Quantum Channel Learning

TL;DR

This work generalizes quantum channel learning from pure-state unitary mappings to learning mappings between density matrices via general quantum channels, cast as a QCQP in Kraus operators with a quadratic fidelity ${\mathcal F}=\sum_s\langle B_s|S|B_s\rangle$. It shows that density-matrix square-root mapping ${\sqrt{\rho}}\to {\sqrt{\varrho}}$ yields exact fidelity for unitary learning and provides a practical surrogate for mixed-state channels, while introducing a hierarchical scheme of partially unitary operators to approximate mixed-unitary channels. The authors develop an eigenvalue-based iterative approach, discuss gauge and trace-preservation constraints, and demonstrate both the potential and limitations of fidelity proxies in constructing quantum channels from data. The framework aims to address quantum inverse problems, variational quantum algorithms, and quantum tomography, with future work on time-dependent extensions, memory effects, and more general channel reconstructions. Overall, the paper advances the representation of knowledge in quantum ML by enabling learning with mixed states and channels, not just pure-state unitary evolutions.

Abstract

The problem of an optimal mapping between Hilbert spaces $IN$ and $OUT$, based on a series of density matrix mapping measurements $ρ^{(l)} \to \varrho^{(l)}$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity $\mathcal{F}=\sum_{l=1}^{M} ω^{(l)} F\left(\varrho^{(l)},\sum_s B_s ρ^{(l)} B^{\dagger}_s\right)$ subject to probability preservation constraints on Kraus operators $B_s$. For $F(\varrho,σ)$ 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 $φ_l=\mathcal{U} ψ_l$ 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 $\varrho^{(l)}=\mathcal{U} ρ^{(l)} \mathcal{U}^{\dagger}$, in this case a quadratic on $\mathcal{U}$ fidelity can be constructed by considering $\sqrt{ρ^{(l)}} \to \sqrt{\varrho^{(l)}}$ 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.

Figures (1)

• Figure 1: For known density matrix mappings ($\mathcal{F}^{prop}=M=1000$) evaluate the total fidelity $\mathcal{F}$ (\ref{['FQuantumChannelMapping']}) (with different definitions of closeness $F$) as a function of the rank $N_r$ of the input density matrix and the quantum channel's Kraus rank $N_s$. Problem dimension is $n=D=20$. (a) Unitary mapping $N_s=1$: The $\mathcal{F}^{\varrho\sigma}$ (\ref{['QpureStateDensityMatrix']}) strongly depends on the rank of the input density matrices. The $\mathcal{F}^{\sqrt{\varrho}\sqrt{\rho}}$ (corresponding to (\ref{['mlproblemVectorDensityMatrixSQRT']}) mapping) and $\mathcal{F}^v$ (\ref{['fidelityScalProduct']}) produce the exact result $\mathcal{F}=M$ in the unitary mapping case. (b) Multiple terms $N_s>1$: The $\mathcal{F}^{\varrho\sigma}$ decreases even more strongly, while $\mathcal{F}^{\sqrt{\varrho}\sqrt{\rho}}$ and $\mathcal{F}^v$ are no longer exact.