Pauli Matrix Functions¶

Aside from the generalised qudit gates, PyQudit also includes functions which output the gate matrices. These matrices are useful in understanding the logic and mathematics behind a gate, as also in using them for building circuits. Parallelly, circuits and gates requiring superposition make direct use of these matrix functions. States ofthe qubit gates are represented by their Pauli Matrices, and the similarity of higher dimensionality matrices coupled with a lack of a different name meant our matrix functions are also termed as Pauli Matrices.

Table of Contents

CX Matrix
- CXd_pauli
- CXd_cstm_pauli
CX-Drag Matrix
- CXDrag_pauli
GXOR Matrix
- GXOR_pauli
Hadamard Matrix
- Hd_pauli
CZ Matrix
- CZd_pauli
X Matrix
- Xd_pauli
Y Matrix
- Yd_pauli
Z Matrix
- Zd_pauli
Toffoli Matrix
- Toffolid_pauli

CX Matrix ¶

The functions for matrix representations of the various implementations of the CX gate.

CXd_pauli ¶

CXd_pauli(d)

Matrix representation of the formulaic version of the CX gate. More efficient on smaller dimensions.

>>> CXd_pauli(4)CXd_cstm_pauli
array([[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0]])

CXd_cstm_pauli ¶

CXd_cstm_pauli(d)

Matrix representation of the custom implementation of the CX gate. More efficient on higher dimensions.

>>> CXd_cstm_pauli(4)
array([[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0]])

CX-Drag Matrix ¶

The functions for the matrix representations of the CX-Drag gate.

CXDrag_pauli ¶

CXDrag_pauli(d)

Matrix representation of the CX-Drag gate, common for both implementations.

>>> CXDrag_pauli(4)
array([[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]])

GXOR Matrix ¶

The functions for the matrix representations of the GXOR gate.

GXOR_pauli ¶

GXOR_pauli(d)

Matrix representation of the GXOR gate, common for both implementations.

>>> GXOR_pauli(4)
array([[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0]])

Hadamard Matrix ¶

The functions for the marix representations of the Hadamard gate. These matrices are expecially useful for understanding and working with superposition.

Hd_pauli ¶

Hd_pauli(d)

Matrix representation of the Hadamard gate, common for both implementations.

>>> Hd_pauli(4)
array([[ 0.5,  0.5,  0.5,  0.5],
       [ 0.5, -0.5,  0.5, -0.5],
       [ 0.5,  0.5, -0.5, -0.5],
       [ 0.5, -0.5, -0.5,  0.5]])

CZ Matrix ¶

The functions for the matrix representations of the CZ gate.

CZd_pauli ¶

CZd_pauli(d)

Matrix representation of the CZ gate implementation.

>>> pq.CZd_pauli(4)
array([[ 1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,
         0.,  0.,  0.],
       [ 0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,
         0.,  0.,  0.],
       [ 0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,
         0.,  0.,  0.],
       [ 0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,
         0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,
         0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0., -1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,
         0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,
         0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0., -1.,  0.,  0.,  0.,  0.,  0.,  0.,
         0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,
         0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,
         0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0., -1.,  0.,  0.,
         0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0., -1.,  0.,
         0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,
         0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,
        -1.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,
         0.,  0., -1.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,
         0.,  1.,  0.]])

X Matrix ¶

The functions for the matrix representations of the X gate.

Xd_pauli ¶

Xd_pauli(d)

Matrix representation of the X gate implementation.

>>> Xd_pauli(4)
array([[0, 0, 0, 1],
       [1, 0, 0, 0],
       [0, 1, 0, 0],
       [0, 0, 1, 0]])

Y Matrix ¶

The functions for the matrix representations of the Y gate.

Yd_pauli ¶

Yd_pauli(d)

Matrix representation of the Y gate implementation.

>>> Yd_pauli(4)
array([[ 0.00000000e+00+0.00000000e+00j,  0.00000000e+00+0.00000000e+00j,
         0.00000000e+00+0.00000000e+00j,  1.00000000e+00-9.80384690e-07j],
       [ 0.00000000e+00+1.00000000e+00j,  0.00000000e+00+0.00000000e+00j,
         0.00000000e+00+0.00000000e+00j,  0.00000000e+00+0.00000000e+00j],
       [ 0.00000000e+00+0.00000000e+00j, -1.00000000e+00+3.26794897e-07j,
         0.00000000e+00+0.00000000e+00j,  0.00000000e+00+0.00000000e+00j],
       [ 0.00000000e+00+0.00000000e+00j,  0.00000000e+00+0.00000000e+00j,
        -6.53589793e-07-1.00000000e+00j,  0.00000000e+00+0.00000000e+00j]])

Z Matrix ¶

The functions for the matrix representations of the Z gate.

Zd_pauli ¶

Zd_pauli(d)

Matrix representation of the Z gate implementation.

>>> Zd_pauli(4)
array([[ 1.00000000e+00+0.00000000e+00j,  0.00000000e+00+0.00000000e+00j,
         0.00000000e+00+0.00000000e+00j,  0.00000000e+00+0.00000000e+00j],
       [ 0.00000000e+00+0.00000000e+00j,  3.26794897e-07+1.00000000e+00j,
         0.00000000e+00+0.00000000e+00j,  0.00000000e+00+0.00000000e+00j],
       [ 0.00000000e+00+0.00000000e+00j,  0.00000000e+00+0.00000000e+00j,
        -1.00000000e+00+6.53589793e-07j,  0.00000000e+00+0.00000000e+00j],
       [ 0.00000000e+00+0.00000000e+00j,  0.00000000e+00+0.00000000e+00j,
         0.00000000e+00+0.00000000e+00j, -9.80384690e-07-1.00000000e+00j]])

Toffoli Matrix ¶

The functions for the matrix representations of the Toffoli gate.

Toffolid_pauli ¶

Toffolid_pauli(d)

Matrix representations of the Toffoli gate implementation.

>>> Toffolid_pauli(4)
array([[1, 0, 0, ..., 0, 0, 0],
       [0, 1, 0, ..., 0, 0, 0],
       [0, 0, 1, ..., 0, 0, 0],
       ...,
       [0, 0, 0, ..., 0, 0, 0],
       [0, 0, 0, ..., 1, 0, 0],
       [0, 0, 0, ..., 0, 1, 0]])