A hybrid ALS method for quantum channel reconstruction

Accurate reconstruction of quantum channels from experimental data is a fundamental challenge in quantum information science. The standard approach via the Choi–Jamiołkowski isomorphism relies on semidefinite programming to enforce physical completely positive and trace-preserving constraints, which becomes computationally prohibitive for high-dimensional systems. To circumvent this dimensionality curse this paper introduces a hybrid alternating least squares framework for quantum channel reconstruction utilizing a mixed unitary parameterization. The proposed approach directly targets the constituent operation elements by alternating between updating the unitary matrices and the associated probability distribution. The probability vector is efficiently resolved via standard constrained least squares. For the unitary elements the framework employs an exact O(n³) singular value decomposition Procrustes update in the single-pair setting, and transitions to a natural gradient flow to manage the simultaneous constraints of multi-pair process tomography. Because the underlying optimization landscape features continuous gauge symmetries and resides on a nonconvex product isospectral manifold establishing theoretical guarantees is highly nontrivial. By embedding the optimization dynamics within the Kurdyka–Łojasiewicz framework and establishing strict sufficient decrease and relative error bounds this work provides a rigorous proof of global convergence to a critical point. Numerical experiments validate these theoretical findings, demonstrating that the framework offers a highly scalable, robust and mathematically principled alternative for recovering physical quantum channels.