(No) Quantum space-time tradeoff for USTCON

TL;DR

This paper demonstrates that quantum algorithms can solve undirected st-connectivity optimally in both time and space, breaking classical tradeoff limitations, and explores conditions under which tradeoffs still exist.

Contribution

It proves the existence of a quantum algorithm achieving optimal time and space simultaneously for USTCON, and analyzes scenarios where tradeoffs persist.

Findings

Quantum algorithm achieves optimal $ ilde{O}(n)$ time and $O( ext{log } n)$ space.

Classical tradeoffs do not apply in the quantum setting for USTCON.

Tradeoffs remain when spectral gap bounds are considered.

Abstract

Undirected $st$-connectivity is important both for its applications in network problems, and for its theoretical connections with logspace complexity. Classically, a long line of work led to a time-space tradeoff of $T=\tilde{O}(n^2/S)$ for any $S$ such that $S=\Omega(\log (n))$ and $S=O(n^2/m)$. Surprisingly, we show that quantumly there is no nontrivial time-space tradeoff: there is a quantum algorithm that achieves both optimal time $\tilde{O}(n)$ and space $O(\log (n))$ simultaneously. This improves on previous results, which required either $O(\log (n))$ space and $\tilde{O}(n^{1.5})$ time, or $\tilde{O}(n)$ space and time. To complement this, we show that there is a nontrivial time-space tradeoff when given a lower bound on the spectral gap of a corresponding random walk.