toqito.matrices.hadamard

toqito.matrices.hadamard(n_param=1)[source]

Produce a 2^{n_param} dimensional Hadamard matrix [WikHad].

The standard Hadamard matrix that is often used in quantum information as a two-qubit quantum gate is defined as

\[\begin{split}H_1 = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\end{split}\]

In general, the Hadamard matrix of dimension 2^{n_param} may be defined as

\[\left( H_n \right)_{i, j} = \frac{1}{2^{\frac{n}{2}}} \left(-1\right)^{i \dot j}\]

Examples

The standard 2-qubit Hadamard matrix can be generated in toqito as

>>> from toqito.matrices import hadamard
>>> hadamard(1)
[[ 0.70710678  0.70710678]
 [ 0.70710678 -0.70710678]]

References