PauliTable

class PauliTable(data)

Bases: BaseOperator, AdjointMixin

DEPRECATED: Symplectic representation of a list Pauli matrices.

Symplectic Representation

The symplectic representation of a single-qubit Pauli matrix is a pair of boolean values \([x, z]\) such that the Pauli matrix is given by \(P = (-i)^{z * x} \sigma_z^z.\sigma_x^x\). The correspondence between labels, symplectic representation, and matrices for single-qubit Paulis are shown in Table 1.

Table 5 Pauli Representations

Label	Symplectic	Matrix
"I"	\([0, 0]\)	\(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)
"X"	\([1, 0]\)	\(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)
"Y"	\([1, 1]\)	\(\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}\)
"Z"	\([0, 1]\)	\(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)

The full Pauli table is a M x 2N boolean matrix:

\[\begin{split}\left(\begin{array}{ccc|ccc} x_{0,0} & ... & x_{0,N-1} & z_{0,0} & ... & z_{0,N-1} \\ x_{1,0} & ... & x_{1,N-1} & z_{1,0} & ... & z_{1,N-1} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ x_{M-1,0} & ... & x_{M-1,N-1} & z_{M-1,0} & ... & z_{M-1,N-1} \end{array}\right)\end{split}\]

where each row is a block vector \([X_i, Z_i]\) with \(X = [x_{i,0}, ..., x_{i,N-1}]\), \(Z = [z_{i,0}, ..., z_{i,N-1}]\) is the symplectic representation of an N-qubit Pauli. This representation is based on reference [1].

PauliTable’s can be created from a list of labels using from_labels(), and converted to a list of labels or a list of matrices using to_labels() and to_matrix() respectively.

Group Product

The Pauli’s in the Pauli table do not represent the full Pauli as they are restricted to having +1 phase. The dot-product for the Pauli’s is defined to discard any phase obtained from matrix multiplication so that we have \(X.Z = Z.X = Y\), etc. This means that for the PauliTable class the operator methods compose() and dot() are equivalent.

A.B	I	X	Y	Z
I	I	X	Y	Z
X	X	I	Z	Y
Y	Y	Z	I	X
Z	Z	Y	X	I

Qubit Ordering

The qubits are ordered in the table such the least significant qubit [x_{i, 0}, z_{i, 0}] is the first element of each of the \(X_i, Z_i\) vector blocks. This is the opposite order to position in string labels or matrix tensor products where the least significant qubit is the right-most string character. For example Pauli "ZX" has "X" on qubit-0 and "Z" on qubit 1, and would have symplectic vectors \(x=[1, 0]\), \(z=[0, 1]\).

Data Access

Subsets of rows can be accessed using the list access [] operator and will return a table view of part of the PauliTable. The underlying Numpy array can be directly accessed using the array property, and the sub-arrays for only the X or Z blocks can be accessed using the X and Z properties respectively.

Iteration

Rows in the Pauli table can be iterated over like a list. Iteration can also be done using the label or matrix representation of each row using the label_iter() and matrix_iter() methods.

References

1. S. Aaronson, D. Gottesman, Improved Simulation of Stabilizer Circuits, Phys. Rev. A 70, 052328 (2004). arXiv:quant-ph/0406196

Initialize the PauliTable.

Veraltet ab Version 0.24.0: The class qiskit.quantum_info.operators.symplectic.pauli_table.PauliTable is deprecated as of qiskit-terra 0.24.0. It will be removed no earlier than 3 months after the release date. Instead, use the class PauliList

Parameter

data (array or str or ScalarOp or PauliTable) – input data.

Verursacht

QiskitError – if input array is invalid shape.

Additional Information:

The input array is not copied so multiple Pauli tables can share the same underlying array.

Methods

adjoint	Return the adjoint of the Operator.
anticommutes_with_all	Return indexes of rows that commute other.
argsort	Return indices for sorting the rows of the table.
commutes	Return list of commutation properties for each row with a Pauli.
commutes_with_all	Return indexes of rows that commute other.
compose	Return the compose output product of two tables.
conjugate	Not implemented.
copy	Make a deep copy of current operator.
delete	Return a copy with Pauli rows deleted from table.
dot	Return the dot output product of two tables.
expand	Return the expand output product of two tables.
from_labels	Construct a PauliTable from a list of Pauli strings.
input_dims	Return tuple of input dimension for specified subsystems.
insert	Insert Pauli's into the table.
label_iter	Return a label representation iterator.
matrix_iter	Return a matrix representation iterator.
output_dims	Return tuple of output dimension for specified subsystems.
power	Return the compose of a operator with itself n times.
reshape	Return a shallow copy with reshaped input and output subsystem dimensions.
sort	Sort the rows of the table.
tensor	Return the tensor output product of two tables.
to_labels	Convert a PauliTable to a list Pauli string labels.
to_matrix	Convert to a list or array of Pauli matrices.
transpose	Not implemented.
unique	Return unique Paulis from the table.

Attributes

X

The X block of the array.

Z

The Z block of the array.

array

The underlying boolean array.

dim

Return tuple (input_shape, output_shape).

num_qubits

Return the number of qubits if a N-qubit operator or None otherwise.

qargs

Return the qargs for the operator.

settings

Return settings.

shape

The full shape of the array()

size

The number of Pauli rows in the table.