GenPauli
| GenPauli | |
| Produces a generalized Pauli operator (sometimes called a Weyl operator) | |
| Other toolboxes required | none |
|---|---|
| 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.
Contents
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 on "expand" to the right to view the MATLAB source code for this function.
%% GENPAULI Produces a generalized Pauli operator% This function has three required arguments:% IND1 (a nonnegative integer from 0 to DIM-1 inclusive)% IND2 (a nonnegative integer from 0 to DIM-1 inclusive)% DIM (a positive integer indicating the dimension)%% P = GenPauli(IND1,IND2,DIM) is a DIM-by-DIM unitary operator. More% specifically, it is the operator X^IND1*Z^IND2, where X and Z are the% "shift" and "clock" operators that naturally generalize the Pauli X and% Z operators. These matrices span the entire space of DIM-by-DIM% matrices as IND1 and IND2 range from 0 to DIM-1, inclusive.%% This function has one optional argument:% SP (default 0)%% P = GenPauli(IND1,IND2,DIM,SP) is as above, with sparsity of the output% determined by the value of SP. If SP = 0 then the output will be full,% if SP = 1 then the output will be sparse.%% URL: http://www.qetlab.com/GenPauli% requires: opt_args.m% author: Nathaniel Johnston (nathaniel@njohnston.ca)% package: QETLAB% last updated: December 18, 2013function p = GenPauli(ind1,ind2,dim,varargin)
% set optional argument defaults: sp=0[sp] = opt_args({ 0 },varargin{:});
w = exp(2i*pi/dim); % primitive root of unity
X = circshift(speye(dim),1); % shift matrix
Z = spdiags((w.^(0:dim-1)).',0,dim,dim); % clock matrix
p = X^ind1*Z^ind2;
if(~sp)
p = full(p);
end
External links
- Generalizations of the Pauli matrices at Wikipedia