RandomUnitary

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

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

Moments of the trace of an orthogonal matrix

It was shown in this MathOverflow thread that if O is a random (according to Haar measure) 3-by-3 orthogonal matrix, then the expectation value of ${\mathrm Tr}(O)^{2k}$ for $k = 1, 2, 3, 4, \ldots$ is $1, 3, 15, 91, \ldots$ (sequence A099251 in the OEIS). We can use the RandomUnitary function to reproduce these values approximately as follows:

>> 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

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.