StabilizerTable

class StabilizerTable(data, phase=None)

Bases: PauliTable, AdjointMixin

DEPRECATED: Symplectic representation of a list Stabilizer matrices.

Symplectic Representation

The symplectic representation of a single-qubit Stabilizer matrix is a pair of boolean values \([x, z]\) and a boolean phase p such that the Stabilizer matrix is given by \(S = (-1)^p \sigma_z^z.\sigma_x^x\). The correspondence between labels, symplectic representation, stabilizer matrices, and Pauli matrices for the single-qubit case is shown in the following table.

Table 8 Table 1: Stabilizer Representations

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

Internally this is stored as a length N boolean phase vector \([p_{N-1}, ..., p_{0}]\) and a PauliTable \(M \times 2N\) 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 \(X_i = [x_{i,0}, ..., x_{i,N-1}]\), \(Z_i = [z_{i,0}, ..., z_{i,N-1}]\) is the symplectic representation of an N-qubit Pauli. This representation is based on reference [1].

StabilizerTable’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 product of the stabilizer elements is defined with respect to the matrix multiplication of the matrices in Table 1. In terms of stabilizes labels the dot product group structure is

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

The dot() method will return the output for row.dot(col) = row.col, while the compose() will return row.compose(col) = col.row from the above table.

Note that while this dot product is different to the matrix product of the PauliTable, it does not change the commutation structure of elements. Hence commutes:() will be the same for the same labels.

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 StabilizerTable. The underlying phase vector and Pauli array can be directly accessed using the phase and array properties respectively. The sub-arrays for only the X or Z blocks can be accessed using the X and Z properties respectively.

The Pauli part of the Stabilizer table can be viewed and accessed as a PauliTable object using the pauli property. Note that this doesn’t copy the underlying array so any changes made to the Pauli table will also change the stabilizer table.

Iteration

Rows in the Stabilizer 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 StabilizerTable.

Obsoleto desde a versão 0.24.0: The class qiskit.quantum_info.operators.symplectic.stabilizer_table.StabilizerTable 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

Parâmetros

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

• phase (array or bool or None) – optional phase vector for input data (Default: None).

Levanta

QiskitError – if input array or phase vector has an invalid shape.

Additional Information:

The input array is not copied so multiple Pauli and Stabilizer 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 PauliTable.
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	Return a copy of the StabilizerTable.
delete	Return a copy with Stabilizer 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 StabilizerTable from a list of Pauli stabilizer strings.
input_dims	Return tuple of input dimension for specified subsystems.
insert	Insert stabilizers'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 StabilizerTable to a list Pauli stabilizer string labels.
to_matrix	Convert to a list or array of Stabilizer matrices.
transpose	Not implemented.
unique	Return unique stabilizers 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.

pauli

Return PauliTable

phase

Return phase vector

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.