Source code for toqito.matrix_props.is_unitary

"""Checks if the matrix is a unitary matrix."""

import numpy as np

from toqito.matrix_props import is_square




def is_unitary(mat: np.ndarray, rtol: float = 1e-05, atol: float = 1e-08) -> bool:
    r"""Check if matrix is unitary [@WikiUniMat].

    A matrix is unitary if its inverse is equal to its conjugate transpose.

    Alternatively, a complex square matrix \(U\) is unitary if its conjugate transpose
    \(U^*\) is also its inverse, that is, if

    \[
        \begin{equation}
            U^* U = U U^* = \mathbb{I},
        \end{equation}
    \]

    where \(\mathbb{I}\) is the identity matrix.

    Examples:
        Consider the following matrix

        \[
            X = \begin{pmatrix}
                0 & 1 \\
                1 & 0
                \end{pmatrix}
        \]

        our function indicates that this is indeed a unitary matrix.

        ```python exec="1" source="above"
        import numpy as np
        from toqito.matrix_props import is_unitary

        A = np.array([[0, 1], [1, 0]])

        print(is_unitary(A))
        ```

        We may also use the `random_unitary` function from `toqito`, and can verify that a randomly
        generated matrix is unitary

        ```python exec="1" source="above"
        from toqito.matrix_props import is_unitary
        from toqito.rand import random_unitary

        mat = random_unitary(2)

        print(is_unitary(mat))
        ```

        Alternatively, the following example matrix \(B\) defined as

        \[
            B = \begin{pmatrix}
                1 & 0 \\
                1 & 1
                \end{pmatrix}
        \]

        is not unitary.

        ```python exec="1" source="above"
        import numpy as np
        from toqito.matrix_props import is_unitary

        B = np.array([[1, 0], [1, 1]])

        print(is_unitary(B))
        ```

    Args:
        mat: Matrix to check.
        rtol: The relative tolerance parameter (default 1e-05).
        atol: The absolute tolerance parameter (default 1e-08).

    Returns:
        Return `True` if matrix is unitary, and `False` otherwise.

    """
    if not is_square(mat):
        return False

    uc_u_mat = mat.conj().T @ mat
    u_uc_mat = mat @ mat.conj().T
    id_mat = np.eye(len(mat))

    # If U^* @ U = I U @ U^*, the matrix "U" is unitary.
    return bool(
        np.allclose(uc_u_mat, id_mat, rtol=rtol, atol=atol) and np.allclose(u_uc_mat, id_mat, rtol=rtol, atol=atol)
    )