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Undirected $st$-connectivity is important both for its applications innetwork 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 nontrivialtime-space tradeoff: there is a quantum algorithm that achieves both optimaltime $\tilde{O}(n)$ and space $O(\log (n))$ simultaneously. This improves onprevious results, which required either $O(\log (n))$ space and$\tilde{O}(n^{1.5})$ time, or $\tilde{O}(n)$ space and time. To complementthis, we show that there is a nontrivial time-space tradeoff when given a lowerbound on the spectral gap of a corresponding random walk.

(No) Quantum space-time tradeoff for USTCON