GenPauli
GenPauli | |
Produces a generalized Pauli operator (sometimes called a Weyl operator) | |
Other toolboxes required | none |
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Related functions | GellMann GenGellMann Pauli |
Function category | Special states, vectors, and operators |
GenPauli is a function that produces generalized Pauli matrices (sometimes called Weyl 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
Source code
Click here to view this function's source code on github.
External links
- Generalizations of the Pauli matrices at Wikipedia