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
External links
- Generalizations of the Pauli matrices at Wikipedia