[Paper Review] Semidefinite Programming for Quantum Channel Learning

Mikhail Gennadievich Belov, Victor V. Dubov+4 more|arXiv (Cornell University)|Jan 18, 2026

Quantum Information and Cryptography0 citations

TL;DR

The paper shows that reconstructing quantum channels from classical data can be formulated as a convex semidefinite programming problem, enabling exact recovery of various channel types, with low Kraus ranks typically sufficient to describe data.

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.

Motivation & Objective

• Motivate learning quantum channels from classical data as an inverse problem within quantum information and ML/AI.
• Formulate quantum channel reconstruction as a constrained optimization problem over Kraus operators or the Choi matrix.
• Demonstrate that the full-rank quantum channel recovery is a convex SDP problem under a quadratic fidelity form.
• Show practical reconstruction results using SDP solvers across unitary and higher Kraus-rank channels.
• Discuss implications for a density-matrix-network computational model and scalability.

Proposed method

• Represent a quantum channel with Kraus operators B_s or the Choi matrix J for the learning task.
• Express the fidelity as a ratio of two quadratic forms in the mapping operators to enable SDP formulation.
• Impose CPTP constraints in Kraus form or linear trace/partition constraints in Choi form, yielding a QCQP that becomes SDP under full rank.
• Reformulate the objective as maximizing Tr(J S) subject to J ⪰ 0 and linear constraints derived from trace preservation or unit-matrix preservation.
• Solve the resulting SDP with interior-point methods and compare with an eigenstructure based approach (Appendix A).
• Optionally use the Choi-matrix representation to linearize fidelity and constraints for SDP readiness.

Figure 1: The rank of the Choi matrix $\mathcal{J}$ ( 16 ) (red), obtained as a solution to the optimization problem ( 22 ) for unitary mapping ( 25 ) data, is shown. This plot demonstrates a perfect reconstruction of the unitary operator from the data for the dimension range $1\leq D=n\leq 30$ . We

Experimental results

Research questions

• RQ1 Can a full-rank quantum channel be reconstructed exactly from classical input/output data using SDP?
• RQ2 How does the Kraus rank affect the difficulty and success of quantum channel learning with SDP versus non-convex methods?
• RQ3 What is the comparative performance of Kraus-operator versus Choi-matrix parametrizations in SDP-based channel learning?
• RQ4 To what extent can unitary learning be recovered as a special case within the SDP framework?
• RQ5 What are the practical limitations and computational resources required for SDP-based quantum channel reconstruction on realistic problem sizes?

Key findings

• Quantum channel reconstruction can be cast as a convex SDP when fidelity is a quadratic form, enabling global optimization.
• The SDP approach can exactly reconstruct unitary channels from information-complete data, yielding a rank-one Choi matrix.
• Even for higher Kraus-rank channels, SDP-based methods perform well, with numerical experiments showing successful recovery.
• Kraus rank of recovered channels is typically a small fraction of the maximal possible rank, suggesting compact representations suffice for observed data.
• Choi-matrix formulation leads to linear objective and linear constraints with a semidefinite constraint, facilitating robust optimization.
• The density-matrix network viewpoint supports decomposing large channels into networks of smaller, tractable components.

Figure 2: The rank of the Choi matrix $\mathcal{J}$ ( 16 ) (red) obtained as a solution to the optimization problem for a random $\psi\to\phi$ mapping ( 6 ): (a) The case $1\leq D=n\leq 30$ . The rank grows slowly with the increase of $n$ ; the rank value typically comprises less than 1.5% of the ma