Difference between revisions of "RandomUnitary"

	RandomUnitary
	Generates a random unitary or orthogonal matrix
Other toolboxes required	opt_args
Related functions	RandomDensityMatrix; RandomStateVector; RandomSuperoperator

Revision as of 00:02, 24 November 2012

RandomUnitary is a function that generates a random unitary or orthogonal matrix, uniformly according to Haar measure.

Syntax

U = RandomUnitary(DIM)
U = RandomUnitary(DIM,RE)\

Argument descriptions

DIM: The number of rows (or equivalently, columns) that U will have.
RE (optional, default 0): A flag (either 0 or 1) indicating that U should only have real entries (RE = 1) or that it is allowed to have complex entries (RE = 1). That is, if you set RE = 1 then U 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:

>> RandomUnitary(2)

ans =

   0.2280 + 0.6126i  -0.2894 - 0.6993i
  -0.3147 + 0.6883i  -0.2501 + 0.6039i

A random orthogonal matrix

To generate a random orthogonal (rather than unitary) matrix, set RE = 1:

>> U = RandomUnitary(3,1)

U =

    0.9805   -0.1869   -0.0603
    0.1678    0.6381    0.7515
   -0.1020   -0.7470    0.6570

To verify that this matrix is indeed orthogonal, we multiply it by its transpose:

>> U'*U

ans =

    1.0000    0.0000    0.0000
    0.0000    1.0000    0.0000
    0.0000    0.0000    1.0000

Notes

The random unitary matrix is generated by constructing a Ginibre ensemble of appropriate size, performing a 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.^[1]

References

↑ Māris Ozols. How to generate a random unitary matrix, 2009.

[1] Māris Ozols. How to generate a random unitary matrix, 2009.

[1]

@@ Line 4: / Line 4: @@
 |req=[[opt_args]]
 |rel=[[RandomDensityMatrix]]<br />[[RandomStateVector]]<br />[[RandomSuperoperator]]
-|upd=November 22, 2012
+|upd=November 23, 2012
-|v=1.00}}
+|v=1.01}}
 <tt>'''RandomUnitary'''</tt> is a [[List of functions|function]] that generates a random [[unitary]] or [[orthogonal matrix]], uniformly according to [[Haar measure]].
 ==Syntax==
 * <tt>U = RandomUnitary(DIM)</tt>
-* <tt>U = RandomUnitary(DIM,RE)</tt>
+* <tt>U = RandomUnitary(DIM,RE)</tt>\
-* <tt>U = RandomUnitary(DIM,RE,DENS)</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 = 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>DENS</tt> (optional, default 1): If equal to 1 then <tt>U</tt> will be a full matrix. If <tt>0 <= DEN < 1</tt> then <tt>U</tt> will be sparse with approximately <tt>DENS*DIM^2</tt> non-zero entries. Note that the number of non-zero entries in this case is *very* approximate. Furthermore, if <tt>DENS < 1</tt> then the distribution of <tt>U</tt> is no longer uniform in any mathematically well-defined sense.
 ==Examples==
@@ Line 52: / Line 50: @@
 .0000    0.0000    1.0000
 </pre>
-===A random sparse unitary matrix===
-To generate an 8-by-8 sparse unitary matrix with approximately 1/4 of its entries non-zero, we can use the following line of code:
-<pre>
->> RandomUnitary(8,0,1/4)
-ans =
-   (2,1)           0 + 0.7140i
-   (7,1)     -0.0000 + 0.7002i
-   (2,2)      0.5448 + 0.0000i
-   (5,2)      0.6281
-   (7,2)     -0.5555 - 0.0000i
-   (2,3)      0.4398 - 0.0000i
-   (5,3)     -0.7781 + 0.0000i
-   (7,3)     -0.4485 + 0.0000i
-   (8,4)           0 - 1.0000i
-   (1,5)      1.0000
-   (3,6)      1.0000
-   (4,7)      1.0000
-   (6,8)      1.0000
-</pre>
-Please remember that constructing random sparse unitary matrices in this way is ''very'' approximate &ndash; 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==