qiskit.quantum_info.StabilizerTable

class StabilizerTable(data, phase=None)

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.

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

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

Initialize the StabilizerTable.

Parameters

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

Raises

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.

init

__init__(data, phase=None)

Initialize the StabilizerTable.

Parameters

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

Raises

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


`__init__`(data[, phase])	Initialize the StabilizerTable.
`adjoint`()	Return the adjoint of the Operator.
`anticommutes_with_all`(other)	Return indexes of rows that commute other.
`argsort`([weight])	Return indices for sorting the rows of the PauliTable.
`commutes`(pauli)	Return list of commutation properties for each row with a Pauli.
`commutes_with_all`(other)	Return indexes of rows that commute other.
`compose`(other[, qargs, front])	Return the compose output product of two tables.
`conjugate`()	Not implemented.
`copy`()	Return a copy of the StabilizerTable.
`delete`(ind[, qubit])	Return a copy with Stabilizer rows deleted from table.
`dot`(other[, qargs])	Return the dot output product of two tables.
`expand`(other)	Return the expand output product of two tables.
`from_labels`(labels)	Construct a StabilizerTable from a list of Pauli stabilizer strings.
`input_dims`([qargs])	Return tuple of input dimension for specified subsystems.
`insert`(ind, value[, qubit])	Insert stabilizers’s into the table.
`label_iter`()	Return a label representation iterator.
`matrix_iter`([sparse])	Return a matrix representation iterator.
`output_dims`([qargs])	Return tuple of output dimension for specified subsystems.
`power`(n)	Return the compose of a operator with itself n times.
`reshape`([input_dims, output_dims, num_qubits])	Return a shallow copy with reshaped input and output subsystem dimensions.
`sort`([weight])	Sort the rows of the table.
`tensor`(other)	Return the tensor output product of two tables.
`to_labels`([array])	Convert a StabilizerTable to a list Pauli stabilizer string labels.
`to_matrix`([sparse, array])	Convert to a list or array of Stabilizer matrices.
`transpose`()	Not implemented.
`unique`([return_index, return_counts])	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.

X

The X block of the array.

Z

The Z block of the array.

adjoint

adjoint()

Return the adjoint of the Operator.

anticommutes_with_all

anticommutes_with_all(other)

Return indexes of rows that commute other.

If other is a multi-row Pauli table the returned vector indexes rows of the current PauliTable that anti-commute with all Pauli’s in other. If no rows satisfy the condition the returned array will be empty.

Parameters

other (PauliTable) – a single Pauli or multi-row PauliTable.

Returns

index array of the anti-commuting rows.

Return type

array

argsort

argsort(weight=False)

Return indices for sorting the rows of the PauliTable.

The default sort method is lexicographic sorting of Paulis by qubit number. By using the weight kwarg the output can additionally be sorted by the number of non-identity terms in the Stabilizer, where the set of all Pauli’s of a given weight are still ordered lexicographically.

This does not sort based on phase values. It will preserve the original order of rows with the same Pauli’s but different phases.

Parameters

weight (bool) – optionally sort by weight if True (Default: False).

Returns

the indices for sorting the table.

Return type

array

The underlying boolean array.

commutes

commutes(pauli)

Return list of commutation properties for each row with a Pauli.

The returned vector is the same length as the size of the table and contains True for rows that commute with the Pauli, and False for the rows that anti-commute.

Parameters

pauli (PauliTable) – a single Pauli row.

Returns

The boolean vector of which rows commute or anti-commute.

Return type

array

Raises

QiskitError – if input is not a single Pauli row.

commutes_with_all

commutes_with_all(other)

Return indexes of rows that commute other.

If other is a multi-row Pauli table the returned vector indexes rows of the current PauliTable that commute with all Pauli’s in other. If no rows satisfy the condition the returned array will be empty.

Parameters

other (PauliTable) – a single Pauli or multi-row PauliTable.

Returns

index array of the commuting rows.

Return type

array

compose

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

Return the compose output product of two tables.

This returns the combination of the compose product of all stabilizers in the current table with all stabilizers in the other table.

The individual stabilizer compose product is given by

`A.compose(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 |

If front=True the composition will be given by the dot() method.

Example

from qiskit.quantum_info.operators import StabilizerTable
 
current = StabilizerTable.from_labels(['+I', '-X'])
other =  StabilizerTable.from_labels(['+X', '-Z'])
print(current.compose(other))

StabilizerTable: ['+X', '-Z', '-I', '-Y']

Parameters

other (StabilizerTable) – another StabilizerTable.
qargs (None or list) – qubits to apply compose product on (Default: None).
front (bool) – If True use dot composition method (default: False).

Returns

the compose outer product table.

Return type

StabilizerTable

Raises

QiskitError – if other cannot be converted to a StabilizerTable.

conjugate

conjugate()

Not implemented.

copy

copy()

Return a copy of the StabilizerTable.

delete

delete(ind, qubit=False)

Return a copy with Stabilizer rows deleted from table.

When deleting qubit columns, qubit-0 is the right-most (largest index) column, and qubit-(N-1) is the left-most (0 index) column of the underlying X and Z arrays.

Parameters

ind (int or list) – index(es) to delete.
qubit (bool) – if True delete qubit columns, otherwise delete Stabilizer rows (Default: False).

Returns

the resulting table with the entries removed.

Return type

StabilizerTable

Raises

QiskitError – if ind is out of bounds for the array size or number of qubits.

dim

Return tuple (input_shape, output_shape).

dot

dot(other, qargs=None)

Return the dot output product of two tables.

This returns the combination of the compose product of all stabilizers in the current table with all stabilizers in the other table.

The individual stabilizer dot product is given by

`A.dot(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 |

Example

from qiskit.quantum_info.operators import StabilizerTable
 
current = StabilizerTable.from_labels(['+I', '-X'])
other =  StabilizerTable.from_labels(['+X', '-Z'])
print(current.dot(other))

StabilizerTable: ['+X', '-Z', '-I', '+Y']

Parameters

other (StabilizerTable) – another StabilizerTable.
qargs (None or list) – qubits to apply dot product on (Default: None).

Returns

the dot outer product table.

Return type

StabilizerTable

Raises

QiskitError – if other cannot be converted to a StabilizerTable.

expand

expand(other)

Return the expand output product of two tables.

This returns the combination of the tensor product of all stabilizers in the other table with all stabilizers in the current table. The current tables qubits will be the least-significant in the returned table. This is the opposite tensor order to tensor().

Example

from qiskit.quantum_info.operators import StabilizerTable
 
current = StabilizerTable.from_labels(['+I', '-X'])
other =  StabilizerTable.from_labels(['-Y', '+Z'])
print(current.expand(other))

StabilizerTable: ['-YI', '+YX', '+ZI', '-ZX']

Parameters

other (StabilizerTable) – another StabilizerTable.

Returns

the expand outer product table.

Return type

StabilizerTable

Raises

QiskitError – if other cannot be converted to a StabilizerTable.

from_labels

classmethod from_labels(labels)

Construct a StabilizerTable from a list of Pauli stabilizer strings.

Pauli Stabilizer string labels are Pauli strings with an optional "+" or "-" character. If there is no +/-sign a + phase is used by default.

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 | $[0, 1]$ | $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$ | $-Z$ |

Parameters

labels (list) – Pauli stabilizer string label(es).

Returns

the constructed StabilizerTable.

Return type

StabilizerTable

Raises

QiskitError – If the input list is empty or contains invalid Pauli stabilizer strings.

input_dims

input_dims(qargs=None)

Return tuple of input dimension for specified subsystems.

insert

insert(ind, value, qubit=False)

Insert stabilizers’s into the table.

When inserting qubit columns, qubit-0 is the right-most (largest index) column, and qubit-(N-1) is the left-most (0 index) column of the underlying X and Z arrays.

Parameters

ind (int) – index to insert at.
value (StabilizerTable) – values to insert.
qubit (bool) – if True delete qubit columns, otherwise delete Pauli rows (Default: False).

Returns

the resulting table with the entries inserted.

Return type

StabilizerTable

Raises

QiskitError – if the insertion index is invalid.

label_iter

label_iter()

Return a label representation iterator.

This is a lazy iterator that converts each row into the string label only as it is used. To convert the entire table to labels use the to_labels() method.

Returns

label iterator object for the StabilizerTable.

Return type

LabelIterator

matrix_iter

matrix_iter(sparse=False)

Return a matrix representation iterator.

This is a lazy iterator that converts each row into the Pauli matrix representation only as it is used. To convert the entire table to matrices use the to_matrix() method.

Parameters

sparse (bool) – optionally return sparse CSR matrices if True, otherwise return Numpy array matrices (Default: False)

Returns

matrix iterator object for the StabilizerTable.

Return type

MatrixIterator

num_qubits

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

output_dims

output_dims(qargs=None)

Return tuple of output dimension for specified subsystems.

pauli

Return PauliTable

phase

Return phase vector

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

Pauli

Raises

QiskitError – if the input and output dimensions of the operator are not equal, or the power is not a positive integer.

qargs

Return the qargs for the operator.

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.

settings

Return settings.

Return type

Dict

shape

The full shape of the array()

size

The number of Pauli rows in the table.

sort

sort(weight=False)

Sort the rows of the table.

The default sort method is lexicographic sorting by qubit number. By using the weight kwarg the output can additionally be sorted by the number of non-identity terms in the Pauli, where the set of all Pauli’s of a given weight are still ordered lexicographically.

This does not sort based on phase values. It will preserve the original order of rows with the same Pauli’s but different phases.

Consider sorting all a random ordering of all 2-qubit Paulis

from numpy.random import shuffle
from qiskit.quantum_info.operators import StabilizerTable
 
# 2-qubit labels
labels = ['+II', '+IX', '+IY', '+IZ', '+XI', '+XX', '+XY', '+XZ',
          '+YI', '+YX', '+YY', '+YZ', '+ZI', '+ZX', '+ZY', '+ZZ',
          '-II', '-IX', '-IY', '-IZ', '-XI', '-XX', '-XY', '-XZ',
          '-YI', '-YX', '-YY', '-YZ', '-ZI', '-ZX', '-ZY', '-ZZ']
# Shuffle Labels
shuffle(labels)
st = StabilizerTable.from_labels(labels)
print('Initial Ordering')
print(st)
 
# Lexicographic Ordering
srt = st.sort()
print('Lexicographically sorted')
print(srt)
 
# Weight Ordering
srt = st.sort(weight=True)
print('Weight sorted')
print(srt)

Initial Ordering
StabilizerTable: ['+ZX', '-YY', '-IZ', '+ZZ', '-XZ', '+II', '+ZI', '-ZI', '-XY', '+IZ', '+ZY', '-ZY', '-YX', '+YX', '-XI', '-IX', '+YI', '+XI', '+YZ', '-ZZ', '-YI', '-II', '-ZX', '+XZ', '+YY', '-YZ', '+IY', '-XX', '-IY', '+IX', '+XX', '+XY']
Lexicographically sorted
StabilizerTable: ['+II', '-II', '-IX', '+IX', '+IY', '-IY', '-IZ', '+IZ', '-XI', '+XI', '-XX', '+XX', '-XY', '+XY', '-XZ', '+XZ', '+YI', '-YI', '-YX', '+YX', '-YY', '+YY', '+YZ', '-YZ', '+ZI', '-ZI', '+ZX', '-ZX', '+ZY', '-ZY', '+ZZ', '-ZZ']
Weight sorted
StabilizerTable: ['+II', '-II', '-IX', '+IX', '+IY', '-IY', '-IZ', '+IZ', '-XI', '+XI', '+YI', '-YI', '+ZI', '-ZI', '-XX', '+XX', '-XY', '+XY', '-XZ', '+XZ', '-YX', '+YX', '-YY', '+YY', '+YZ', '-YZ', '+ZX', '-ZX', '+ZY', '-ZY', '+ZZ', '-ZZ']

Parameters

weight (bool) – optionally sort by weight if True (Default: False).

Returns

a sorted copy of the original table.

Return type

StabilizerTable

tensor

tensor(other)

Return the tensor output product of two tables.

This returns the combination of the tensor product of all stabilizers in the current table with all stabilizers in the other table. The other tables qubits will be the least-significant in the returned table. This is the opposite tensor order to tensor().

Example

from qiskit.quantum_info.operators import StabilizerTable
 
current = StabilizerTable.from_labels(['+I', '-X'])
other =  StabilizerTable.from_labels(['-Y', '+Z'])
print(current.tensor(other))

StabilizerTable: ['-IY', '+IZ', '+XY', '-XZ']

Parameters

other (StabilizerTable) – another StabilizerTable.

Returns

the tensor outer product table.

Return type

StabilizerTable

Raises

QiskitError – if other cannot be converted to a StabilizerTable.

to_labels

to_labels(array=False)

Convert a StabilizerTable to a list Pauli stabilizer string labels.

For large StabilizerTables converting using the array=True kwarg will be more efficient since it allocates memory for the full Numpy array of labels in advance.

Label	Phase	Symplectic	Matrix	Pauli

Parameters

array (bool) – return a Numpy array if True, otherwise return a list (Default: False).

Returns

The rows of the StabilizerTable in label form.

Return type

list or array

to_matrix

to_matrix(sparse=False, array=False)

Convert to a list or array of Stabilizer matrices.

For large StabilizerTables converting using the array=True kwarg will be more efficient since it allocates memory for the full rank-3 Numpy array of matrices in advance.

Label	Phase	Symplectic	Matrix	Pauli

Parameters

sparse (bool) – if True return sparse CSR matrices, otherwise return dense Numpy arrays (Default: False).
array (bool) – return as rank-3 numpy array if True, otherwise return a list of Numpy arrays (Default: False).

Returns

A list of dense Pauli matrices if array=False and sparse=False. list: A list of sparse Pauli matrices if array=False and sparse=True. array: A dense rank-3 array of Pauli matrices if array=True.

Return type

list

transpose

transpose()

Not implemented.

unique

unique(return_index=False, return_counts=False)

Return unique stabilizers from the table.

Example

from qiskit.quantum_info.operators import StabilizerTable
 
st = StabilizerTable.from_labels(['+X', '+I', '-I', '-X', '+X', '-X', '+I'])
unique = st.unique()
print(unique)

StabilizerTable: ['+X', '+I', '-I', '-X']

Parameters

return_index (bool) – If True, also return the indices that result in the unique array. (Default: False)
return_counts (bool) – If True, also return the number of times each unique item appears in the table.

Returns

unique

the table of the unique rows.

unique_indices: np.ndarray, optional

The indices of the first occurrences of the unique values in the original array. Only provided if return_index is True.

unique_counts: np.array, optional

The number of times each of the unique values comes up in the original array. Only provided if return_counts is True.

Return type

StabilizerTable

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