Pauli

class qiskit.quantum_info.Pauli(data=None)

Bases: 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:

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 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-2.

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 $k$ -th qubit corresponds to the $k$ -th entry in the $z$ and $x$ arrays

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

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

from qiskit.quantum_info import Pauli
 
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]))

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 ortuple orPauli 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 $\mathbb{Z}_4$ . 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.

Attributes

dim

Return tuple (input_shape, output_shape).

name

Unique string identifier for operation type.

num_clbits

Number of classical bits.

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.

x

The x vector for the Pauli.

z

The z vector for the Pauli.

Methods

adjoint

adjoint()

Pauli

Attributes

dim

name

num_clbits

num_qubits

phase

qargs

settings

x

z

Methods

adjoint

anticommutes

apply_layout

commutes

compose

conjugate

copy

delete

dot

equiv

evolve

expand

input_dims

insert

inverse

output_dims

power

reshape

set_truncation

tensor

to_instruction

to_label

to_matrix

transpose