A quantum analogue of convex optimization

• URL: http://arxiv.org/abs/2510.02151v2
• Date: Fri, 07 Nov 2025 14:24:53 GMT
• Title: A quantum analogue of convex optimization
• Authors: Eunou Lee,
• Abstract summary: We introduce a quantum analogue of unconstrained convex optimization.<n>We compute the minimum eigenvalue of a Schr"odinger operator $h = -Delta + V with convex potential.<n>We analyze with novel techniques that allow us to focus on the low-energy space.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 Schr\"odinger operator $h = -\Delta + 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 $\epsilon$ in polynomial time in $n$, $1/\epsilon$, 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 $n$, $m$, $1/\epsilon$ and the radius $R$ of a ball encompassing the region.

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