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PauliTable

class PauliTable(data)[소스]

기반 클래스: BaseOperator, 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.

Table 5 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.

참조

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

Initialize the PauliTable.

매개변수

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

예외 발생

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

adjoint

Return the adjoint of the Operator.

anticommutes_with_all

Return indexes of rows that commute other.

argsort

Return indices for sorting the rows of the table.

commutes

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

commutes_with_all

Return indexes of rows that commute other.

compose

Return the compose output product of two tables.

conjugate

Not implemented.

copy

Make a deep copy of current operator.

delete

Return a copy with Pauli rows deleted from table.

dot

Return the dot output product of two tables.

expand

Return the expand output product of two tables.

from_labels

Construct a PauliTable from a list of Pauli strings.

input_dims

Return tuple of input dimension for specified subsystems.

insert

Insert Pauli's into the table.

label_iter

Return a label representation iterator.

matrix_iter

Return a matrix representation iterator.

output_dims

Return tuple of output dimension for specified subsystems.

power

Return the compose of a operator with itself n times.

reshape

Return a shallow copy with reshaped input and output subsystem dimensions.

sort

Sort the rows of the table.

tensor

Return the tensor output product of two tables.

to_labels

Convert a PauliTable to a list Pauli string labels.

to_matrix

Convert to a list or array of Pauli matrices.

transpose

Not implemented.

unique

Return unique Paulis 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.

qargs

Return the qargs for the operator.

settings

Return settings.

반환 형식

Dict

shape

The full shape of the array()

size

The number of Pauli rows in the table.