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# PauliTable

class PauliTable(data)

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Bases: qiskit.quantum_info.operators.base_operator.BaseOperator, qiskit.quantum_info.operators.mixins.adjoint.AdjointMixin

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.

LabelSymplecticMatrix
"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.BIXYZ
IIXYZ
XXIZY
YYZIX
ZZYXI

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(opens in a new tab)

Initialize the PauliTable.

Parameters

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

Raises

QiskitError – if input array is invalid shape.

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

## Methods

PauliTable.adjoint()

Return the adjoint of the Operator.

### anticommutes_with_all

PauliTable.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

PauliTable.argsort(weight=False)

Return indices for sorting 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.

Parameters

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

Returns

the indices for sorting the table.

Return type

array

### commutes

PauliTable.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

PauliTable.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

PauliTable.compose(other, qargs=None, front=True)

Return the compose output product of two tables.

This returns the combination of the dot product of all Paulis in the current table with all Pauli’s in the other table and discards the complex phase from the product. Note that for PauliTables this method is equivalent to dot() and hence the front kwarg does not change the output.

Example

from qiskit.quantum_info.operators import PauliTable

current = PauliTable.from_labels(['I', 'X'])
other =  PauliTable.from_labels(['Y', 'Z'])
print(current.compose(other))
PauliTable: ['Y', 'Z', 'Z', 'Y']

Parameters

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

Returns

the compose outer product table.

Return type

PauliTable

Raises

QiskitError – if other cannot be converted to a PauliTable.

### conjugate

PauliTable.conjugate()

Not implemented.

### copy

PauliTable.copy()

Make a deep copy of current operator.

### delete

PauliTable.delete(ind, qubit=False)

Return a copy with Pauli rows deleted from table.

When deleting qubits the qubit index is the same as the column index 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 Pauli rows (Default: False).

Returns

the resulting table with the entries removed.

Return type

PauliTable

Raises

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

### dot

PauliTable.dot(other, qargs=None)

Return the dot output product of two tables.

This returns the combination of the dot product of all Paulis in the current table with all Pauli’s in the other table and discards the complex phase from the product. Note that for PauliTables this method is equivalent to compose().

Example

from qiskit.quantum_info.operators import PauliTable

current = PauliTable.from_labels(['I', 'X'])
other =  PauliTable.from_labels(['Y', 'Z'])
print(current.dot(other))
PauliTable: ['Y', 'Z', 'Z', 'Y']

Parameters

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

Returns

the dot outer product table.

Return type

PauliTable

Raises

QiskitError – if other cannot be converted to a PauliTable.

### expand

PauliTable.expand(other)

Return the expand output product of two tables.

This returns the combination of the tensor product of all Paulis in the other table with all Pauli’s in the current table, with the current tables qubits being the least-significant in the returned table. This is the opposite tensor order to tensor().

Example

from qiskit.quantum_info.operators import PauliTable

current = PauliTable.from_labels(['I', 'X'])
other =  PauliTable.from_labels(['Y', 'Z'])
print(current.expand(other))
PauliTable: ['YI', 'YX', 'ZI', 'ZX']

Parameters

other (PauliTable) – another PauliTable.

Returns

the expand outer product table.

Return type

PauliTable

Raises

QiskitError – if other cannot be converted to a PauliTable.

### from_labels

classmethod PauliTable.from_labels(labels)

Construct a PauliTable from a list of Pauli strings.

Parameters

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

Returns

the constructed PauliTable.

Return type

PauliTable

Raises

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

### input_dims

PauliTable.input_dims(qargs=None)

Return tuple of input dimension for specified subsystems.

### insert

PauliTable.insert(ind, value, qubit=False)

Insert Pauli’s into the table.

When inserting qubits the qubit index is the same as the column index of the underlying X and Z arrays.

Parameters

• ind (int) – index to insert at.
• value (PauliTable) – 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

PauliTable

Raises

QiskitError – if the insertion index is invalid.

### label_iter

PauliTable.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 PauliTable.

Return type

LabelIterator

### matrix_iter

PauliTable.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 PauliTable.

Return type

MatrixIterator

### output_dims

PauliTable.output_dims(qargs=None)

Return tuple of output dimension for specified subsystems.

### power

PauliTable.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

PauliTable.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.

### sort

PauliTable.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.

Example

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

from numpy.random import shuffle
from qiskit.quantum_info.operators import PauliTable

# 2-qubit labels
labels = ['II', 'IX', 'IY', 'IZ', 'XI', 'XX', 'XY', 'XZ',
'YI', 'YX', 'YY', 'YZ', 'ZI', 'ZX', 'ZY', 'ZZ']
# Shuffle Labels
shuffle(labels)
pt = PauliTable.from_labels(labels)
print('Initial Ordering')
print(pt)

# Lexicographic Ordering
srt = pt.sort()
print('Lexicographically sorted')
print(srt)

# Weight Ordering
srt = pt.sort(weight=True)
print('Weight sorted')
print(srt)
Initial Ordering
PauliTable: [
'IZ', 'XZ', 'ZY', 'YI', 'YZ', 'IX', 'II', 'ZI', 'IY', 'XY', 'XI', 'YY', 'ZX',
'XX', 'ZZ', 'YX'
]
Lexicographically sorted
PauliTable: [
'II', 'IX', 'IY', 'IZ', 'XI', 'XX', 'XY', 'XZ', 'YI', 'YX', 'YY', 'YZ', 'ZI',
'ZX', 'ZY', 'ZZ'
]
Weight sorted
PauliTable: [
'II', 'IX', 'IY', 'IZ', 'XI', 'YI', 'ZI', 'XX', 'XY', 'XZ', 'YX', 'YY', 'YZ',
'ZX', 'ZY', 'ZZ'
]

Parameters

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

Returns

a sorted copy of the original table.

Return type

PauliTable

### tensor

PauliTable.tensor(other)

Return the tensor output product of two tables.

This returns the combination of the tensor product of all Paulis in the current table with all Pauli’s in the other table, with the other tables qubits being the least-significant in the returned table. This is the opposite tensor order to expand().

Example

from qiskit.quantum_info.operators import PauliTable

current = PauliTable.from_labels(['I', 'X'])
other =  PauliTable.from_labels(['Y', 'Z'])
print(current.tensor(other))
PauliTable: ['IY', 'IZ', 'XY', 'XZ']

Parameters

other (PauliTable) – another PauliTable.

Returns

the tensor outer product table.

Return type

PauliTable

Raises

QiskitError – if other cannot be converted to a PauliTable.

### to_labels

PauliTable.to_labels(array=False)

Convert a PauliTable to a list Pauli string labels.

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

LabelSymplecticMatrix

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

Parameters

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

Returns

The rows of the PauliTable in label form.

Return type

list or array

### to_matrix

PauliTable.to_matrix(sparse=False, array=False)

Convert to a list or array of Pauli matrices.

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

LabelSymplecticMatrix

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

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

PauliTable.transpose()

Not implemented.

### unique

PauliTable.unique(return_index=False, return_counts=False)

Return unique Paulis from the table.

Example

from qiskit.quantum_info.operators import PauliTable

pt = PauliTable.from_labels(['X', 'Y', 'X', 'I', 'I', 'Z', 'X', 'Z'])
unique = pt.unique()
print(unique)
PauliTable: ['X', 'Y', 'I', 'Z']

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

PauliTable

## 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.

Return type

Dict

### shape

The full shape of the array()

### size

The number of Pauli rows in the table.