One Coordinate at a Time: Convergence Guarantees for Rotosolve in Variational Quantum Algorithms

TL;DR

This paper provides the first formal convergence analysis of the Rotosolve algorithm for variational quantum algorithms, establishing conditions and rates under which it converges.

Contribution

We rigorously prove convergence guarantees for Rotosolve, a popular interpolation-based coordinate descent method, under non-convex and smooth landscapes, and with the PL condition.

Findings

Rotosolve converges to ε-stationary points in non-convex smooth landscapes.

Under the PL condition, Rotosolve converges to ε-suboptimal points.

Explicit worst-case convergence rates are derived for finite quantum measurement regimes.

Abstract

In this paper, we resolve an open question in the field of optimization algorithms for training parametrized quantum circuits: Does the popular Rotosolve algorithm converge? Until now, interpolation-based coordinate descent methods such as Rotosolve have mostly been treated as heuristics, lacking any formal convergence guarantees. We rigorously analyze Rotosolve, and show that it converges to $\varepsilon$-stationary points if the optimization landscape is non-convex and smooth; and to $\varepsilon$-suboptimal points if the objective function additionally obeys the Polyak-Lojasiewicz (PL) condition. Further, we derive explicit worst-case rates of convergence in the finite quantum measurement regime. These rates are contrasted against those from a similar coordinate-based method: Randomized Coordinate Descent (RCD). Although in the worst case their rates are, prima facie, equivalent, we present arguments for a more nuanced comparison between the two. We highlight that Rotosolve is hyperparameter-free, and implicitly uses first and second derivatives in its updates. Finally, we supplement our theoretical findings with numerical experiments from Quantum Machine Learning; and compare the performance of Rotosolve against RCD, Stochastic Gradient Descent, Simultaneous Perturbation Stochastic Approximation, and Randomized Stochastic Gradient Free methods.