RandomUnitary - Revision history

Ian Auckes at 03:57, 15 April 2019

2019-04-15T03:57:17Z

Nathaniel at 19:53, 6 January 2016

2016-01-06T19:53:58Z

Nathaniel at 16:39, 30 September 2014

2014-09-30T16:39:32Z

Nathaniel at 16:38, 30 September 2014

2014-09-30T16:38:54Z

Nathaniel at 15:36, 25 September 2014

2014-09-25T15:36:27Z

Nathaniel: /* Moments of the trace of an orthogonal matrix */ typo

2014-09-10T14:28:16Z

Moments of the trace of an orthogonal matrix: typo

Nathaniel at 17:39, 9 September 2014

2014-09-09T17:39:45Z

Nathaniel at 00:02, 24 November 2012

2012-11-24T00:02:44Z

Nathaniel: Uploaded v1.01

2012-11-24T00:02:21Z

Uploaded v1.01

Nathaniel at 19:38, 22 November 2012

2012-11-22T19:38:13Z

@@ Line 14: / Line 14: @@
 ==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 = 1</tt>). That is, if you set <tt>RE = 1</tt> then <tt>U</tt> will be an orthogonal matrix, not just a unitary matrix.
+* <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==

@@ Line 2: / Line 2: @@
 |name=RandomUnitary
 |desc=Generates a random unitary or orthogonal matrix
-|rel=[[RandomDensityMatrix]]<br />[[RandomStateVector]]<br />[[RandomSuperoperator]]
+|rel=[[RandomDensityMatrix]]<br />[[RandomPOVM]]<br />[[RandomStateVector]]<br />[[RandomSuperoperator]]
 |cat=[[List of functions#Random_things|Random things]]
 |upd=September 30, 2014

@@ Line 17: / Line 17: @@
 ==Examples==
-===A random qubit [[gate]]===
+===A random qubit gate===
 To generate a random quantum gate that acts on qubits, you could use the following code:
 <syntaxhighlight>
@@ Line 68: / Line 68: @@
 ==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>
+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}}

@@ Line 4: / Line 4: @@
 |rel=[[RandomDensityMatrix]]<br />[[RandomStateVector]]<br />[[RandomSuperoperator]]
 |cat=[[List of functions#Random_things|Random things]]
-|upd=November 23, 2012
+|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 [[Haar measure]].
+<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==

@@ Line 1: / Line 1: @@
 {{Function
 |name=RandomUnitary
-|desc=Generates a random [[unitary]] or [[orthogonal matrix]]
+|desc=Generates a random unitary or orthogonal matrix
 |rel=[[RandomDensityMatrix]]<br />[[RandomStateVector]]<br />[[RandomSuperoperator]]
 |upd=November 23, 2012
-|v=1.01}}
+|v=0.50}}
 <tt>'''RandomUnitary'''</tt> is a [[List of functions|function]] that generates a random [[unitary]] or [[orthogonal matrix]], uniformly according to [[Haar measure]].
@@ Line 19: / Line 19: @@
 ===A random qubit [[gate]]===
 To generate a random quantum gate that acts on qubits, you could use the following code:
-<pre>
+<syntaxhighlight>
 >> RandomUnitary(2)
@@ Line 26: / Line 26: @@
 .2280 + 0.6126i  -0.2894 - 0.6993i
    -0.3147 + 0.6883i  -0.2501 + 0.6039i
-</pre>
+</syntaxhighlight>
 ===A random orthogonal matrix===
 To generate a random orthogonal (rather than unitary) matrix, set <tt>RE = 1</tt>:
-<pre>
+<syntaxhighlight>
 >> U = RandomUnitary(3,1)
@@ Line 38: / Line 38: @@
 .1678    0.6381    0.7515
     -0.1020   -0.7470    0.6570
-</pre>
+</syntaxhighlight>
 To verify that this matrix is indeed orthogonal, we multiply it by its transpose:
-<pre>
+<syntaxhighlight>
 >> U'*U
@@ Line 49: / Line 49: @@
 .0000    1.0000    0.0000
 .0000    0.0000    1.0000
-</pre>
+</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:
-<pre>
+<syntaxhighlight>
 >> s = 10^5;
     ct = zeros(1,4);
@@ Line 65: / Line 65: @@
 .9997    2.9917   14.9298   90.4464
-</pre>
+</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>
 ==References==
 <references />

@@ Line 10: / Line 10: @@
 ==Syntax==
 * <tt>U = RandomUnitary(DIM)</tt>
-* <tt>U = RandomUnitary(DIM,RE)</tt>\
+* <tt>U = RandomUnitary(DIM,RE)</tt>
 ==Argument descriptions==

← Older revision		Revision as of 19:38, 22 November 2012
Line 76:		Line 76:

	Please remember that constructing random sparse unitary matrices in this way is ''very'' approximate – it is not uniformly distributed in any meaningful sense and the number of non-zero elements will sometimes not be very close to <tt>DENS*DIM^2</tt>.		Please remember that constructing random sparse unitary matrices in this way is ''very'' approximate – it is not uniformly distributed in any meaningful sense and the number of non-zero elements will sometimes not be very close to <tt>DENS*DIM^2</tt>.
		+
		+	==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>
		+
		+	==References==
		+	<references />