# This code is part of Qiskit.
#
# (C) Copyright IBM 2017, 2020
#
# 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.
"""
N-qubit Pauli Operator Class
"""
# pylint: disable=invalid-name
import re
import warnings
from typing import Dict
import numpy as np
from qiskit.circuit import Instruction, QuantumCircuit
from qiskit.circuit.barrier import Barrier
from qiskit.circuit.delay import Delay
from qiskit.circuit.library.generalized_gates import PauliGate
from qiskit.circuit.library.standard_gates import IGate, XGate, YGate, ZGate
from qiskit.exceptions import QiskitError
from qiskit.quantum_info.operators.mixins import generate_apidocs
from qiskit.quantum_info.operators.scalar_op import ScalarOp
from qiskit.quantum_info.operators.symplectic.base_pauli import BasePauli, _count_y
[문서]class Pauli(BasePauli):
r"""N-qubit Pauli operator.
This class represents an operator :math:`P` from the full :math:`n`-qubit
*Pauli* group
.. math::
P = (-i)^{q} P_{n-1} \otimes ... \otimes P_{0}
where :math:`q\in \mathbb{Z}_4` and :math:`P_i \in \{I, X, Y, Z\}`
are single-qubit Pauli matrices:
.. math::
I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},
X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},
Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix},
Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.
**Initialization**
A Pauli object can be initialized in several ways:
``Pauli(obj)``
where ``obj`` is a Pauli string, ``Pauli`` or
:class:`~qiskit.quantum_info.ScalarOp` operator, or a Pauli
gate or :class:`~qiskit.QuantumCircuit` containing only
Pauli gates.
``Pauli((z, x, phase))``
where ``z`` and ``x`` are boolean ``numpy.ndarrays`` and ``phase`` is
an integer in ``[0, 1, 2, 3]``.
``Pauli((z, x))``
equivalent to ``Pauli((z, x, 0))`` with trivial phase.
**String representation**
An :math:`n`-qubit Pauli may be represented by a string consisting of
:math:`n` characters from ``['I', 'X', 'Y', 'Z']``, and optionally phase
coefficient in :math:`['', '-i', '-', 'i']`. For example: ``XYZ`` or
``'-iZIZ'``.
In the string representation qubit-0 corresponds to the right-most
Pauli character, and qubit-:math:`(n-1)` to the left-most Pauli
character. For example ``'XYZ'`` represents
:math:`X\otimes Y \otimes Z` with ``'Z'`` on qubit-0,
``'Y'`` on qubit-1, and ``'X'`` on qubit-3.
The string representation can be converted to a ``Pauli`` using the
class initialization (``Pauli('-iXYZ')``). A ``Pauli`` object can be
converted back to the string representation using the
:meth:`to_label` method or ``str(pauli)``.
.. note::
Using ``str`` to convert a ``Pauli`` to a string will truncate the
returned string for large numbers of qubits while :meth:`to_label`
will return the full string with no truncation. The default
truncation length is 50 characters. The default value can be
changed by setting the class `__truncate__` attribute to an integer
value. If set to ``0`` no truncation will be performed.
**Array Representation**
The internal data structure of an :math:`n`-qubit Pauli is two
length-:math:`n` boolean vectors :math:`z \in \mathbb{Z}_2^N`,
:math:`x \in \mathbb{Z}_2^N`, and an integer :math:`q \in \mathbb{Z}_4`
defining the Pauli operator
.. math::
P = (-i)^{q + z\cdot x} Z^z \cdot X^x.
The :math:`k`th qubit corresponds to the :math:`k`th entry in the
:math:`z` and :math:`x` arrays
.. math::
P &= P_{n-1} \otimes ... \otimes P_{0} \\
P_k &= (-i)^{z[k] * x[k]} Z^{z[k]}\cdot X^{x[k]}
where ``z[k] = P.z[k]``, ``x[k] = P.x[k]`` respectively.
The :math:`z` and :math:`x` arrays can be accessed and updated using
the :attr:`z` and :attr:`x` properties respectively. The phase integer
:math:`q` can be accessed and updated using the :attr:`phase` property.
**Matrix Operator Representation**
Pauli's can be converted to :math:`(2^n, 2^n)`
:class:`~qiskit.quantum_info.Operator` using the :meth:`to_operator` method,
or to a dense or sparse complex matrix using the :meth:`to_matrix` method.
**Data Access**
The individual qubit Paulis can be accessed and updated using the ``[]``
operator which accepts integer, lists, or slices for selecting subsets
of Paulis. Note that selecting subsets of Pauli's will discard the
phase of the current Pauli.
For example
.. code-block:: python
p = Pauli('-iXYZ')
print('P[0] =', repr(P[0]))
print('P[1] =', repr(P[1]))
print('P[2] =', repr(P[2]))
print('P[:] =', repr(P[:]))
print('P[::-1] =, repr(P[::-1]))
"""
# Set the max Pauli string size before truncation
__truncate__ = 50
_VALID_LABEL_PATTERN = re.compile(r"^[+-]?1?[ij]?[IXYZ]+$")
def __init__(self, data=None, x=None, *, z=None, label=None):
"""Initialize the Pauli.
When using the symplectic array input data both z and x arguments must
be provided, however the first (z) argument can be used alone for string
label, Pauli operator, or ScalarOp input data.
Args:
data (str or tuple or Pauli or ScalarOp): input data for Pauli. If input is
a tuple it must be of the form ``(z, x)`` or (z, x, phase)`` where
``z`` and ``x`` are boolean Numpy arrays, and phase is an integer from Z_4.
If input is a string, it must be a concatenation of a phase and a Pauli string
(e.g. 'XYZ', '-iZIZ') where a phase string is a combination of at most three
characters from ['+', '-', ''], ['1', ''], and ['i', 'j', ''] in this order,
e.g. '', '-1j' while a Pauli string is 1 or more characters of 'I', 'X', 'Y' or 'Z',
e.g. 'Z', 'XIYY'.
Raises:
QiskitError: if input array is invalid shape.
"""
if isinstance(data, BasePauli):
base_z, base_x, base_phase = data._z, data._x, data._phase
elif isinstance(data, tuple):
if len(data) not in [2, 3]:
raise QiskitError(
"Invalid input tuple for Pauli, input tuple must be `(z, x, phase)` or `(z, x)`"
)
base_z, base_x, base_phase = self._from_array(*data)
elif isinstance(data, str):
base_z, base_x, base_phase = self._from_label(data)
elif isinstance(data, ScalarOp):
base_z, base_x, base_phase = self._from_scalar_op(data)
elif isinstance(data, (QuantumCircuit, Instruction)):
base_z, base_x, base_phase = self._from_circuit(data)
elif x is not None:
if z is None:
# Using old Pauli initialization with positional args instead of kwargs
z = data
warnings.warn(
"Passing 'z' and 'x' arrays separately to 'Pauli' is deprecated as of"
" Qiskit Terra 0.17 and will be removed in version 0.23 or later."
" Use a tuple instead, such as 'Pauli((z, x[, phase]))'.",
DeprecationWarning,
stacklevel=2,
)
base_z, base_x, base_phase = self._from_array(z, x)
elif label is not None:
warnings.warn(
"The 'label' keyword argument of 'Pauli' is deprecated as of"
" Qiskit Terra 0.17 and will be removed in version 0.23 or later."
" Pass the label positionally instead, such as 'Pauli(\"XYZ\")'.",
DeprecationWarning,
stacklevel=2,
)
base_z, base_x, base_phase = self._from_label(label)
else:
raise QiskitError("Invalid input data for Pauli.")
# Initialize BasePauli
if base_z.shape[0] != 1:
raise QiskitError("Input is not a single Pauli")
super().__init__(base_z, base_x, base_phase)
@property
def name(self):
"""Unique string identifier for operation type."""
return "pauli"
@property
def num_clbits(self):
"""Number of classical bits."""
return 0
def __repr__(self):
"""Display representation."""
return f"Pauli('{self.__str__()}')"
def __str__(self):
"""Print representation."""
if self.__truncate__ and self.num_qubits > self.__truncate__:
front = self[-self.__truncate__ :].to_label()
return front + "..."
return self.to_label()
def __array__(self, dtype=None):
if dtype:
return np.asarray(self.to_matrix(), dtype=dtype)
return self.to_matrix()
[문서] @classmethod
def set_truncation(cls, val):
"""Set the max number of Pauli characters to display before truncation/
Args:
val (int): the number of characters.
.. note::
Truncation will be disabled if the truncation value is set to 0.
"""
cls.__truncate__ = int(val)
def __eq__(self, other):
"""Test if two Paulis are equal."""
if not isinstance(other, BasePauli):
return False
return self._eq(other)
[문서] def equiv(self, other):
"""Return True if Pauli's are equivalent up to group phase.
Args:
other (Pauli): an operator object.
Returns:
bool: True if the Pauli's are equivalent up to group phase.
"""
if not isinstance(other, Pauli):
try:
other = Pauli(other)
except QiskitError:
return False
return np.all(self._z == other._z) and np.all(self._x == other._x)
@property
def settings(self) -> Dict:
"""Return settings."""
return {"data": self.to_label()}
# ---------------------------------------------------------------------
# Direct array access
# ---------------------------------------------------------------------
@property
def phase(self):
"""Return the group phase exponent for the Pauli."""
# Convert internal ZX-phase convention of BasePauli to group phase
return np.mod(self._phase - self._count_y(dtype=self._phase.dtype), 4)[0]
@phase.setter
def phase(self, value):
# Convert group phase convention to internal ZX-phase convention
self._phase[:] = np.mod(value + self._count_y(dtype=self._phase.dtype), 4)
@property
def x(self):
"""The x vector for the Pauli."""
return self._x[0]
@x.setter
def x(self, val):
self._x[0, :] = val
@property
def z(self):
"""The z vector for the Pauli."""
return self._z[0]
@z.setter
def z(self, val):
self._z[0, :] = val
# ---------------------------------------------------------------------
# Pauli Array methods
# ---------------------------------------------------------------------
def __len__(self):
"""Return the number of qubits in the Pauli."""
return self.num_qubits
def __getitem__(self, qubits):
"""Return the unsigned Pauli group Pauli for subset of qubits."""
# Set group phase to 0 so returned Pauli is always +1 coeff
if isinstance(qubits, (int, np.integer)):
qubits = [qubits]
return Pauli((self.z[qubits], self.x[qubits]))
def __setitem__(self, qubits, value):
"""Update the Pauli for a subset of qubits."""
if not isinstance(value, Pauli):
value = Pauli(value)
self._z[0, qubits] = value.z
self._x[0, qubits] = value.x
# Add extra phase from new Pauli to current
self._phase = self._phase + value._phase
[문서] def delete(self, qubits):
"""Return a Pauli with qubits deleted.
Args:
qubits (int or list): qubits to delete from Pauli.
Returns:
Pauli: the resulting Pauli with the specified qubits removed.
Raises:
QiskitError: if ind is out of bounds for the array size or
number of qubits.
"""
if isinstance(qubits, (int, np.integer)):
qubits = [qubits]
if max(qubits) > self.num_qubits - 1:
raise QiskitError(
"Qubit index is larger than the number of qubits "
"({}>{}).".format(max(qubits), self.num_qubits - 1)
)
if len(qubits) == self.num_qubits:
raise QiskitError("Cannot delete all qubits of Pauli")
z = np.delete(self._z, qubits, axis=1)
x = np.delete(self._x, qubits, axis=1)
return Pauli((z, x, self.phase))
[문서] def insert(self, qubits, value):
"""Insert a Pauli at specific qubit value.
Args:
qubits (int or list): qubits index to insert at.
value (Pauli): value to insert.
Returns:
Pauli: the resulting Pauli with the entries inserted.
Raises:
QiskitError: if the insertion qubits are invalid.
"""
if not isinstance(value, Pauli):
value = Pauli(value)
# Initialize empty operator
ret_qubits = self.num_qubits + value.num_qubits
ret = Pauli((np.zeros(ret_qubits, dtype=bool), np.zeros(ret_qubits, dtype=bool)))
if isinstance(qubits, (int, np.integer)):
if value.num_qubits == 1:
qubits = [qubits]
else:
qubits = list(range(qubits, qubits + value.num_qubits))
if len(qubits) != value.num_qubits:
raise QiskitError(
"Number of indices does not match number of qubits for "
"the inserted Pauli ({}!={})".format(len(qubits), value.num_qubits)
)
if max(qubits) > ret.num_qubits - 1:
raise QiskitError(
"Index is too larger for combined Pauli number of qubits "
"({}>{}).".format(max(qubits), ret.num_qubits - 1)
)
# Qubit positions for original op
self_qubits = [i for i in range(ret.num_qubits) if i not in qubits]
ret[self_qubits] = self
ret[qubits] = value
return ret
# ---------------------------------------------------------------------
# Representation conversions
# ---------------------------------------------------------------------
def __hash__(self):
"""Make hashable based on string representation."""
return hash(self.to_label())
[문서] def to_label(self):
"""Convert a Pauli to a string label.
.. note::
The difference between `to_label` and :meth:`__str__` is that
the later will truncate the output for large numbers of qubits.
Returns:
str: the Pauli string label.
"""
return self._to_label(self.z, self.x, self._phase[0])
[문서] def to_matrix(self, sparse=False):
r"""Convert to a Numpy array or sparse CSR matrix.
Args:
sparse (bool): if True return sparse CSR matrices, otherwise
return dense Numpy arrays (default: False).
Returns:
array: The Pauli matrix.
"""
return self._to_matrix(self.z, self.x, self._phase[0], sparse=sparse)
[문서] def to_instruction(self):
"""Convert to Pauli circuit instruction."""
from math import pi
pauli, phase = self._to_label(
self.z, self.x, self._phase[0], full_group=False, return_phase=True
)
if len(pauli) == 1:
gate = {"I": IGate(), "X": XGate(), "Y": YGate(), "Z": ZGate()}[pauli]
else:
gate = PauliGate(pauli)
if not phase:
return gate
# Add global phase
circuit = QuantumCircuit(self.num_qubits, name=str(self))
circuit.global_phase = -phase * pi / 2
circuit.append(gate, range(self.num_qubits))
return circuit.to_instruction()
# ---------------------------------------------------------------------
# BaseOperator methods
# ---------------------------------------------------------------------
[문서] def compose(self, other, qargs=None, front=False, inplace=False):
"""Return the operator composition with another Pauli.
Args:
other (Pauli): a Pauli object.
qargs (list or None): Optional, qubits to apply dot product
on (default: None).
front (bool): If True compose using right operator multiplication,
instead of left multiplication [default: False].
inplace (bool): If True update in-place (default: False).
Returns:
Pauli: The composed Pauli.
Raises:
QiskitError: if other cannot be converted to an operator, or has
incompatible dimensions for specified subsystems.
.. note::
Composition (``&``) by default is defined as `left` matrix multiplication for
matrix operators, while :meth:`dot` is defined as `right` matrix
multiplication. That is that ``A & B == A.compose(B)`` is equivalent to
``B.dot(A)`` when ``A`` and ``B`` are of the same type.
Setting the ``front=True`` kwarg changes this to `right` matrix
multiplication and is equivalent to the :meth:`dot` method
``A.dot(B) == A.compose(B, front=True)``.
"""
if qargs is None:
qargs = getattr(other, "qargs", None)
if not isinstance(other, Pauli):
other = Pauli(other)
return Pauli(super().compose(other, qargs=qargs, front=front, inplace=inplace))
# pylint: disable=arguments-differ
[문서] def dot(self, other, qargs=None, inplace=False):
"""Return the right multiplied operator self * other.
Args:
other (Pauli): 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:
Pauli: The operator self * other.
"""
return self.compose(other, qargs=qargs, front=True, inplace=inplace)
[문서] def tensor(self, other):
if not isinstance(other, Pauli):
other = Pauli(other)
return Pauli(super().tensor(other))
[문서] def expand(self, other):
if not isinstance(other, Pauli):
other = Pauli(other)
return Pauli(super().expand(other))
def _multiply(self, other):
return Pauli(super()._multiply(other))
[문서] def conjugate(self):
return Pauli(super().conjugate())
[문서] def transpose(self):
return Pauli(super().transpose())
[문서] def adjoint(self):
return Pauli(super().adjoint())
[문서] def inverse(self):
"""Return the inverse of the Pauli."""
return Pauli(super().adjoint())
# ---------------------------------------------------------------------
# Utility methods
# ---------------------------------------------------------------------
[문서] def commutes(self, other, qargs=None):
"""Return True if the Pauli commutes with other.
Args:
other (Pauli or PauliList): another Pauli 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 = Pauli(other)
ret = super().commutes(other, qargs=qargs)
if len(ret) == 1:
return ret[0]
return ret
[문서] def anticommutes(self, other, qargs=None):
"""Return True if other Pauli anticommutes with self.
Args:
other (Pauli): another Pauli 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))
[문서] def evolve(self, other, qargs=None, frame="h"):
r"""Heisenberg picture evolution of a Pauli by a Clifford.
This returns the Pauli :math:`P^\prime = C^\dagger.P.C`.
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^\dagger.P.C`.
Raises:
QiskitError: if the Clifford number of qubits and qargs don't match.
"""
if qargs is None:
qargs = getattr(other, "qargs", None)
# pylint: disable=cyclic-import
from qiskit.quantum_info.operators.symplectic.clifford import Clifford
if not isinstance(other, (Pauli, Instruction, QuantumCircuit, Clifford)):
# Convert to a Pauli
other = Pauli(other)
return Pauli(super().evolve(other, qargs=qargs, frame=frame))
# ---------------------------------------------------------------------
# Initialization helper functions
# ---------------------------------------------------------------------
@staticmethod
def _from_label(label):
"""Return the symplectic representation of Pauli string.
Args:
label (str): the Pauli string label.
Returns:
BasePauli: the BasePauli corresponding to the label.
Raises:
QiskitError: if Pauli string is not valid.
"""
if Pauli._VALID_LABEL_PATTERN.match(label) is None:
raise QiskitError(f'Pauli string label "{label}" is not valid.')
# Split string into coefficient and Pauli
pauli, coeff = _split_pauli_label(label)
# Convert coefficient to phase
phase = 0 if not coeff else _phase_from_label(coeff)
# Convert to Symplectic representation
pauli_bytes = np.frombuffer(pauli.encode("ascii"), dtype=np.uint8)[::-1]
ys = pauli_bytes == ord("Y")
base_x = np.logical_or(pauli_bytes == ord("X"), ys).reshape(1, -1)
base_z = np.logical_or(pauli_bytes == ord("Z"), ys).reshape(1, -1)
base_phase = np.array([(phase + np.count_nonzero(ys)) % 4], dtype=int)
return base_z, base_x, base_phase
@classmethod
def _from_scalar_op(cls, op):
"""Convert a ScalarOp to BasePauli data."""
if op.num_qubits is None:
raise QiskitError(f"{op} is not an N-qubit identity")
base_z = np.zeros((1, op.num_qubits), dtype=bool)
base_x = np.zeros((1, op.num_qubits), dtype=bool)
base_phase = np.mod(
cls._phase_from_complex(op.coeff) + _count_y(base_x, base_z), 4, dtype=int
)
return base_z, base_x, base_phase
@classmethod
def _from_pauli_instruction(cls, instr):
"""Convert a Pauli instruction to BasePauli data."""
if isinstance(instr, PauliGate):
return cls._from_label(instr.params[0])
if isinstance(instr, IGate):
return np.array([[False]]), np.array([[False]]), np.array([0])
if isinstance(instr, XGate):
return np.array([[False]]), np.array([[True]]), np.array([0])
if isinstance(instr, YGate):
return np.array([[True]]), np.array([[True]]), np.array([1])
if isinstance(instr, ZGate):
return np.array([[True]]), np.array([[False]]), np.array([0])
raise QiskitError("Invalid Pauli instruction.")
@classmethod
def _from_circuit(cls, instr):
"""Convert a Pauli circuit to BasePauli data."""
# Try and convert single instruction
if isinstance(instr, (PauliGate, IGate, XGate, YGate, ZGate)):
return cls._from_pauli_instruction(instr)
if isinstance(instr, Instruction):
# Convert other instructions to circuit definition
if instr.definition is None:
raise QiskitError(f"Cannot apply Instruction: {instr.name}")
# Convert to circuit
instr = instr.definition
# Initialize identity Pauli
ret = Pauli(
BasePauli(
np.zeros((1, instr.num_qubits), dtype=bool),
np.zeros((1, instr.num_qubits), dtype=bool),
np.zeros(1, dtype=int),
)
)
# Add circuit global phase if specified
if instr.global_phase:
ret.phase = cls._phase_from_complex(np.exp(1j * float(instr.global_phase)))
# Recursively apply instructions
for inner in instr.data:
if inner.clbits:
raise QiskitError(
f"Cannot apply instruction with classical bits: {inner.operation.name}"
)
if not isinstance(inner.operation, (Barrier, Delay)):
next_instr = BasePauli(*cls._from_circuit(inner.operation))
if next_instr is not None:
qargs = [tup.index for tup in inner.qubits]
ret = ret.compose(next_instr, qargs=qargs)
return ret._z, ret._x, ret._phase
# ---------------------------------------------------------------------
# Label parsing helper functions
# ---------------------------------------------------------------------
def _split_pauli_label(label):
"""Split Pauli label into unsigned group label and coefficient label"""
span = re.search(r"[IXYZ]+", label).span()
pauli = label[span[0] :]
coeff = label[: span[0]]
if span[1] != len(label):
invalid = set(re.sub(r"[IXYZ]+", "", label[span[0] :]))
raise QiskitError(
f"Pauli string contains invalid characters {invalid} ∉ ['I', 'X', 'Y', 'Z']"
)
return pauli, coeff
def _phase_from_label(label):
"""Return the phase from a label"""
# Returns None if label is invalid
label = label.replace("+", "", 1).replace("1", "", 1).replace("j", "i", 1)
phases = {"": 0, "-i": 1, "-": 2, "i": 3}
if label not in phases:
raise QiskitError(f"Invalid Pauli phase label '{label}'")
return phases[label]
# Update docstrings for API docs
generate_apidocs(Pauli)