🤖 AI Summary
This study addresses the problem of reconstructing quantum channels from classical data samples, focusing on scenarios where the overall fidelity can be expressed as the ratio of two quadratic forms. The work proposes an optimization approach based on semidefinite programming (SDP), leveraging the Choi matrix representation and Kraus operator decomposition to efficiently solve the channel reconstruction problem. It is the first to systematically apply SDP to a variety of quantum channel learning settings, achieving high-precision reconstructions using off-the-shelf solvers. Experimental results reveal that the reconstructed channels typically exhibit Kraus ranks amounting to only a few percent of their theoretical maximum, indicating that real-world quantum processes possess remarkably low intrinsic complexity. This insight substantially enhances both the efficiency and practicality of quantum channel reconstruction.
📝 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.