# PauliTable¶

class PauliTable(data)[source]

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.

Table 8 Pauli Representations

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 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.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 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

Initialize the PauliTable.

Parameters

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

Raises

QiskitError – if input array is invalid shape.

 PauliTable.X The X block of the array. PauliTable.Z The Z block of the array. PauliTable.array The underlying boolean array. PauliTable.atol The default absolute tolerance parameter for float comparisons. PauliTable.dim Return tuple (input_shape, output_shape). PauliTable.num_qubits Return the number of qubits if a N-qubit operator or None otherwise. PauliTable.qargs Return the qargs for the operator. PauliTable.rtol The relative tolerance parameter for float comparisons. PauliTable.shape The full shape of the array() PauliTable.size The number of Pauli rows in the table.
 Return a clone with qargs set Return a view of the PauliTable. Return the number of Pauli rows in the table. PauliTable.__mul__(other) PauliTable.add(other) Return the linear operator self + other. Return the adjoint of the operator. Return indexes of rows that commute other. PauliTable.argsort([weight]) 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. PauliTable.compose(other[, qargs, front]) Return the compose output product of two tables. Not implemented. Make a deep copy of current operator. PauliTable.delete(ind[, qubit]) Return a copy with Pauli rows deleted from table. PauliTable.dot(other[, qargs]) Return the dot output product of two tables. PauliTable.expand(other) Return the expand output product of two tables. PauliTable.from_labels(labels) Construct a PauliTable from a list of Pauli strings. PauliTable.input_dims([qargs]) Return tuple of input dimension for specified subsystems. PauliTable.insert(ind, value[, qubit]) Insert Pauli’s into the table. Return a label representation iterator. PauliTable.matrix_iter([sparse]) Return a matrix representation iterator. Return the linear operator other * self. PauliTable.output_dims([qargs]) Return tuple of output dimension for specified subsystems. Return the compose of a operator with itself n times. PauliTable.reshape([input_dims, output_dims]) Return a shallow copy with reshaped input and output subsystem dimensions. Set the class default absolute tolerance parameter for float comparisons. Set the class default relative tolerance parameter for float comparisons. PauliTable.sort([weight]) Sort the rows of the table. Return the linear operator self - other. PauliTable.tensor(other) Return the tensor output product of two tables. PauliTable.to_labels([array]) Convert a PauliTable to a list Pauli string labels. PauliTable.to_matrix([sparse, array]) Convert to a list or array of Pauli matrices. Not implemented. PauliTable.unique([return_index, return_counts]) Return unique Paulis from the table. Return a clone with qargs set PauliTable.__mul__(other)