli._x[:, qubit].T.astype(base_pauli._phase.dtype)"," zp = base_pauli._z[:, qubit].T.astype(base_pauli._phase.dtype)"," base_pauli._phase += 2 * (xp + zp)"," return base_pauli","","","def _evolve_z(base_pauli, qubit):"," \"\"\"Update P -> Z.P.Z\"\"\""," base_pauli._phase += 2 * base_pauli._x[:, qubit].T.astype(base_pauli._phase.dtype)"," return base_pauli","","","def _evolve_cx(base_pauli, qctrl, qtrgt):"," \"\"\"Update P -> CX.P.CX\"\"\""," base_pauli._x[:, qtrgt] ^= base_pauli._x[:, qctrl]"," base_pauli._z[:, qctrl] ^= base_pauli._z[:, qtrgt]"," return base_pauli","","","def _evolve_cz(base_pauli, q1, q2):"," \"\"\"Update P -> CZ.P.CZ\"\"\""," x1 = base_pauli._x[:, q1].copy()"," x2 = base_pauli._x[:, q2].copy()"," base_pauli._z[:, q1] ^= x2"," base_pauli._z[:, q2] ^= x1"," base_pauli._phase += 2 * np.logical_and(x1, x2).T.astype(base_pauli._phase.dtype)"," return base_pauli","","","def _evolve_cy(base_pauli, qctrl, qtrgt):"," \"\"\"Update P -> CY.P.CY\"\"\""," x1 = base_pauli._x[:, qctrl].copy()"," x2 = base_pauli._x[:, qtrgt].copy()"," z2 = base_pauli._z[:, qtrgt].copy()"," base_pauli._x[:, qtrgt] ^= x1"," base_pauli._z[:, qtrgt] ^= x1"," base_pauli._z[:, qctrl] ^= np.logical_xor(x2, z2)"," base_pauli._phase += x1 + 2 * np.logical_and(x1, x2).T.astype(base_pauli._phase.dtype)"," return base_pauli","","","def _evolve_swap(base_pauli, q1, q2):"," \"\"\"Update P -> SWAP.P.SWAP\"\"\""," x1 = base_pauli._x[:, q1].copy()"," z1 = base_pauli._z[:, q1].copy()"," base_pauli._x[:, q1] = base_pauli._x[:, q2]"," base_pauli._z[:, q1] = base_pauli._z[:, q2]"," base_pauli._x[:, q2] = x1"," base_pauli._z[:, q2] = z1"," 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