cirq.choi_to_superoperator

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Returns the superoperator matrix of a quantum channel specified via the Choi matrix.

cirq.choi_to_superoperator(
    choi: np.ndarray
) -> np.ndarray

Quantum channel E: L(H1) -> L(H2) may be specified by its Choi matrix J(E) defined as

$$
J(E) = (E \otimes I)(|\phi\rangle\langle\phi|)
$$

where \(|\phi\rangle = \sum_i|i\rangle|i\rangle\) is the unnormalized maximally entangled state and I: L(H1) -> L(H1) is the identity map. Choi matrix is unique for a given channel. Alternatively, E may be specified by its superoperator matrix K(E) defined so that

$$
K(E) vec(\rho) = vec(E(\rho))
$$

where the vectorization map \(vec\) rearranges d-by-d matrices into d**2-dimensional vectors. Superoperator matrix is unique for a given channel. It is also called the natural representation of a quantum channel.

A quantum channel can be viewed as a tensor with four indices. Different ways of grouping the indices into two pairs yield different matrix representations of the channel, including the superoperator and Choi representations. Hence, the conversion between the superoperator and Choi matrices is a permutation of matrix elements effected by reshaping the array and swapping its axes. Therefore, its cost is O(d**4) where d is the dimension of the input and output Hilbert space.