Semidefinite Programming for Quantum Channel Learning

TL;DR

This paper presents a convex semidefinite programming approach for reconstructing quantum channels from classical data, demonstrating efficient solutions and low Kraus rank channels in various scenarios.

Contribution

It introduces a novel SDP-based method for quantum channel learning that efficiently reconstructs channels with low Kraus rank from classical data.

Findings

SDP effectively reconstructs quantum channels from classical data.

Reconstructed channels typically have low Kraus rank.

The method applies to various types of quantum channels and operators.

Abstract

The problem of reconstructing a quantum channel from a sample of classical data is considered. When the total fidelity can be represented as a ratio of two quadratic forms (e.g., in the case of mapping a mixed state to a pure state, projective operators, unitary learning, and others), Semidefinite Programming (SDP) can be applied to solve the fidelity optimization problem with respect to the Choi matrix. A remarkable feature of SDP is that the optimization is convex, which allows the problem to be efficiently solved by a variety of numerical algorithms. We have tested several commercially available SDP solvers, all of which allowed for the reconstruction of quantum channels of different forms. A notable feature is that the Kraus rank of the obtained quantum channel typically comprises less than a few percent of its maximal possible value. This suggests that a relatively small Kraus rank quantum channel is typically sufficient to describe experimentally observed classical data. The theory was also applied to the problem of reconstructing projective operators from data. Finally, we discuss a classical computational model based on quantum channel transformation, performed and calculated on a classical computer, possibly hardware-optimized.