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Pauli

Pauli(data=None, x=None, *, z=None, label=None) GitHub(opens in a new tab)

Bases: qiskit.quantum_info.operators.symplectic.base_pauli.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).\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 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-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 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 and xx arrays

P=Pn1...P0Pk=(i)z[k]x[k]Zz[k]Xx[k]\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 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 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’.

Raises

QiskitError – if input array is invalid shape.


Methods

adjoint

Pauli.adjoint() GitHub(opens in a new tab)

Return the adjoint of the Operator.

anticommutes

Pauli.anticommutes(other, qargs=None) GitHub(opens in a new tab)

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

commutes

Pauli.commutes(other, qargs=None) GitHub(opens in a new tab)

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) GitHub(opens in a new tab)

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() GitHub(opens in a new tab)

Return the conjugate of each Pauli in the list.

copy

Pauli.copy()

Make a deep copy of current operator.

delete

Pauli.delete(qubits) GitHub(opens in a new tab)

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.

dot

Pauli.dot(other, qargs=None, inplace=False) GitHub(opens in a new tab)

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) GitHub(opens in a new tab)

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, frame='h') GitHub(opens in a new tab)

Heisenberg picture evolution of a Pauli by a Clifford.

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

By choosing the parameter frame=’s’, this function returns the Schrödinger evolution of the Pauli P=C.P.CP^\prime = C.P.C^\dagger. This option yields a faster calculation.

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 or ‘s’ for Schrödinger framework.

Returns

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

Return type

Pauli

Raises

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

expand

Pauli.expand(other) GitHub(opens in a new tab)

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

Pauli.input_dims(qargs=None)

Return tuple of input dimension for specified subsystems.

insert

Pauli.insert(qubits, value) GitHub(opens in a new tab)

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.

inverse

Pauli.inverse() GitHub(opens in a new tab)

Return the inverse of the Pauli.

output_dims

Pauli.output_dims(qargs=None)

Return tuple of output dimension for specified subsystems.

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.

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) GitHub(opens in a new tab)

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.

tensor

Pauli.tensor(other) GitHub(opens in a new tab)

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

Pauli.to_instruction() GitHub(opens in a new tab)

Convert to Pauli circuit instruction.

to_label

Pauli.to_label() GitHub(opens in a new tab)

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) GitHub(opens in a new tab)

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

transpose

Pauli.transpose() GitHub(opens in a new tab)

Return the transpose of each Pauli in the list.


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.

Return type

Dict

x

The x vector for the Pauli.

z

The z vector for the Pauli.

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