# PauliList

class qiskit.quantum_info.PauliList(data)

GitHub

Bases: 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.

Parameters

data (Pauli orlist) – 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.

## 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.

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()

GitHub

Return the adjoint of each Pauli in the list.

### anticommutes

anticommutes(other, qargs=None)

GitHub

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 Paulis anticommute, False if they commute.

Return type

bool

### anticommutes_with_all

anticommutes_with_all(other)

GitHub

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.

Parameters

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

Returns

index array of the anti-commuting rows.

Return type

array

### argsort

argsort(weight=False, phase=False)

GitHub

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.

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

commutes(other, qargs=None)

GitHub

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 Paulis commute, False if they anti-commute.

Return type

bool

### commutes_with_all

commutes_with_all(other)

GitHub

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.

Parameters

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

Returns

index array of the commuting rows.

Return type

array

### compose

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

GitHub

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

conjugate()

GitHub

Return the conjugate of each Pauli in the list.

### copy

copy()

GitHub

Make a deep copy of current operator.

### delete

delete(ind, qubit=False)

GitHub

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 orlist) – 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.

### dot

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

GitHub

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

equiv(other)

GitHub

Entrywise comparison of Pauli equivalence up to global phase.

Parameters

other (PauliList orPauli) – a comparison object.

Returns

An array of True or False for entrywise equivalence

of the current table.

Return type

np.ndarray

### evolve

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

GitHub

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 $P^\prime = C.P.C^\dagger$.

Parameters

• other (Pauli orClifford orQuantumCircuit) – 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.

Returns

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

Return type

PauliList

Raises

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

### expand

expand(other)

GitHub

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.

### from_symplectic

classmethod from_symplectic(z, x, phase=0)

GitHub

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

### group_commuting

group_commuting(qubit_wise=False)

GitHub

Partition a PauliList into sets of commuting Pauli strings.

Parameters

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'])]

Returns

List of PauliLists where each PauliList contains commuting Pauli operators.

Return type

### group_qubit_wise_commuting

group_qubit_wise_commuting()

GitHub

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

Returns

List of PauliLists where each PauliList contains commutable Pauli operators.

Return type

### input_dims

input_dims(qargs=None)

GitHub

Return tuple of input dimension for specified subsystems.

### insert

insert(ind, value, qubit=False)

GitHub

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.

Parameters

• ind (int) – index to insert at.
• value (PauliList) – values to insert.
• qubit (bool) – if True insert qubit columns, otherwise insert Pauli rows (Default: False).

Returns

the resulting table with the entries inserted.

Return type

PauliList

Raises

QiskitError – if the insertion index is invalid.

### inverse

inverse()

GitHub

Return the inverse of each Pauli in the list.

### label_iter

label_iter()

GitHub

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

matrix_iter(sparse=False)

GitHub

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

### noncommutation_graph

noncommutation_graph(qubit_wise)

GitHub

Create the non-commutation graph of this PauliList.

This transforms the measurement operator grouping problem into graph coloring problem. The constructed graph contains one node for each Pauli. The nodes will be connecting for any two Pauli terms that do _not_ commute.

Parameters

qubit_wise (bool) – whether the commutation rule is applied to the whole operator, or on a per-qubit basis.

Returns

the non-commutation graph with nodes for each Pauli and edges

indicating a non-commutation relation. Each node will hold the index of the Pauli term it corresponds to in its data. The edges of the graph hold no data.

Return type

rustworkx.PyGraph

### output_dims

output_dims(qargs=None)

GitHub

Return tuple of output dimension for specified subsystems.

### power

power(n)

GitHub

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

Clifford

Raises

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

### reshape

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

GitHub

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.

### sort

sort(weight=False, phase=False)

GitHub

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']

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

tensor(other)

GitHub

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

to_labels(array=False)

GitHub

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.

LabelSymplecticMatrix

| "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

to_matrix(sparse=False, array=False)

GitHub

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.

LabelSymplecticMatrix

| "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

transpose()

GitHub

Return the transpose of each Pauli in the list.

### unique

unique(return_index=False, return_counts=False)

GitHub

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