PAC global optimization for VQE in low-curvature geometric regimes

TL;DR

This paper establishes noise-robust PAC guarantees for the global optimization of VQE in low-curvature geometric regimes, showing sample complexity depends on curvature dimension rather than total parameters.

Contribution

It introduces a geometric framework with PAC guarantees for VQE, demonstrating that optimization complexity depends on low-curvature structure rather than full parameter space.

Findings

Sample complexity scales with curvature dimension r, not total parameters p.

Algorithm finds an ε-optimal region with high probability, near the global minimum.

Complexity is quasi-polynomial in p and ε^{-1}, logarithmic in δ^{-1}.

Abstract

We give noise-robust, Probably Approximately Correct (PAC) guarantees of global $\varepsilon$-optimality for the Variational Quantum Eigensolver under explicit geometric conditions. For periodic ansatzes with bounded generators -- yielding a globally Lipschitz cost landscape on a toroidal parameter space -- we assume that the low-energy region containing the global minimum is a Morse--Bott submanifold whose normal Hessian has rank $r = O(\log p)$ for $p$ parameters, and which satisfies polynomial fiber regularity with respect to coordinate-aligned, embedded flats. This low-curvature-dimensional structure serves as a model for regimes in which only a small number of directions control energy variation, and is consistent with mechanisms such as strong parameter tying together with locality in specific multiscale and tied shallow architectures. Under this assumption, the sample complexity required to find an $\varepsilon$-optimal region with confidence $1-\delta$ scales with the curvature dimension $r$ rather than the ambient dimension $p$. With probability at least $1-\delta$, the algorithm outputs a region in which all points are $\varepsilon$-optimal, and at least one lies within a bounded neighborhood of the global minimum. The resulting complexity is quasi-polynomial in $p$ and $\varepsilon^{-1}$ and logarithmic in $\delta^{-1}$. This identifies a geometric regime in which high-probability global optimization remains feasible despite shot noise.