GenPauli

GenPauli
Produces a generalized Pauli operator
Other toolboxes required	opt_args
Related functions	GellMann Pauli

GenPauli is a function that produces generalized Pauli matrices. More specifically, it produces a unitary matrix of the form $X^j Z^k$, where $X$ and $Z$ are the $d \times d$ "shift" and "clock" matrices defined by: \[X = \begin{bmatrix} 0 & 0 & 0 & \cdots &0 & 1\\ 1 & 0 & 0 & \cdots & 0 & 0\\ 0 & 1 & 0 & \cdots & 0 & 0\\ 0 & 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots &\vdots &\vdots \\ 0 & 0 &0 & \cdots & 1 & 0\\ \end{bmatrix} \quad \text{and} \quad Z = \begin{bmatrix} 1 & 0 & 0 & \cdots & 0\\ 0 & \omega & 0 & \cdots & 0\\ 0 & 0 &\omega ^2 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & \omega ^{d-1} \end{bmatrix}, \] and $\omega = \exp(2\pi i/d)$ is a primitive root of unity.

Syntax

• P = GenPauli(IND1,IND2,DIM)
• P = GenPauli(IND1,IND2,DIM,SP)

Argument descriptions

• IND1: The exponent of $X$, the shift matrix (this was called $j$ above). Should be an integer from 0 to DIM-1, inclusive.
• IND2: The exponent of $Z$, the clock matrix (this was called $k$ above). Should be an integer from 0 to DIM-1, inclusive.
• DIM: The size of the output matrix (this was called $d$ above).
• SP (optional, default 0): A flag (either 1 or 0) indicating that the generalized Pauli matrix produced should or should not be sparse.

Examples

Gives the Pauli operators when DIM = 2

>> GenPauli(1,0,2) % Pauli X operator

ans =

     0     1
     1     0

>> GenPauli(0,1,2) % Pauli Z operator

ans =

   1.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i  -1.0000 + 0.0000i

>> GenPauli(1,1,2) % Pauli Y operator (up to global phase)

ans =

   0.0000 + 0.0000i  -1.0000 + 0.0000i
   1.0000 + 0.0000i   0.0000 + 0.0000i

>> GenPauli(0,0,2) % identity operator

ans =

     1     0
     0     1

In Higher Dimensions

>> GenPauli(1,0,3) % generalized Pauli X

ans =

     0     0     1
     1     0     0
     0     1     0

>> GenPauli(0,1,3) % generalized Pauli Z

ans =

   1.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i  -0.5000 + 0.8660i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i  -0.5000 - 0.8660i

>> GenPauli(2,3,4,1) % sparse 4-dimensional generalized Pauli

ans =

   (3,1)      1.0000 + 0.0000i
   (4,2)     -0.0000 - 1.0000i
   (1,3)     -1.0000 + 0.0000i
   (2,4)      0.0000 + 1.0000i

External links

• Generalizations of the Pauli matrices at Wikipedia