Problem
Research questions and friction points this paper is trying to address.
convex body volume estimation
quantum query complexity
high-dimensional geometry
quantum membership oracle
Improved Upper and Lower Bounds for Quantum Convex-Body Volume Estimation
🤖 AI Summary
This study addresses the high quantum query complexity in volume estimation of high-dimensional convex bodies by proposing an optimized quantum algorithm. Methodologically, it integrates input-dependent spectral gap analysis with normalized ratio estimation, employing a single quantum estimator to replace conventional multi-stage procedures while combining quantum walks, simulated annealing, and classical bridge sampling techniques. The primary contribution lies in reducing the upper bound of quantum query complexity to O(d^{5/2} + d^{3/2}/ε) while establishing an Ω(d) lower bound. This advancement significantly enhances the quantum speedup, surpassing the best existing results in the literature.
📝 Abstract
Estimating the volume of a high-dimensional convex body is a fundamental problem in theoretical computer science. Given a quantum membership oracle for a convex body $K\subset\mathbb{R}^d$, we show that estimating $\mathrm{vol}(K)$ to relative error $\varepsilon$ takes $\widetilde O(d^{5/2}+d^{3/2}/\varepsilon)$ queries. This improves the previous quantum upper bound $\widetilde O(d^3+d^{9/4}/\varepsilon)$ of Chakrabarti et al. (ACM TQC 2023) and Cornelissen--Hamoudi (SODA 2023), and achieves a larger quantum speedup over the currently best classical upper bound $\widetilde O(d^{7/2}+d^3/\varepsilon^2)$ of Jia et al. (JACM 2026). Our quantum algorithm combines two new ingredients: an input-dependent effective spectral-gap analysis of quantum walks based on uniform classical warm-start mixing bounds, and a single quantum estimator for products of normalizer ratios in simulated annealing based on classical bridge sampling. We also prove an $\Omega(d)$ quantum query lower bound for constant relative error, improving the previous $\Omega(\sqrt d)$ lower bound.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Quantum volume estimation
Quantum walk
Spectral gap analysis
Bridge sampling
Query complexity
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