The problem of an optimal mapping between Hilbert spaces $IN$ and $OUT$, based on a series of density matrix mapping measurements $\rho^{ (l)} \to \varrho^{ (l)}$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity $\mathcal{ F}=\sum_{ l=1}^{ M} \omega^{ (l)} F\left(\varrho^{ (l)},\sum_s B_s \rho^{ (l)} B^{ \dagger}_s\right)$ subject to probability preservation constraints on Kraus operators $B_s$. For $F(\varrho,\sigma)$ in the form that total fidelity can be represented as a quadratic form with superoperator $\mathcal{ F}=\sum_s\left\langle B_s\middle|S\middle| B_s \right\rangle$ (either exactly or as an approximation) an iterative algorithm is developed. The work introduces two important generalizations of unitary learning: 1. $IN$/$OUT$ states are represented as density matrices. 2. The mapping itself is formulated as a mixed unitary quantum channel $A^{ OUT}=\sum_s |w_s|^2 \mathcal{ U}_s A^{ IN} \mathcal{ U}_s^{ \dagger}$ (no general quantum channel yet). This marks a crucial advancement from the commonly studied unitary mapping of pure states $\phi_l=\mathcal{ U} \psi_l$ to a quantum channel, what allows us to distinguish probabilistic mixture of states and their superposition. An application of the approach is demonstrated on unitary learning of density matrix mapping $\varrho^{ (l)}=\mathcal{ U} \rho^{ (l)} \mathcal{ U}^{ \dagger}$, in this case a quadratic on $\mathcal{ U}$ fidelity can be constructed by considering $\sqrt{ \rho^{ (l)}} \to \sqrt{ \varrho^{ (l)}}$ mapping, and on a quantum channel, where quadratic on $B_s$ fidelity is an approximation -- a quantum channel is then obtained as a hierarchy of unitary mappings, a mixed unitary channel. The approach can be applied to studying quantum inverse problems, variational quantum algorithms, quantum tomography, and more.
Categories: cs.LG cs.NA math.NA quant-ph