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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2510.02151 |
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| _version_ | 1866915603695009792 |
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| author | Lee, Eunou |
| author_facet | Lee, Eunou |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02151 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A quantum analogue of convex optimization Lee, Eunou Quantum Physics Numerical Analysis 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. |
| title | A quantum analogue of convex optimization |
| topic | Quantum Physics Numerical Analysis |
| url | https://arxiv.org/abs/2510.02151 |