# Semidefinite Programming for Quantum Channel Learning

arXiv: https://arxiv.org/abs/2601.12502 · HTML: https://mlsanity.com/paper/2601.12502/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 41 references that are in the map.
- First posted on arXiv 18 Jan 2026.

## Builds on (most cited first, 9 of 9)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Learning Unitary Operators with Help From u(n)](https://mlsanity.com/paper/1607.04903/) (2016)
- [Optimal quantum dataset for learning a unitary transformation](https://mlsanity.com/paper/2203.00546/) (2022)
- [On The Radon-Nikodym Spectral Approach With Optimal Clustering](https://mlsanity.com/paper/1906.00460/) (2019)
- [Partially Unitary Learning](https://mlsanity.com/paper/2405.10263/) (2024)
- [Quantum Channel Learning](https://mlsanity.com/paper/2407.04406/) (2024)
- [Superstate Quantum Mechanics](https://mlsanity.com/paper/2502.00037/) (2025)

## Abstract (arXiv)

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.