Conditions for global optimality in quantum optimization algorithms
Conditions for Global Optimality in Quantum Arimoto-Blahut Algorithms
Information Theory
Summary
Quantum Arimoto-Blahut (AB) algorithms help solve complex information problems. The authors identify exact mathematical conditions that show when these algorithms reach the best possible solution. They also introduce ways to check if the solution found is truly optimal using only simple tests. Their work helps understand why sometimes these algorithms get stuck and how to detect it. This improves confidence in using AB methods for quantum information tasks.
What this means in practice
- •For quantum information engineers: Verify that quantum algorithms for optimizing communication channels have reached the true global optimum using new computable criteria.
- •For algorithm developers in optimization: Detect and avoid suboptimal stable points in iterative quantum optimization methods by employing direction-based optimality certificates.
Authors
Geng Liu, Masahito Hayashi
Abstract
Generalized Arimoto--Blahut (AB) algorithms are widely used in information theory and quantum optimization, but monotonic objective decrease and numerical stabilization do not guarantee global optimality. We establish necessary and sufficient conditions for the global optimality of full-rank AB fixed points for convex differentiable objectives under linear constraints. An AB fixed point is globally optimal if and only if the AB update direction and the objective gradient differ by an element of the constraint normal space at that point. The same compatibility condition characterizes agreement between individual AB and mirror-descent (MD) updates, whereas pathwise equivalence requires it along the entire common trajectory. Thus, an AB algorithm may follow a trajectory different from MD and still reach the global optimum. We also develop a posteriori optimality certificates based on feasible directional derivatives, including finite-difference upper bounds on the objective gap that require only objective evaluations. For channel relative entropy between dephasing and depolarizing channels, we analytically identify the global minimizer as the unique full-rank AB fixed point, although pathwise equivalence fails. Numerical experiments illustrate convergence of AB and MD along different trajectories and validate the certificates. By contrast, an amplitude-damping example shows that a monotone AB iteration can stabilize at a suboptimal fixed point, whose nonoptimality is detected by the finite-difference certificate. These results provide structural and computable criteria for assessing global optimality in quantum AB algorithms.