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arXiv:2510.02151

A quantum analogue of convex optimization

Eunou Lee

Published Oct 2, 2025• 0 citations
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Abstract

Convex optimization is the powerhouse behind the theory and practice of optimization. We introduce a quantum analogue of unconstrained convex optimization: computing the minimum eigenvalue of a Schrödinger operator $h = -Δ+ V $ with convex potential $V:\mathbb R^n \rightarrow \mathbb R_{\ge 0}$ such that $V(x)\rightarrow\infty $ as $\|x\|\rightarrow\infty$. For this problem, we present an efficient quantum algorithm, called the Fundamental Gap Algorithm (FGA), that computes the minimum eigenvalue of $h$ up to error $ε$ in polynomial time in $n$, $1/ε$, and parameters that depend on $V$. Adiabatic evolution of the ground state is used as a key subroutine, which we analyze with novel techniques that allow us to focus on the low-energy space. We apply the FGA to give the first known polynomial-time algorithm for finding the lowest frequency of an $n$-dimensional convex drum, or mathematically, the minimum eigenvalue of the Dirichlet Laplacian on an $n$-dimensional region that is defined by $m$ linear constraints in polynomial time in $n$, $m$, $1/ε$ and the radius $R$ of a ball encompassing the region.

DOI
10.48550/arxiv.2510.02151

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Convex optimization is the powerhouse behind the theory and practice of optimization. We introduce a quantum analogue of unconstrained convex optimization: computing the minimum eigenvalue of a Schrödinger operator h=−Δ+Vh = -Δ+ V h=−Δ+V with convex potential V:Rn→R≥0V:\mathbb R^n \rightarrow \mathbb R_{\ge 0}V:Rn→R≥0​ such that V(x)→∞V(x)\rightarrow\infty V(x)→∞ as ∥x∥→∞\|x\|\rightarrow\infty∥x∥→∞. For this problem, we present an efficient quantum algorithm, called the Fundamental Gap Algorithm (FGA), that computes the minimum eigenvalue of hhh up to error εεε in polynomial time in nnn, 1/ε1/ε1/ε, and parameters that depend on VVV. Adiabatic evolution of the ground state is used as a key subroutine, which we analyze with novel techniques that allow us to focus on the low-energy space. We apply the FGA to give the first known polynomial-time algorithm for finding the lowest frequency of an nnn-dimensional convex drum, or mathematically, the minimum eigenvalue of the Dirichlet Laplacian on an nnn-dimensional region that is defined by mmm linear constraints in polynomial time in nnn, mmm, 1/ε1/ε1/ε and the radius RRR of a ball encompassing the region.
Spectral gaps play a fundamental role in many areas of mathematics, computer science, and physics. In quantum mechanics, the spectral gap of Schrödinger operators has a long history of study due to its physical relevance, while in quantum computing spectral gaps are an important proxy for efficiency, such as in the quantum adiabatic algorithm. Motivated by convex optimization, we study Schrödinger operators associated with self-concordant barriers over convex domains and prove non-asymptotic lower bounds on the spectral gap for this class of operators. Significantly, we find that the spectral gap does not display any condition-number dependence when the usual Laplacian is replaced by the Laplace--Beltrami operator, which uses second-order information of the barrier and hence can take the curvature of the barrier into account. As an algorithmic application, we construct a novel quantum interior point method that applies to arbitrary self-concordant barriers and shows no condition-number dependence. To achieve this we combine techniques from semiclassical analysis, convex optimization, and quantum annealing.
This work investigates the possibility of quantum speedups for continuous optimization through quantum Hamiltonian simulation. We establish the first rigorous query complexity bounds for unconstrained convex optimization via digital quantum annealing, based on the non-adiabatic Quantum Hamiltonian Descent (QHD) framework. In the process, we derive rigorous resource estimates for digital quantum simulation of Schrödinger operators depending only on input simulation parameters, given black-box evaluation access to separable, Lipschitz continuous potential b(t) f(x). We apply these simulation bounds to assess the complexity of high-dimensional convex optimization. Our annealing schedule achieves arbitrarily fast convergence rates in the evolution time, with computational time determined solely by the cost of discretization. We show that a G-Lipschitz convex function can be optimized to an error of ε with 𝒪̃(d^{1.5} G² R²/ε²) queries, given a starting point that is Euclidean distance R from optimal. Under reasonable assumptions such as the complexity of simulating Schrödinger operators, we show that Ω̃(d/ε²) queries are necessary. This suggests QHD does not offer improvements over classical methods in the noiseless zeroth order setting. However, we show that the QHD algorithm can tolerate 𝒪̃(ε³ /d^{1.5} G² R²) noise in function evaluation, and as a result, provides a super-quadratic query advantage over the best existing noise-tolerant classical algorithms in the high-dimensional setting. We leverage this to design a quantum algorithm for stochastic convex optimization that offers a super-quadratic speedup over all known classical and quantum algorithms in this regime. To our knowledge, these results represent the first rigorous quantum speedups for convex optimization obtained through a dynamical algorithm.
In this paper, we present a direct quantum adaptation of the classical shifted power method. The method is very similar to the iterative phase estimation algorithm; however, it does not require any initial estimate of an eigenvector and as in the classical case its convergence (the success probability) is directly related to the eigengap. If the amount of the gap is polynomial in the number of qubits nnn, then the algorithm can converge to the dominant eigenvalue in O(poly(n))O(poly(n))O(poly(n)) time. The method can be potentially used in any eigenvalue related problems and to find minimum/maximum of a quantum state in lieu of Grover's search algorithm. In addition, if the solution space of an optimization problem with nnn parameters can be encoded as the eigenspace of an 2n2^n2n dimensional unitary operator in O(poly(n))O(poly(n))O(poly(n)) time, then the solution for such a problem can be found in O(poly(n))O(poly(n))O(poly(n)) if the eigengap is polynomial. As an example, using the quantum gates, we show how to generate the solution space of the quadratic unconstrained binary optimization as eigenvectors of a diagonal unitary matrix and find the solution for the problem.
Quantum adiabatic optimization (QAO) is performed using a time-dependent Hamiltonian H(s)H(s)H(s) with spectral gap γ(s)γ(s)γ(s). Assuming the existence of an oracle ΓΓΓ such that $γ_\min = Θ\left(\min_sΓ(s)\right)$, we provide an algorithm that reliably performs QAO in time $O\left(γ_\min^{-1}\right)$ with $O\left(\log(γ_\min^{-1})\right)$ oracle queries, where $γ_\min = \min_s γ(s)$. Our strategy is not heuristic and does not require guessing time parameters or annealing paths. Rather, our algorithm naturally produces an annealing path such that ∥dH/ds∥≈γ(s)\|dH/ds\| \approx γ(s)∥dH/ds∥≈γ(s) and chooses its own runtime to be as close as possible to optimal while promising convergence to the ground state. We then demonstrate the feasibility of this approach in practice by explicitly constructing a gap oracle ΓΓΓ for the problem of finding the minimum point m=argminuW(u)m = \mathrm{argmin}_u W(u)m=argminu​W(u) of the cost function W:V⟶[0,1]W:\mathcal{V}\longrightarrow [0,1]W:V⟶[0,1], restricting ourselves to computational basis measurements and driving Hamiltonian H(0)=I−∣V∣−1∑u,v∈V∣u⟩⟨v∣H(0)=I - |\mathcal{V}|^{-1}\sum_{u,v \in \mathcal{V}}\vert{u}\rangle\langle{v}\vertH(0)=I−∣V∣−1∑u,v∈V​∣u⟩⟨v∣. Requiring only that WWW have a constant lower bound on its spectral gap and upper bound κκκ on its spectral ratio, our QAO algorithm returns mmm with probability (1−ε)(1−e−1/ε)(1-ε)(1-e^{-1/ε})(1−ε)(1−e−1/ε) in time O~(ε−1[∣V∣+(κ−1)2/3∣V∣2/3])\widetilde{\mathcal{O}}(ε^{-1}[\sqrt{|\mathcal{V}|} + (κ-1)^{2/3}|\mathcal{V}|^{2/3}])O(ε−1[∣V∣​+(κ−1)2/3∣V∣2/3]). This achieves a quantum advantage for all κκκ, and recovers Grover scaling up to logarithmic factors when κ≈1κ\approx 1κ≈1. We implement the algorithm as a subroutine in an optimization procedure that produces mmm with exponentially small failure probability and expected runtime O~(ε−1[∣V∣+(κ−1)2/3∣V∣2/3])\widetilde{\mathcal{O}}(ε^{-1}[\sqrt{|\mathcal{V}|} + (κ-1)^{2/3}|\mathcal{V}|^{2/3}])O(ε−1[∣V∣​+(κ−1)2/3∣V∣2/3]) even when κκκ is not known beforehand.
Several previous works have investigated the circumstances under which quantum adiabatic optimization algorithms can tunnel out of local energy minima that trap simulated annealing or other classical local search algorithms. Here we investigate the even more basic question of whether adiabatic optimization algorithms always succeed in polynomial time for trivial optimization problems in which there are no local energy minima other than the global minimum. Surprisingly, we find a counterexample in which the potential is a single basin on a graph, but the eigenvalue gap is exponentially small as a function of the number of vertices. In this counterexample, the ground state wave-function consists of two lobes separated by a region of exponentially small amplitude. Conversely, we prove if the ground state wavefunction is single-peaked then the eigenvalue gap scales at worst as one over the square of the number of vertices.
Several previous works have investigated the circumstances under which quantum adiabatic optimization algorithms can tunnel out of local energy minima that trap simulated annealing or other classical local search algorithms. Here we investigate the even more basic question of whether adiabatic optimization algorithms always succeed in polynomial time for trivial optimization problems in which there are no local energy minima other than the global minimum. Surprisingly, we find a counterexample in which the potential is a single basin on a graph, but the eigenvalue gap is exponentially small as a function of the number of vertices. In this counterexample, the ground state wavefunction consists of two "lobes" separated by a region of exponentially small amplitude. Conversely, we prove if the ground state wavefunction is single-peaked then the eigenvalue gap scales at worst as one over the square of the number of vertices.