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Pauli

qiskit.quantum_info.Pauli(data=None)GitHub(opens in a new tab)

Bases: BasePauli

N-qubit Pauli operator.

This class represents an operator PP from the full nn-qubit Pauli group

P=(i)qPn1...P0P = (-i)^{q} P_{n-1} \otimes ... \otimes P_{0}

where qZ4q\in \mathbb{Z}_4 and Pi{I,X,Y,Z}P_i \in \{I, X, Y, Z\} are single-qubit Pauli matrices:

I=(1001),X=(0110),Y=(0ii0),Z=(1001).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 nn-qubit Pauli may be represented by a string consisting of nn characters from ['I', 'X', 'Y', 'Z'], and optionally phase coefficient in [,i,,i]['', '-i', '-', 'i']. For example: XYZ or '-iZIZ'.

In the string representation qubit-0 corresponds to the right-most Pauli character, and qubit-(n1)(n-1) to the left-most Pauli character. For example 'XYZ' represents XYZX\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 nn-qubit Pauli is two length-nn boolean vectors zZ2Nz \in \mathbb{Z}_2^N, xZ2Nx \in \mathbb{Z}_2^N, and an integer qZ4q \in \mathbb{Z}_4 defining the Pauli operator

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

The kk-th qubit corresponds to the kk-th entry in the zz and xx arrays

P=Pn1...P0Pk=(i)z[k]x[k]Zz[k]Xx[k]\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 zz and xx arrays can be accessed and updated using the z and x properties respectively. The phase integer qq can be accessed and updated using the phase property.

Matrix Operator Representation

Pauli’s can be converted to (2n,2n)(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

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(opens in a new tab) ortuple(opens in a new tab) 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 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()

Return the adjoint of the Operator.

anticommutes

anticommutes(other, qargs=None)

Return True if other Pauli anticommutes with self.

Parameters

Returns

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

Return type

bool(opens in a new tab)

commutes

commutes(other, qargs=None)

Return True if the Pauli commutes with other.

Parameters

Returns

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

Return type

bool(opens in a new tab)

compose

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

Return the operator composition with another Pauli.

Parameters

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

conjugate()

Return the conjugate of each Pauli in the list.

copy

copy()

Make a deep copy of current operator.

delete

delete(qubits)

Return a Pauli with qubits deleted.

Parameters

qubits (int(opens in a new tab) orlist(opens in a new tab)) – 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.

dot

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

Return the right multiplied operator self * other.

Parameters

Returns

The operator self * other.

Return type

Pauli

equiv

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(opens in a new tab)

evolve

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

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

Parameters

Returns

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

Return type

Pauli

Raises

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

expand

expand(other)

Return the reverse-order tensor product with another Pauli.

Parameters

other (Pauli) – a Pauli object.

Returns

the tensor product bab \otimes a, where aa

is the current Pauli, and bb is the other Pauli.

Return type

Pauli

input_dims

input_dims(qargs=None)

Return tuple of input dimension for specified subsystems.

insert

insert(qubits, value)

Insert a Pauli at specific qubit value.

Parameters

Returns

the resulting Pauli with the entries inserted.

Return type

Pauli

Raises

QiskitError – if the insertion qubits are invalid.

inverse

inverse()

Return the inverse of the Pauli.

output_dims

output_dims(qargs=None)

Return tuple of output dimension for specified subsystems.

power

power(n)

Return the compose of a operator with itself n times.

Parameters

n (int(opens in a new tab)) – 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)

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

Parameters

  • input_dims (None or tuple(opens in a new tab)) – new subsystem input dimensions. If None the original input dims will be preserved [Default: None].
  • output_dims (None or tuple(opens in a new tab)) – new subsystem output dimensions. If None the original output dims will be preserved [Default: None].
  • num_qubits (None or int(opens in a new tab)) – 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 set_truncation(val)

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

Parameters

val (int(opens in a new tab)) – the number of characters.

Note

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

tensor

tensor(other)

Return the tensor product with another Pauli.

Parameters

other (Pauli) – a Pauli object.

Returns

the tensor product aba \otimes b, where aa

is the current Pauli, and bb 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

to_instruction()

Convert to Pauli circuit instruction.

to_label

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(opens in a new tab)

to_matrix

to_matrix(sparse=False)

Convert to a Numpy array or sparse CSR matrix.

Parameters

sparse (bool(opens in a new tab)) – if True return sparse CSR matrices, otherwise return dense Numpy arrays (default: False).

Returns

The Pauli matrix.

Return type

array

transpose

transpose()

Return the transpose of each Pauli in the list.

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