Difference between revisions of "GenPauli"
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|name=GenPauli | |name=GenPauli | ||
|desc=Produces a generalized Pauli operator | |desc=Produces a generalized Pauli operator | ||
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|rel=[[GellMann]]<br />[[GenGellMann]]<br />[[Pauli]] | |rel=[[GellMann]]<br />[[GenGellMann]]<br />[[Pauli]] | ||
| + | |cat=[[List of functions#Special_states,_vectors,_and_operators|Special states, vectors, and operators]] | ||
|upd=December 18, 2013 | |upd=December 18, 2013 | ||
|v=1.00}} | |v=1.00}} | ||
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==Examples== | ==Examples== | ||
===Gives the Pauli operators when <tt>DIM = 2</tt>=== | ===Gives the Pauli operators when <tt>DIM = 2</tt>=== | ||
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>> GenPauli(1,0,2) % Pauli X operator | >> GenPauli(1,0,2) % Pauli X operator | ||
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1 0 | 1 0 | ||
0 1 | 0 1 | ||
| − | </ | + | </syntaxhighlight> |
===In Higher Dimensions=== | ===In Higher Dimensions=== | ||
| − | < | + | <syntaxhighlight> |
>> GenPauli(1,0,3) % generalized Pauli X | >> GenPauli(1,0,3) % generalized Pauli X | ||
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(1,3) -1.0000 + 0.0000i | (1,3) -1.0000 + 0.0000i | ||
(2,4) 0.0000 + 1.0000i | (2,4) 0.0000 + 1.0000i | ||
| − | </ | + | </syntaxhighlight> |
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| + | {{SourceCode|name=GenPauli}} | ||
==External links== | ==External links== | ||
* [http://en.wikipedia.org/wiki/Generalizations_of_the_Pauli_matrices Generalizations of the Pauli matrices] at Wikipedia | * [http://en.wikipedia.org/wiki/Generalizations_of_the_Pauli_matrices Generalizations of the Pauli matrices] at Wikipedia | ||
Revision as of 15:17, 29 September 2014
| GenPauli | |
| Produces a generalized Pauli 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. 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 1In 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 + 1.0000iSource code
Click here to view this function's source code on github.
External links
- Generalizations of the Pauli matrices at Wikipedia