Semidefinite Programming for Quantum Channel Learning

TL;DR

The paper tackles learning a quantum channel from classical IN/OUT data by casting fidelity as a ratio of two quadratic forms in the channel operators, turning the reconstruction into a convex SDP over the Choi matrix $\mathcal{J}$ (or Kraus operators $\{B_s\}$). It demonstrates that SDP-based learning can exactly recover unitary dynamics and rank-constrained channels, while empirical results reveal a persistent low Kraus rank phenomenon across random and data-driven samples, suggesting small-rank channels suffices for practical data modeling. The work further shows the method’s versatility by reconstructing rank-1 projections, offering a data-driven path to projective operator learning, and discusses approximate inverses and density-matrix-network interpretations to scale the approach in AI/ML contexts. Overall, the convex SDP framework provides a robust, scalable avenue for quantum-channel learning from classical data, with potential implications for quantum-inspired classical computation and ML architectures.

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.

Figures (4)

• Figure 1: The rank of the Choi matrix $\mathcal{J}$ (\ref{['Jmatrix']}) (red), obtained as a solution to the optimization problem (\ref{['SDPobjective']}) for unitary mapping (\ref{['SampleU']}) 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 made similar plots in belov2024partiallyPRE using our original software. The presented plot was generated using the off-the-shelf SDP software: CVXPY and SDPA, both of which reconstruct the unitary channel exactly. The fidelity (green) of the reconstructed channel is exact because the quantum channel was reconstructed perfectly, and the original unitary mapping maps a pure state to a pure state (\ref{['unitaryDynamcs']}), yielding the exact fidelity (\ref{['FidelityBS']}). The purpose of this plot is to demonstrate the application of the convex SDP optimization approach to the problem of unitary learning.
• Figure 2: The rank of the Choi matrix $\mathcal{J}$ (\ref{['Jmatrix']}) (red) obtained as a solution to the optimization problem for a random $\psi\to\phi$ mapping (\ref{['generalWavefunctionsMapping']}): (a) The case $1\le D=n \le 30$. The rank grows slowly with the increase of $n$; the rank value typically comprises less than 1.5% of the matrix dimension $Dn$. (b) The case where $n=15$ is fixed and $1\le D\le 30$ is varied. The maximal obtained Choi matrix rank is $15$ for $D=1$ (trace calculation); the rank decreases slowly with an increase in $D$. The green line represents relative fidelity $\mathcal{F}/\sum_l \omega^{(l)}$. Since the sample is random, the fidelity decreases with the increase in problem size, as it becomes less likely for two random states to match as the dimension increases. For $D=1$, the fidelity is exact: $\mathcal{F}=\sum_l \omega^{(l)}$; this is simply a trace calculation. These plots show that it is difficult to obtain a high-rank Choi matrix as the solution to the quantum channel reconstruction optimization problem. Quantum channels reconstructed from classical data usually exhibit a low Kraus rank.
• Figure 3: The rank of the Choi matrix $\mathcal{J}_{\mathbf{i};\mathbf{i}^{\prime}}$ (\ref{['Jmatrix']}) obtained as a solution to the optimization problem for a case with random $S_{\mathbf{i};\mathbf{i}^{\prime}}$, without any internal structure, is shown. Similarly to Fig. \ref{['resRankn']}, we also obtain a low-rank Choi matrix $\mathcal{J}$. This leads us to conclude that the low rank of the solution (a few percent of the matrix size $Dn$) is an intrinsic property of the optimization problem (\ref{['SDPobjective']}).
• Figure 4: The rank of the Choi matrix $\mathcal{J}$ (\ref{['Jmatrix']}) (red) obtained as a solution to the optimization problem for a sample calculated as a random pure state passed through the quantum channel, with the eigenvector corresponding to the maximal eigenvalue of the output density matrix taken as the result. The fidelity $\mathcal{F}_{init}/\sum_l \omega^{(l)}$ (green) is the fidelity of this initial map on the generated sample. The fidelity $\mathcal{F}/\sum_l \omega^{(l)}$ (olive) is the fidelity of the quantum channel obtained as a result of the optimization problem (\ref{['SDPobjective']}) with $S_{jk;j^{\prime}k^{\prime}}$ (\ref{['SexamplePurePureExact']}) from this sample. At high dimensions, it is several times higher than the fidelity of the original map. The low Kraus rank feature is also confirmed for this sample. (a) The case $1\le D=n \le 30$. (b) The case where $n = 15$ is fixed and $1\le D\le 30$ is varied. When $D=1$, trace preservation reduces to the trace-calculating quantum channel, which has Kraus rank $N_s=n$.