Semidefinite Programming for Quantum Channel Learning

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.

💡 Research Summary

This paper addresses the problem of reconstructing a quantum channel from classical input‑output data, focusing on cases where the overall fidelity can be expressed as a ratio of two quadratic forms. The authors begin by converting classical vector‑to‑vector mappings into quantum state mappings, constructing density matrices for inputs and outputs. By restricting attention to scenarios where mixed input states are mapped to pure output states, the fidelity between the channel’s output and the target pure state becomes exactly quadratic in the channel’s Kraus operators.

The central technical contribution is the reformulation of the channel learning task as a semidefinite program (SDP). Starting from the Kraus representation (\mathcal{E}(\rho)=\sum_s B_s\rho B_s^\dagger), both the objective (total fidelity) and the physical constraints (complete positivity, trace‑preserving or unit‑matrix‑preserving) are quadratic in the operators (B_s). By introducing the Choi matrix (J) – defined as (J_{jk;j’k’}=\sum_s B_{jk}^{s*}B_{j’k’}^{s}) – these quadratic expressions become linear: the fidelity is (\operatorname{Tr}(JS)) for a data‑dependent matrix (S), while the CPTP constraints turn into partial‑trace equalities (\operatorname{Tr}{\text{out}}(J)=\mathbb{I}{\text{in}}) (or (\operatorname{Tr}{\text{in}}(J)=\mathbb{I}{\text{out}})). The only remaining non‑linear requirement is the positivity of the Choi matrix, which is precisely the semidefinite constraint (J\succeq0). Consequently, the learning problem becomes the convex SDP:

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