Quantum Space-Time Tradeoffs for TSP via Extremal Set Systems

2026-07-14

AI Summary

This work investigates quantum space–time trade-offs for permutation-based minimization problems, such as the Traveling Salesman Problem. It introduces the first general quantum trade-off framework tailored to this class of problems, combining classical extremal set systems with quantum minimum-finding to achieve improved complexity in the QRAM model. By leveraging a novel construction of extremal sets and a refined analysis of normalized chain densities, the proposed algorithm significantly outperforms the prior quantum approach by Caroppo et al. across the memory range $1 < S \leq 1.657$, yielding an explicit quantum space–time trade-off curve that advances the state of the art.

Abstract

Recent work of Ameli, Nederlof and Wang and of Dallant and Kozma introduced a framework for improving classical space--time tradeoffs for the Traveling Salesman Problem (TSP) and related permutation problems via extremal set systems with many maximal chains. In this note we observe that, for so called permutation problems whose outer aggregation is a minimum (such as TSP), the same framework admits a simple quantum analogue: instead of iterating over the covering family of set systems, we apply quantum minimum finding over the family. More precisely, let $P_S$ denote the optimal inverse normalized chain density among set systems of normalized size at most $S$. Then TSP admits a bounded-error quantum algorithm using $\widetilde O(S^n)$ QRAM space and \[ \widetilde O\!\left((S\sqrt{P_S})^n\right) \] time. The same argument applies to other minimization problems over permutations with a similar structure to TSP. Combining this observation with improved extremal set-system constructions of Andoni, Dallant, Kozma and Yu gives an explicit quantum space--time tradeoff curve, which beats the known quantum tradeoff by Caroppo et al. for all $1<S \leq 1.657$.

Problem

• Quantum algorithms
• Space-time tradeoffs
• Traveling Salesman Problem
• Permutation problems
• Extremal set systems

Innovation

• quantum algorithm
• space-time tradeoff
• Traveling Salesman Problem
• extremal set systems
• quantum minimum finding