RandomUnitary

2019-04-15T03:57:17Z

Ian Auckes:

{{Function
|name=RandomUnitary
|desc=Generates a random unitary or orthogonal matrix
|rel=[[RandomDensityMatrix]]<br />[[RandomPOVM]]<br />[[RandomStateVector]]<br />[[RandomSuperoperator]]
|cat=[[List of functions#Random_things|Random things]]
|upd=September 30, 2014
|v=0.50}}
<tt>'''RandomUnitary'''</tt> is a [[List of functions|function]] that generates a random unitary or orthogonal matrix, uniformly according to [http://en.wikipedia.org/wiki/Haar_measure Haar measure].

==Syntax==
* <tt>U = RandomUnitary(DIM)</tt>
* <tt>U = RandomUnitary(DIM,RE)</tt>

==Argument descriptions==
* <tt>DIM</tt>: The number of rows (or equivalently, columns) that <tt>U</tt> will have.
* <tt>RE</tt> (optional, default 0): A flag (either 0 or 1) indicating that <tt>U</tt> should only have real entries (<tt>RE = 1</tt>) or that it is allowed to have complex entries (<tt>RE = 0</tt>). That is, if you set <tt>RE = 1</tt> then <tt>U</tt> will be an orthogonal matrix, not just a unitary matrix.

==Examples==
===A random qubit gate===
To generate a random quantum gate that acts on qubits, you could use the following code:
<syntaxhighlight>
>> RandomUnitary(2)

ans =

0.2280 + 0.6126i -0.2894 - 0.6993i
-0.3147 + 0.6883i -0.2501 + 0.6039i
</syntaxhighlight>

===A random orthogonal matrix===
To generate a random orthogonal (rather than unitary) matrix, set <tt>RE = 1</tt>:
<syntaxhighlight>
>> U = RandomUnitary(3,1)

U =

0.9805 -0.1869 -0.0603
0.1678 0.6381 0.7515
-0.1020 -0.7470 0.6570
</syntaxhighlight>

To verify that this matrix is indeed orthogonal, we multiply it by its transpose:
<syntaxhighlight>
>> U'*U

ans =

1.0000 0.0000 0.0000
0.0000 1.0000 0.0000
0.0000 0.0000 1.0000
</syntaxhighlight>

===Moments of the trace of an orthogonal matrix===
It was shown in [http://mathoverflow.net/questions/180110/moments-of-the-trace-of-orthogonal-matrices this MathOverflow thread] that if O is a random (according to Haar measure) 3-by-3 orthogonal matrix, then the expectated value of ${\mathrm Tr}(O)^{2k}$ for $k = 1, 2, 3, 4, \ldots$ is $1, 3, 15, 91, \ldots$ (sequence [https://oeis.org/A099251 A099251] in the OEIS). We can use the <tt>RandomUnitary</tt> function to reproduce these values approximately as follows:
<syntaxhighlight>
>> s = 10^5;
ct = zeros(1,4);
for j = 1:s
trO = trace(RandomUnitary(3,1))^2;
ct = ct + trO.^(1:4);
end
ct/s

ans =

0.9997 2.9917 14.9298 90.4464
</syntaxhighlight>

==Notes==
The random unitary matrix is generated by constructing a Ginibre ensemble of appropriate size, performing a [http://en.wikipedia.org/wiki/QR_decomposition QR decomposition] on that ensemble, and then multiplying the columns of the unitary matrix Q by the sign of the corresponding diagonal entries of R.<ref>Māris Ozols. ''[http://home.lu.lv/~sd20008/papers/essays/Random%20unitary%20%5Bpaper%5D.pdf How to generate a random unitary matrix]'', 2009.</ref>

{{SourceCode|name=RandomUnitary}}

==References==
<references />

QETLAB - User contributions [en]

RandomUnitary