PAC global optimization for VQE in low-curvature geometric regimes
Benjamin Asch
TL;DR
The paper addresses the challenge of globally optimizing the Variational Quantum Eigensolver (VQE) in high-dimensional parameter spaces by exploiting a low-curvature geometric regime: a Morse–Bott low-energy dent with normal Hessian rank $r=O(\log p)$ and polynomial fiber regularity on a torus parameter space. It introduces Adaptive Lipschitz Elimination on the Torus (ALeT), which performs randomized slicing of the dent onto $(r{+}1)$-dimensional flats and uses noisy Lipschitz certificates to certify suboptimal regions under shot noise, yielding PAC guarantees. Under these structural assumptions, the authors show a quasi-polynomial sample complexity in $p$ and $1/\varepsilon$ and logarithmic dependence on $1/\delta$, making high-probability global optimization feasible in regimes where curvature concentrates in a small number of directions. The approach provides a geometric-probabilistic framework that links circuit architecture (tying, locality, shallow multiscale patterns) to tractable optimization, offering practical warm-starts and robustness insights for NISQ-era quantum devices. Overall, the work delineates a conditional but scalable pathway to reliable global optimization in VQE by leveraging intrinsic geometric dimensionality rather than relying on global convexity or surjective properties.
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 $1-δ$ scales with the curvature dimension $r$ rather than the ambient dimension $p$. With probability at least $1-δ$, 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 $δ^{-1}$. This identifies a geometric regime in which high-probability global optimization remains feasible despite shot noise.