The problem of minimizing the maximum of $N$ convex, Lipschitz functions plays significant roles in optimization and machine learning. It has a series of results, with the most recent one requiring $O(Nε^{-2/3} + ε^{-8/3})$ queries to a first-order oracle to compute an $ε$-suboptimal point. On the other hand, quantum algorithms for optimization are rapidly advancing with speedups shown on many important optimization problems. In this paper, we conduct a systematic study for quantum algorithms and lower bounds for minimizing the maximum of $N$ convex, Lipschitz functions. On one hand, we develop quantum algorithms with an improved complexity bound of $\tilde{O}(\sqrt{N}ε^{-5/3} + ε^{-8/3})$. On the other hand, we prove that quantum algorithms must take $\tildeΩ(\sqrt{N}ε^{-2/3})$ queries to a first order quantum oracle, showing that our dependence on $N$ is optimal up to poly-logarithmic factors.
The problem of minimizing the maximum of N convex, Lipschitz functions plays significant roles in optimization and machine learning. It has a series of results, with the most recent one requiring O(Nε−2/3+ε−8/3) queries to a first-order oracle to compute an ε-suboptimal point. On the other hand, quantum algorithms for optimization are rapidly advancing with speedups shown on many important optimization problems. In this paper, we conduct a systematic study for quantum algorithms and lower bounds for minimizing the maximum of N convex, Lipschitz functions. On one hand, we develop quantum algorithms with an improved complexity bound of O~(Nε−5/3+ε−8/3). On the other hand, we prove that quantum algorithms must take Ω~(Nε−2/3) queries to a first order quantum oracle, showing that our dependence on N is optimal up to poly-logarithmic factors.
Finite-sum optimization has wide applications in machine learning, covering important problems such as support vector machines, regression, etc. In this paper, we initiate the study of solving finite-sum optimization problems by quantum computing. Specifically, let f1,…,fn:Rd→R be ℓ-smooth convex functions and ψ:Rd→R be a μ-strongly convex proximal function. The goal is to find an ε-optimal point for F(x)=n1∑i=1nfi(x)+ψ(x). We give a quantum algorithm with complexity O~(n+d+ℓ/μ(n1/3d1/3+n−2/3d5/6)), improving the classical tight bound Θ~(n+nℓ/μ). We also prove a quantum lower bound Ω~(n+n3/4(ℓ/μ)1/4) when d is large enough. Both our quantum upper and lower bounds can extend to the cases where ψ is not necessarily strongly convex, or each fi is Lipschitz but not necessarily smooth. In addition, when F is nonconvex, our quantum algorithm can find an ε-critial point using O~(n+ℓ(d1/3n1/3+d)/ε2) queries.
Recent results suggest that quantum computers possess the potential to speed up nonconvex optimization problems. However, a crucial factor for the implementation of quantum optimization algorithms is their robustness against experimental and statistical noises. In this paper, we systematically study quantum algorithms for finding an ε-approximate second-order stationary point (ε-SOSP) of a d-dimensional nonconvex function, a fundamental problem in nonconvex optimization, with noisy zeroth- or first-order oracles as inputs. We first prove that, up to noise of O(ε10/d5), accelerated perturbed gradient descent with quantum gradient estimation takes O(logd/ε1.75) quantum queries to find an ε-SOSP. We then prove that perturbed gradient descent is robust to the noise of O(ε6/d4) and O(ε/d0.5+ζ) for ζ>0 on the zeroth- and first-order oracles, respectively, which provides a quantum algorithm with poly-logarithmic query complexity. We then propose a stochastic gradient descent algorithm using quantum mean estimation on the Gaussian smoothing of noisy oracles, which is robust to O(ε1.5/d) and O(ε/d) noise on the zeroth- and first-order oracles, respectively. The quantum algorithm takes O(d2.5/ε3.5) and O(d2/ε3) queries to the two oracles, giving a polynomial speedup over the classical counterparts. Moreover, we characterize the domains where quantum algorithms can find an ε-SOSP with poly-logarithmic, polynomial, or exponential number of queries in d, or the problem is information-theoretically unsolvable even by an infinite number of queries. In addition, we prove an Ω(ε−12/7) lower bound in ε for any randomized classical and quantum algorithm to find an ε-SOSP using either noisy zeroth- or first-order oracles.
We study the quantum query complexity of maximizing a non-negative submodular function, considering both the unconstrained setting and, for monotone functions, a cardinality constraint k on an n-element ground set. In the exact reversible digital value-oracle model, our unconstrained algorithm achieves an expected (1/2−ε)-approximation using only Oε(logn) queries. In contrast, any classical randomized algorithm that attains a fixed expected ratio above 1/4 requires Ω(n/logn) queries (Li, Feldman, Kazemi, and Karbasi, 2022), establishing an exponential separation in query complexity. For cardinality-constrained maximization, we give a bounded-error quantum algorithm that achieves a (1−1/e−ε)-approximation using Oε(min{n,n/k}) queries. When k=o(n), our algorithm achieves at least a quadratic speedup up to logarithmic factors over classical randomized algorithms (Mirzasoleiman, Badanidiyuru, Karbasi, Vondrák, and Krause, 2015; Peng and Rubinstein, 2025). Moreover, when k=cn for any fixed rational c<1−1/e−ε, the query complexity reduces to Oε,c(logn), yielding an exponential separation from the classical Ω(n/logn) lower bound (Li, Feldman, Kazemi, and Karbasi, 2022). We further prove quantum lower bounds of exp(Ω(ε2n)) queries for achieving a ratio beyond 1/2+ε without constraints, and exp(Ω(ε2k)) queries for exceeding 1−1/e+ε when k/n≤ε. These barriers demonstrate that quantum computation offers no exponential speedup at these approximation thresholds.
Quantum algorithms for optimization problems are of general interest. Despite recent progress in classical lower bounds for nonconvex optimization under different settings and quantum lower bounds for convex optimization, quantum lower bounds for nonconvex optimization are still widely open. In this paper, we conduct a systematic study of quantum query lower bounds on finding ε-approximate stationary points of nonconvex functions, and we consider the following two important settings: 1) having access to p-th order derivatives; or 2) having access to stochastic gradients. The classical query lower bounds is Ω(ε−p1+p) regarding the first setting, and Ω(ε−4) regarding the second setting (or Ω(ε−3) if the stochastic gradient function is mean-squared smooth). In this paper, we extend all these classical lower bounds to the quantum setting. They match the classical algorithmic results respectively, demonstrating that there is no quantum speedup for finding ε-stationary points of nonconvex functions with p-th order derivative inputs or stochastic gradient inputs, whether with or without the mean-squared smoothness assumption. Technically, our quantum lower bounds are obtained by showing that the sequential nature of classical hard instances in all these settings also applies to quantum queries, preventing any quantum speedup other than revealing information of the stationary points sequentially.
This paper considers the problem for finding the (δ,ε)-Goldstein stationary point of Lipschitz continuous objective, which is a rich function class to cover a great number of important applications. We construct a zeroth-order quantum estimator for the gradient of the smoothed surrogate. Based on such estimator, we propose a novel quantum algorithm that achieves a query complexity of O~(d3/2δ−1ε−3) on the stochastic function value oracle, where d is the dimension of the problem. We also enhance the query complexity to O~(d3/2δ−1ε−7/3) by introducing a variance reduction variant. Our findings demonstrate the clear advantages of utilizing quantum techniques for non-convex non-smooth optimization, as they outperform the optimal classical methods on the dependency of ε by a factor of ε−2/3.
We establish a 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−1/2) requires Ω(n) membership queries, where the ellipsoids are contained in the unit ball. The same lower bound can be shown to hold if the returned point is only required to be approximately feasible, within Θ(n−3/2) distance from the feasible set. This resolves 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×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×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 an Ω(min{κ,n}) gradient-query lower bound for constant-accuracy optimization of 1-strongly convex and κ-smooth functions, which is optimal up to a logarithmic factor.