Quantum limits on searching smooth shapes for optimal points revealed
Near-Optimal Quantum Lower Bounds for Convex Optimization via Fourier Rank
Data Structures and Algorithms
Summary
Finding the best spot inside a smooth, curved shape, called a convex set, is important in many problems. The authors discovered that even with quantum computers, you need nearly as many checks of the shape as its size to find a very precise best spot. They built this by studying how hard it is to calculate certain matrix properties using quantum methods. This work answers a question about the power and limits of quantum algorithms for these optimization tasks.
Quantum query complexityConvex optimizationMembership oracleEllipsoidsFourier rankMatrix determinantMinimum eigenvaluePhase-query modelSmooth convex functions
Authors
Brandon Augustino, Shouvanik Chakrabarti, Enrico Fontana, Dylan Herman, Junhyung Lyle Kim, Guneykan Ozgul, Nadezhda Voronova
Abstract
We establish a near-linear quantum query lower bound for high-accuracy convex optimization over an explicit family of $n$-dimensional ellipsoids. We focus on linear optimization with an explicitly given objective, where the feasible set is accessed through a membership oracle. We show that any algorithm that, for every unit linear objective, returns an exactly feasible point with additive objective error $Θ(n^{-2})$ requires $Ω\!\left(\frac{n}{\log n\,\log\log n}\right)$ membership queries. The same lower bound can be shown to hold if the returned point is only required to be approximately feasible, within $Θ(n^{-2})$ distance from the feasible set. This resolves, up to logarithmic factors, an open question posed by Chakrabarti, Childs, Li, and Wu~(\textit{Quantum}, 2020) and by van Apeldoorn, Gilyén, Gribling, and de Wolf~(\textit{Quantum}, 2020). Coupled with the upper bounds in these papers, the query complexity of high-accuracy convex optimization is characterized tightly up to logarithmic factors. The proof is built around a lower bound for determinant computation that is derived via a novel polynomial method based on Fourier-rank. In the continuous matrix phase-query model, computing the determinant of a real $n\times n$ matrix requires at least $n/2$ matrix-vector product queries. The construction also yields an $Ω(n)$ phase-query lower bound for estimating the minimum eigenvalue of a real symmetric $n\times n$ matrix to additive accuracy $Θ(n^{-2})$. These results extend the determinant and minimum-eigenvalue lower bounds of Childs, Hung, and Li~(ICALP 2021) from finite fields to the real-valued setting. Based on the same constructions, we also prove a near-optimal gradient-query lower bound for constant-accuracy optimization of smooth and strongly convex functions.