Pauli

class Pauli(data=None, x=None, *, z=None, label=None)

Bases: qiskit.quantum_info.operators.symplectic.base_pauli.BasePauli

N-qubit Pauli operator.

This class represents an operator $P$ from the full $n$ -qubit Pauli group

P = (-i)^{q} P_{n-1} \otimes ... \otimes P_{0}

where $q\in \mathbb{Z}_4$ and $P_i \in \{I, X, Y, Z\}$ are single-qubit Pauli matrices:

\begin{split}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}.\end{split}

Initialization

A Pauli object can be initialized in several ways:

Pauli(obj)

where obj is a Pauli string, Pauli or ScalarOp operator, or a Pauli gate or 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 $n$ -qubit Pauli may be represented by a string consisting of $n$ characters from ['I', 'X', 'Y', 'Z'], and optionally phase coefficient in $['', '-i', '-', 'i']$ . For example: XYZ or '-iZIZ'.

In the string representation qubit-0 corresponds to the right-most Pauli character, and qubit- $(n-1)$ to the left-most Pauli character. For example 'XYZ' represents $X\otimes Y \otimes Z$ 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 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 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 $n$ -qubit Pauli is two length- $n$ boolean vectors $z \in \mathbb{Z}_2^N$ , $x \in \mathbb{Z}_2^N$ , and an integer $q \in \mathbb{Z}_4$ defining the Pauli operator

P = (-i)^{q + z\cdot x} Z^z \cdot X^x.

The $k$ and $x$ arrays

\begin{split}P &= P_{n-1} \otimes ... \otimes P_{0} \\ P_k &= (-i)^{z[k] * x[k]} Z^{z[k]}\cdot X^{x[k]}\end{split}

where z[k] = P.z[k], x[k] = P.x[k] respectively.

The $z$ and $x$ arrays can be accessed and updated using the z and x properties respectively. The phase integer $q$ can be accessed and updated using the phase property.

Matrix Operator Representation

Pauli’s can be converted to $(2^n, 2^n)$ Operator using the to_operator() method, or to a dense or sparse complex matrix using the 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

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.

Parameters

data (str or tuple or Pauli orScalarOp) – 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’.
x (np.ndarray) – DEPRECATED, symplectic x vector.
z (np.ndarray) – DEPRECATED, symplectic z vector.
label (str) – DEPRECATED, string label.

Raises

QiskitError – if input array is invalid shape.

Methods

adjoint

Pauli.adjoint()

Return the adjoint of the Operator.

anticommutes

Pauli.anticommutes(other, qargs=None)

Return True if other Pauli anticommutes with self.

Parameters

other (Pauli) – another Pauli operator.
qargs (list) – qubits to apply dot product on (default: None).

Returns

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

Return type

bool

append_paulis

Pauli.append_paulis(paulis=None, pauli_labels=None)

DEPRECATED: Append pauli at the end.

Parameters

paulis (Pauli) – the to-be-inserted or appended pauli
pauli_labels (list[str]) – the to-be-inserted or appended pauli label

Returns

self

Return type

Pauli

commutes

Pauli.commutes(other, qargs=None)

Return True if the Pauli commutes with other.

Parameters

other (Pauli orPauliList) – another Pauli 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

compose

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

Return the operator composition with another Pauli.

Parameters

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

The composed Pauli.

Return type

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 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 dot() method A.dot(B) == A.compose(B, front=True).

conjugate

Pauli.conjugate()

Return the conjugate of each Pauli in the list.

copy

Pauli.copy()

Make a deep copy of current operator.

delete

Pauli.delete(qubits)

Return a Pauli with qubits deleted.

Parameters

qubits (int or list) – qubits to delete from Pauli.

Returns

the resulting Pauli with the specified qubits removed.

Return type

Pauli

Raises

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

delete_qubits

Pauli.delete_qubits(indices)

DEPRECATED: Delete pauli at the indices.

This function is deprecated. Equivalent functionality can be obtained using the delete() method.

Parameters

indices (list[int]) – the indices of to-be-deleted paulis

Returns

self

Return type

Pauli

dot

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

Return the right multiplied operator self * other.

Parameters

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

The operator self * other.

Return type

Pauli

equiv

Pauli.equiv(other)

Return True if Pauli’s are equivalent up to group phase.

Parameters

other (Pauli) – an operator object.

Returns

True if the Pauli’s are equivalent up to group phase.

Return type

bool

evolve

Pauli.evolve(other, qargs=None)

Heisenberg picture evolution of a Pauli by a Clifford.

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

Parameters

other (Pauli orClifford orQuantumCircuit) – The Clifford operator to evolve by.
qargs (list) – a list of qubits to apply the Clifford to.

Returns

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

Return type

Pauli

Raises

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

expand

Pauli.expand(other)

Return the reverse-order tensor product with another Pauli.

Parameters

other (Pauli) – a Pauli object.

Returns

the tensor product $b \otimes a$ , where $a$

is the current Pauli, and $b$ is the other Pauli.

Return type

Pauli

from_label

static Pauli.from_label(*args, **kwargs)

DEPRECATED: Construct a Pauli from a string label.

This function is deprecated use Pauli(label) instead.

Parameters

label (str) – Pauli string label.

Returns

the constructed Pauli.

Return type

Pauli

Raises

QiskitError – If the input list is empty or contains invalid
Pauli strings. –

input_dims

Pauli.input_dims(qargs=None)

Return tuple of input dimension for specified subsystems.

insert

Pauli.insert(qubits, value)

Insert a Pauli at specific qubit value.

Parameters

qubits (int or list) – qubits index to insert at.
value (Pauli) – value to insert.

Returns

the resulting Pauli with the entries inserted.

Return type

Pauli

Raises

QiskitError – if the insertion qubits are invalid.

insert_paulis

Pauli.insert_paulis(indices=None, paulis=None, pauli_labels=None)

DEPRECATED: Insert or append pauli to the targeted indices.

This function is deprecated. Similar functionality can be obtained using the insert() method.

If indices is None, it means append at the end.

Parameters

indices (list[int]) – the qubit indices to be inserted
paulis (Pauli) – the to-be-inserted or appended pauli
pauli_labels (list[str]) – the to-be-inserted or appended pauli label

Note

the indices refers to the location of original paulis, e.g. if indices = [0, 2], pauli_labels = [‘Z’, ‘I’] and original pauli = ‘ZYXI’ the pauli will be updated to ZY’I’XI’Z’ ‘Z’ and ‘I’ are inserted before the qubit at 0 and 2.

Returns

self

Return type

Pauli

Raises

QiskitError – provide both paulis and pauli_labels at the same time

inverse

Pauli.inverse()

Return the inverse of the Pauli.

kron

Pauli.kron(other)

DEPRECATED: Kronecker product of two paulis.

This function is deprecated. Use expand() instead.

Order is $P_2 (other) otimes P_1 (self)$

Parameters

other (Pauli) – P2

Returns

self

Return type

Pauli

output_dims

Pauli.output_dims(qargs=None)

Return tuple of output dimension for specified subsystems.

pauli_single

classmethod Pauli.pauli_single(num_qubits, index, pauli_label)

DEPRECATED: Generate single qubit pauli at index with pauli_label with length num_qubits.

Parameters

num_qubits (int) – the length of pauli
index (int) – the qubit index to insert the single qubit
pauli_label (str) – pauli

Returns

single qubit pauli

Return type

Pauli

power

Pauli.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.

random

classmethod Pauli.random(num_qubits, seed=None)

DEPRECATED: Return a random Pauli on number of qubits.

This function is deprecated use random_pauli() instead.

Parameters

num_qubits (int) – the number of qubits
seed (int) – Optional. To set a random seed.

Returns

the random pauli

Return type

Pauli

reshape

Pauli.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.

set_truncation

classmethod Pauli.set_truncation(val)

Set the max number of Pauli characters to display before truncation/

Parameters

val (int) – the number of characters.

Note

Truncation will be disabled if the truncation value is set to 0.

sgn_prod

static Pauli.sgn_prod(*args, **kwargs)

DEPRECATED: Multiply two Paulis and track the phase.

This function is deprecated. The Pauli class now handles full Pauli group multiplication using compose() or dot().

$P_3 = P_1 otimes P_2$: X*Y

Parameters

p1 (Pauli) – pauli 1
p2 (Pauli) – pauli 2

Returns

the multiplied pauli (without phase) complex: the sign of the multiplication, 1, -1, 1j or -1j

Return type

Pauli

tensor

Pauli.tensor(other)

Return the tensor product with another Pauli.

Parameters

other (Pauli) – a Pauli object.

Returns

the tensor product $a \otimes b$ , where $a$

is the current Pauli, and $b$ is the other Pauli.

Return type

Pauli

Note

The tensor product can be obtained using the ^ binary operator. Hence a.tensor(b) is equivalent to a ^ b.

to_instruction

Pauli.to_instruction()

Convert to Pauli circuit instruction.

to_label

Pauli.to_label()

Convert a Pauli to a string label.

Note

The difference between to_label and __str__() is that the later will truncate the output for large numbers of qubits.

Returns

the Pauli string label.

Return type

str

to_matrix

Pauli.to_matrix(sparse=False)

Convert to a Numpy array or sparse CSR matrix.

Parameters

sparse (bool) – if True return sparse CSR matrices, otherwise return dense Numpy arrays (default: False).

Returns

The Pauli matrix.

Return type

array

to_spmatrix

Pauli.to_spmatrix()

DEPRECATED Convert Pauli to a sparse matrix representation (CSR format).

This function is deprecated. Use to_matrix() with kwarg sparse=True instead.

Returns

a sparse matrix with CSR format that represents the pauli.

Return type

scipy.sparse.csr_matrix

transpose

Pauli.transpose()

Return the transpose of each Pauli in the list.

update_x

Pauli.update_x(x, indices=None)

DEPRECATED: Update partial or entire x.

This function is deprecated. Use the setter for X instead.

Parameters

x (numpy.ndarray or list) – to-be-updated x
indices (numpy.ndarray or list or optional) – to-be-updated qubit indices

Returns

self

Return type

Pauli

Raises

QiskitError – when updating whole x, the number of qubits must be the same.

update_z

Pauli.update_z(z, indices=None)

DEPRECATED: Update partial or entire z.

This function is deprecated. Use the setter for Z instead.

Parameters

z (numpy.ndarray or list) – to-be-updated z
indices (numpy.ndarray or list or optional) – to-be-updated qubit indices

Returns

self

Return type

Pauli

Raises

QiskitError – when updating whole z, the number of qubits must be the same.

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 group phase exponent for the Pauli.

qargs

Return the qargs for the operator.

settings

Return settings.

Return type

Dict

x

The x vector for the Pauli.

z

The z vector for the Pauli.

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