# qiskit.quantum_info.PauliList¶

class PauliList(data)[source]

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)
print("4. ", pauli_list)

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


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.

Parameters

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.

The input array is not copied so multiple Pauli tables can share the same underlying array.

__init__(data)[source]

Initialize the PauliList.

Parameters

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.

The input array is not copied so multiple Pauli tables can share the same underlying array.

Methods

 __init__(data) Initialize the PauliList. Return the adjoint of each Pauli in the list. anticommutes(other[, qargs]) Return True if other Pauli that anticommutes with other. Return indexes of rows that commute other. argsort([weight, phase]) Return indices for sorting the rows of the table. commutes(other[, qargs]) Return True for each Pauli that commutes with other. commutes_with_all(other) Return indexes of rows that commute other. compose(other[, qargs, front, inplace]) Return the composition self∘other for each Pauli in the list. Return the conjugate of each Pauli in the list. Make a deep copy of current operator. delete(ind[, qubit]) Return a copy with Pauli rows deleted from table. dot(other[, qargs, inplace]) Return the composition other∘self for each Pauli in the list. equiv(other) Entrywise comparison of Pauli equivalence up to global phase. evolve(other[, qargs]) Evolve the Pauli by a Clifford. expand(other) Return the expand product of each Pauli in the list. from_symplectic(z, x[, phase]) Construct a PauliList from a symplectic data. input_dims([qargs]) Return tuple of input dimension for specified subsystems. insert(ind, value[, qubit]) Insert Pauli’s into the table. Return the inverse of each Pauli in the list. Return a label representation iterator. matrix_iter([sparse]) Return a matrix representation iterator. output_dims([qargs]) Return tuple of output dimension for specified subsystems. Return the compose of a operator with itself n times. reshape([input_dims, output_dims, num_qubits]) Return a shallow copy with reshaped input and output subsystem dimensions. sort([weight, phase]) Sort the rows of the table. tensor(other) Return the tensor product with each Pauli in the list. to_labels([array]) Convert a PauliList to a list Pauli string labels. to_matrix([sparse, array]) Convert to a list or array of Pauli matrices. Return the transpose of each Pauli in the list. unique([return_index, return_counts]) Return unique Paulis from the table.

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 operator 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.
adjoint()[source]

Return the adjoint of each Pauli in the list.

anticommutes(other, qargs=None)[source]

Return True if other Pauli that anticommutes with other.

Parameters
• other (PauliList) – another PauliList operator.

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

Returns

True if Pauli’s anticommute, False if they commute.

Return type

bool

anticommutes_with_all(other)[source]

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.

Parameters

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

Returns

index array of the anti-commuting rows.

Return type

array

argsort(weight=False, phase=False)[source]

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.

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

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

Returns

the indices for sorting the table.

Return type

array

commutes(other, qargs=None)[source]

Return True for each Pauli that commutes with other.

Parameters
• other (PauliList) – another PauliList operator.

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

Returns

True if Pauli’s commute, False if they anti-commute.

Return type

bool

commutes_with_all(other)[source]

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.

Parameters

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

Returns

index array of the commuting rows.

Return type

array

compose(other, qargs=None, front=False, inplace=False)[source]

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

Parameters
• 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

the list of composed Paulis.

Return type

PauliList

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.

conjugate()[source]

Return the conjugate of each Pauli in the list.

copy()

Make a deep copy of current operator.

delete(ind, qubit=False)[source]

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.

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

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

Returns

the resulting table with the entries removed.

Return type

PauliList

Raises

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

property dim

Return tuple (input_shape, output_shape).

dot(other, qargs=None, inplace=False)[source]

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

Parameters
• 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

the list of composed Paulis.

Return type

PauliList

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.

equiv(other)[source]

Entrywise comparison of Pauli equivalence up to global phase.

Parameters

other (PauliList or Pauli) – a comparison object.

Returns

An array of True or False for entrywise equivalence

of the current table.

Return type

np.ndarray

evolve(other, qargs=None)[source]

Evolve the Pauli by a Clifford.

This returns the Pauli $$P^\prime = C.P.C^\dagger$$.

Parameters
• other (Pauli or Clifford or QuantumCircuit) – The Clifford operator to evolve by.

• qargs (list) – a list of qubits to apply the Clifford to.

Returns

the Pauli $$C.P.C^\dagger$$.

Return type

Pauli

Raises

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

expand(other)[source]

Return the expand product of each Pauli in the list.

Parameters

other (PauliList) – another PauliList.

Returns

the list of tensor product Paulis.

Return type

PauliList

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.

classmethod from_symplectic(z, x, phase=0)[source]

Construct a PauliList from a symplectic data.

Parameters
• 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

the constructed PauliList.

Return type

PauliList

input_dims(qargs=None)

Return tuple of input dimension for specified subsystems.

insert(ind, value, qubit=False)[source]

Insert Pauli’s into the table.

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

Parameters
• 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

the resulting table with the entries inserted.

Return type

PauliList

Raises

QiskitError – if the insertion index is invalid.

inverse()[source]

Return the inverse of each Pauli in the list.

label_iter()[source]

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.

Returns

label iterator object for the PauliList.

Return type

LabelIterator

matrix_iter(sparse=False)[source]

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.

Parameters

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

Returns

matrix iterator object for the PauliList.

Return type

MatrixIterator

property num_qubits

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

output_dims(qargs=None)

Return tuple of output dimension for specified subsystems.

property phase

Return the phase exponent of the PauliList.

power(n)

Return the compose of a operator with itself n times.

Parameters

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

Returns

the n-times composed operator.

Return type

Pauli

Raises

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

property qargs

Return the qargs for the operator.

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

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

Parameters
• 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].

Returns

returns self with reshaped input and output dimensions.

Return type

BaseOperator

Raises

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

property settings

Return operator settings.

property shape

The full shape of the array()

property size

The number of Pauli rows in the table.

sort(weight=False, phase=False)[source]

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

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

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

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

Returns

a sorted copy of the original table.

Return type

PauliList

tensor(other)[source]

Return the tensor product with each Pauli in the list.

Parameters

other (PauliList) – another PauliList.

Returns

the list of tensor product Paulis.

Return type

PauliList

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.

to_labels(array=False)[source]

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 8 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"

$$[0, 1]$$

$$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$

Parameters

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

Returns

The rows of the PauliList in label form.

Return type

list or array

to_matrix(sparse=False, array=False)[source]

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 9 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"

$$[0, 1]$$

$$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$

Parameters
• 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

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.

Return type

list

transpose()[source]

Return the transpose of each Pauli in the list.

unique(return_index=False, return_counts=False)[source]

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']

Parameters
• 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

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.

Return type

PauliList

property x

The x array for the symplectic representation.

property z

The z array for the symplectic representation.