PauliTable
class qiskit.quantum_info.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 such that the Pauli matrix is given by . The correspondence between labels, symplectic representation, and matrices for single-qubit Paulis are shown in Table 1.
| Label | Symplectic | Matrix |
|---|---|---|
"I" | ||
"X" | ||
"Y" | ||
"Z" |
The full Pauli table is a M x 2N boolean matrix:
where each row is a block vector with , 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 , 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 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 , .
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
- S. Aaronson, D. Gottesman, Improved Simulation of Stabilizer Circuits, Phys. Rev. A 70, 052328 (2004). arXiv:quant-ph/0406196
Initialize the PauliTable.
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
Parameters
data (array or str orScalarOp orPauliTable) – input data.
Raises
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.
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.
Methods
adjoint
adjoint()
Return the adjoint of the Operator.
Return type
Self
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 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
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=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
Raises
QiskitError – if other cannot be converted to a PauliTable.
conjugate
conjugate()
Not implemented.
copy
copy()
Make a deep copy of current operator.
delete
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 orlist) – 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
Raises
QiskitError – if ind is out of bounds for the array size or number of qubits.
dot
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
Raises
QiskitError – if other cannot be converted to a PauliTable.
expand
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
Raises
QiskitError – if other cannot be converted to a PauliTable.
from_labels
classmethod 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
Raises
- QiskitError – If the input list is empty or contains invalid
- Pauli strings. –
input_dims
input_dims(qargs=None)
Return tuple of input dimension for specified subsystems.
insert
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
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 PauliTable.
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 PauliTable.
Return type
MatrixIterator
output_dims
output_dims(qargs=None)
Return tuple of output dimension for specified subsystems.
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.
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.
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
tensor
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
Raises
QiskitError – if other cannot be converted to a PauliTable.
to_labels
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.
| Label | Symplectic | Matrix |
|---|
| "I" | | |
| "X" | | |
| "Y" | | |
| "Z" | | |
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
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.
| Label | Symplectic | Matrix |
|---|
| "I" | | |
| "X" | | |
| "Y" | | |
| "Z" | | |
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
transpose
transpose()
Not implemented.
unique
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