PauliList

class qiskit.quantum_info.PauliList(data)

ベースクラス: BasePauli, LinearMixin, GroupMixin

List of N-qubit Pauli operators.

This class is an efficient representation of a list of Pauli operators. It supports 1D numpy array indexing returning a Pauli for integer indexes or a 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 Pauli.

PauliList(Pauli) and PauliList(list[Pauli])

where Pauli is 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,

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)

1.  ['II', 'ZI', '-iYY']
2.  ['iXI']
3.  ['iXI', 'iZZ']
4.  ['YZ', '-iIX']

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.

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]))

Integer:  Pauli('ZZ')
List:  PauliList(['XX', 'IZ'])
Slice:  PauliList(['XX', 'ZZ'])

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 label_iter() and matrix_iter() methods.

Initialize the PauliList.

パラメータ:

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.

例外:

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.

Attributes

dim

Return tuple (input_shape, output_shape).

num_qubits

Return the number of qubits if a N-qubit operator or None otherwise.

phase

Return the phase exponent of the PauliList.

qargs

Return the qargs for the operator.

settings

Return settings.

shape

The full shape of the array()

size

The number of Pauli rows in the table.

x

The x array for the symplectic representation.

z

The z array for the symplectic representation.

Methods

adjoint()

Return the adjoint of each Pauli in the list.

anticommutes(other, qargs=None)

Return True if other Pauli that anticommutes with other.

パラメータ:

• other (PauliList) – another PauliList operator.

• qargs (list) – qubits to apply dot product on (default: None).

戻り値:

True if Paulis anticommute, False if they commute.

戻り値の型:

bool

anticommutes_with_all(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 Paulis in other. If no rows satisfy the condition the returned array will be empty.

パラメータ:

other (PauliList) – a single Pauli or multi-row PauliList.

戻り値:

index array of the anti-commuting rows.

戻り値の型:

array

argsort(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 Paulis of a given weight are still ordered lexicographically.

パラメータ:

• weight (bool) – Optionally sort by weight if True (Default: False).

• phase (bool) – Optionally sort by phase before weight or order (Default: False).

戻り値:

the indices for sorting the table.

戻り値の型:

array

commutes(other, qargs=None)

Return True for each Pauli that commutes with other.

パラメータ:

• other (PauliList) – another PauliList operator.

• qargs (list) – qubits to apply dot product on (default: None).

戻り値:

True if Paulis commute, False if they anti-commute.

戻り値の型:

bool

commutes_with_all(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 Paulis in other. If no rows satisfy the condition the returned array will be empty.

パラメータ:

other (PauliList) – a single Pauli or multi-row PauliList.

戻り値:

index array of the commuting rows.

戻り値の型:

array

compose(other, qargs=None, front=False, inplace=False)

Return the composition self∘other for each Pauli in the list.

パラメータ:

• 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).

戻り値:

the list of composed Paulis.

戻り値の型:

PauliList

例外:

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.

conjugate()

Return the conjugate of each Pauli in the list.

copy()

Make a deep copy of current operator.

delete(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 X and Z arrays.

パラメータ:

• ind (int or list) – index(es) to delete.

• qubit (bool) – if True delete qubit columns, otherwise delete Pauli rows (Default: False).

戻り値:

the resulting table with the entries removed.

戻り値の型:

PauliList

例外:

QiskitError – if ind is out of bounds for the array size or number of qubits.

dot(other, qargs=None, inplace=False)

Return the composition other∘self for each Pauli in the list.

パラメータ:

• 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).

戻り値:

the list of composed Paulis.

戻り値の型:

PauliList

例外:

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.

equiv(other)

Entrywise comparison of Pauli equivalence up to global phase.

パラメータ:

other (PauliList or Pauli) – a comparison object.

戻り値:

An array of True or False for entrywise equivalence

of the current table.

戻り値の型:

np.ndarray

evolve(other, qargs=None, frame='h')

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 \(P^\prime = C^\dagger.P.C\).

Schrödinger picture evolves the Pauli as \(P^\prime = C.P.C^\dagger\).

パラメータ:

• 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 (default) or 's' for Schrödinger framework.

戻り値:

the Pauli \(C^\dagger.P.C\) (Heisenberg picture) or the Pauli \(C.P.C^\dagger\) (Schrödinger picture).

戻り値の型:

PauliList

例外:

QiskitError – if the Clifford number of qubits and qargs don’t match.

expand(other)

Return the expand product of each Pauli in the list.

パラメータ:

other (PauliList) – another PauliList.

戻り値:

the list of tensor product Paulis.

戻り値の型:

PauliList

例外:

QiskitError – if other cannot be converted to a PauliList, does not have either 1 or the same number of Paulis as the current list.

classmethod from_symplectic(z, x, phase=0)

Construct a PauliList from a symplectic data.

パラメータ:

• 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.

戻り値:

the constructed PauliList.

戻り値の型:

PauliList

group_commuting(qubit_wise=False)

Partition a PauliList into sets of commuting Pauli strings.

パラメータ:

qubit_wise (bool) –

whether the commutation rule is applied to the whole operator, or on a per-qubit basis. For example:

>>> 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'])]

戻り値:

List of PauliLists where each PauliList contains commuting Pauli operators.

戻り値の型:

list[PauliList]

group_qubit_wise_commuting()

Partition a PauliList into sets of mutually qubit-wise commuting Pauli strings.

戻り値:

List of PauliLists where each PauliList contains commutable Pauli operators.

戻り値の型:

list[PauliList]

input_dims(qargs=None)

Return tuple of input dimension for specified subsystems.

insert(ind, value, qubit=False)

Insert Paulis into the table.

When inserting qubits the qubit index is the same as the column index of the underlying X and Z arrays.

パラメータ:

• ind (int) – index to insert at.

• value (PauliList) – values to insert.

• qubit (bool) – if True insert qubit columns, otherwise insert Pauli rows (Default: False).

戻り値:

the resulting table with the entries inserted.

戻り値の型:

PauliList

例外:

QiskitError – if the insertion index is invalid.

inverse()

Return the inverse of each Pauli in the list.

label_iter()

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 to_labels() method.

戻り値:

label iterator object for the PauliList.

戻り値の型:

LabelIterator

matrix_iter(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 to_matrix() method.

パラメータ:

sparse (bool) – optionally return sparse CSR matrices if True, otherwise return Numpy array matrices (Default: False)

戻り値:

matrix iterator object for the PauliList.

戻り値の型:

MatrixIterator

output_dims(qargs=None)

Return tuple of output dimension for specified subsystems.

power(n)

Return the compose of a operator with itself n times.

パラメータ:

n (int) – the number of times to compose with self (n>0).

戻り値:

the n-times composed operator.

戻り値の型:

Pauli

例外:

QiskitError – if the input and output dimensions of the operator are not equal, or the power is not a positive integer.

reshape(input_dims=None, output_dims=None, num_qubits=None)

Return a shallow copy with reshaped input and output subsystem dimensions.

パラメータ:

• input_dims (None or tuple) – new subsystem input dimensions. If None the original input dims will be preserved [Default: None].

• output_dims (None or tuple) – new subsystem output dimensions. If None the original output dims will be preserved [Default: None].

• num_qubits (None or int) – reshape to an N-qubit operator [Default: None].

戻り値:

returns self with reshaped input and output dimensions.

戻り値の型:

BaseOperator

例外:

QiskitError – if combined size of all subsystem input dimension or subsystem output dimensions is not constant.

sort(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 Paulis of a given weight are still ordered lexicographically.

Example

Consider sorting all a random ordering of all 2-qubit Paulis

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)

Initial Ordering
['YX', 'ZZ', 'XZ', 'YI', 'YZ', 'II', 'XX', 'XI', 'XY', 'YY', 'IX', 'IZ',
 'ZY', 'ZI', 'ZX', 'IY']
Lexicographically sorted
['II', 'IX', 'IY', 'IZ', 'XI', 'XX', 'XY', 'XZ', 'YI', 'YX', 'YY', 'YZ',
 'ZI', 'ZX', 'ZY', 'ZZ']
Weight sorted
['II', 'IX', 'IY', 'IZ', 'XI', 'YI', 'ZI', 'XX', 'XY', 'XZ', 'YX', 'YY',
 'YZ', 'ZX', 'ZY', 'ZZ']

パラメータ:

• weight (bool) – optionally sort by weight if True (Default: False).

• phase (bool) – Optionally sort by phase before weight or order (Default: False).

戻り値:

a sorted copy of the original table.

戻り値の型:

PauliList

tensor(other)

Return the tensor product with each Pauli in the list.

パラメータ:

other (PauliList) – another PauliList.

戻り値:

the list of tensor product Paulis.

戻り値の型:

PauliList

例外:

QiskitError – if other cannot be converted to a PauliList, does not have either 1 or the same number of Paulis as the current list.

to_labels(array=False)

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.

Table 3 Pauli Representations

Label	Symplectic	Matrix
"I"	\([0, 0]\)	\(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)
"X"	\([1, 0]\)	\(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)
"Y"	\([1, 1]\)	\(\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}\)
"Z"	\([0, 1]\)	\(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)

パラメータ:

array (bool) – return a Numpy array if True, otherwise return a list (Default: False).

戻り値:

The rows of the PauliList in label form.

戻り値の型:

list or array

to_matrix(sparse=False, array=False)

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.

Table 4 Pauli Representations

Label	Symplectic	Matrix
"I"	\([0, 0]\)	\(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)
"X"	\([1, 0]\)	\(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)
"Y"	\([1, 1]\)	\(\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}\)
"Z"	\([0, 1]\)	\(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)

パラメータ:

• 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).

戻り値:

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.

戻り値の型:

list

transpose()

Return the transpose of each Pauli in the list.

unique(return_index=False, return_counts=False)

Return unique Paulis from the table.

Example

from qiskit.quantum_info.operators import PauliList

pt = PauliList(['X', 'Y', '-X', 'I', 'I', 'Z', 'X', 'iZ'])
unique = pt.unique()
print(unique)

['X', 'Y', '-X', 'I', 'Z', 'iZ']

パラメータ:

• 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.

戻り値:

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.

戻り値の型:

PauliList