Quantum Channel Learning

The problem of an optimal mapping between Hilbert spaces IN and OUT, based on a series of density matrix mapping measurements ρ(l) → varrho(l), l=1dots M, is formulated as an optimization problem maximizing the total fidelity F=suml=1M ω(l) Fleft(varrho(l),sums Bs ρ(l) Bdaggersright) subject to probability preservation constraints on Kraus operators Bs. For F(varrho,σ) in the form that total fidelity can be represented as a quadratic form with superoperator F=sumsleftlangle Bsmiddle|Smiddle| Bs rightrangle (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 AOUT=sums |ws|2 Us AIN Usdagger (no general quantum channel yet). This marks a crucial advancement from the commonly studied unitary mapping of pure states φl=U psil 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)=U ρ(l) Udagger, in this case a quadratic on U fidelity can be constructed by considering √ρ(l) → √varrho(l) mapping, and on a quantum channel, where quadratic on Bs 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.