Pauli
class qiskit.quantum_info.Pauli(data=None)
Bases: BasePauli
N-qubit Pauli operator.
This class represents an operator from the full -qubit Pauli group
where and are single-qubit Pauli matrices:
Initialization
A Pauli object can be initialized in several ways:
Pauli(obj)
where
obj
is a Pauli string,Pauli
orScalarOp
operator, or a Pauli gate orQuantumCircuit
containing only Pauli gates.
Pauli((z, x, phase))
where
z
andx
are booleannumpy.ndarrays
andphase
is an integer in[0, 1, 2, 3]
.
Pauli((z, x))
equivalent to
Pauli((z, x, 0))
with trivial phase.
String representation
An -qubit Pauli may be represented by a string consisting of 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- to the left-most Pauli character. For example 'XYZ'
represents 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)
.
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 -qubit Pauli is two length- boolean vectors , , and an integer defining the Pauli operator
The -th qubit corresponds to the -th entry in the and arrays
where z[k] = P.z[k]
, x[k] = P.x[k]
respectively.
The and arrays can be accessed and updated using the z
and x
properties respectively. The phase integer can be accessed and updated using the phase
property.
Matrix Operator Representation
Pauli’s can be converted to 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 . 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
anticommutes
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
apply_layout
apply_layout(layout, num_qubits=None)
Apply a transpiler layout to this Pauli
Parameters
- layout (TranspileLayout |list[int] | None) – Either a
TranspileLayout
, a list of integers or None. If both layout and num_qubits are none, a copy of the operator is returned. - num_qubits (int | None) – The number of qubits to expand the operator to. If not provided then if
layout
is aTranspileLayout
the number of the transpiler output circuit qubits will be used by default. Iflayout
is a list of integers the permutation specified will be applied without any expansion. If layout is None, the operator will be expanded to the given number of qubits.
Returns
A new Pauli
with the provided layout applied
Return type
commutes
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
compose
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
Raises
QiskitError – if other cannot be converted to an operator, or has incompatible dimensions for specified subsystems.
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
copy
delete
delete(qubits)
Return a Pauli with qubits deleted.
Parameters
qubits (int orlist) – qubits to delete from Pauli.
Returns
the resulting Pauli with the specified qubits removed.
Return type
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
- 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
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
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 .
Schrödinger picture evolves the Pauli as .
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 - framework. (Schrödinger) –
Returns
the Pauli (Heisenberg picture) or the Pauli (Schrödinger picture).
Return type
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 , where
is the current Pauli, and is the other Pauli.
Return type
input_dims
insert
insert(qubits, value)
Insert a Pauli at specific qubit value.
Parameters
Returns
the resulting Pauli with the entries inserted.
Return type
Raises
QiskitError – if the insertion qubits are invalid.
inverse
output_dims
power
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
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) – 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 set_truncation(val)
Set the max number of Pauli characters to display before truncation/
Parameters
val (int) – the number of characters.
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 , where
is the current Pauli, and is the other Pauli.
Return type
The tensor product can be obtained using the ^
binary operator. Hence a.tensor(b)
is equivalent to a ^ b
.
to_instruction
to_label
to_label()
Convert a Pauli to a string label.
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
to_matrix
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