Quantum Channel Learning

TL;DR

This paper formulates an optimization approach for learning quantum channels that map density matrices between Hilbert spaces, introducing a quadratic fidelity-based iterative algorithm and generalizing from pure states to mixed states and channels.

Contribution

It introduces a novel framework for quantum channel learning using density matrices and mixed unitary channels, advancing beyond pure state unitary mappings.

Findings

Developed an iterative algorithm for optimizing quantum channel fidelity.

Generalized unitary learning to mixed states and channels.

Demonstrated application to density matrix and quantum channel learning.

Abstract

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.