PauliTable¶
- class PauliTable(data)[source]¶
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.
¶ 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 usingto_labels()
andto_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()
anddot()
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 thearray
property, and the sub-arrays for only the X or Z blocks can be accessed using theX
andZ
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()
andmatrix_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.
- Parameters
data (array or str or ScalarOp or PauliTable) – 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.
Methods
Return the adjoint of the Operator.
Return indexes of rows that commute other.
Return indices for sorting the rows of the table.
Return list of commutation properties for each row with a Pauli.
Return indexes of rows that commute other.
Return the compose output product of two tables.
Not implemented.
Make a deep copy of current operator.
Return a copy with Pauli rows deleted from table.
Return the dot output product of two tables.
Return the expand output product of two tables.
Construct a PauliTable from a list of Pauli strings.
Return tuple of input dimension for specified subsystems.
Insert Pauli's into the table.
Return a label representation iterator.
Return a matrix representation iterator.
Return tuple of output dimension for specified subsystems.
Return the compose of a operator with itself n times.
Return a shallow copy with reshaped input and output subsystem dimensions.
Sort the rows of the table.
Return the tensor output product of two tables.
Convert a PauliTable to a list Pauli string labels.
Convert to a list or array of Pauli matrices.
Not implemented.
Return unique Paulis from the table.
Attributes
- 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
- size¶
The number of Pauli rows in the table.