(No) Quantum space-time tradeoff for USTCON
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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=Õ(n^2/S) for any S such that S=Ω(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 Õ(n) and space O(log (n)) simultaneously. This improves on previous results, which required either O(log (n)) space and Õ(n^1.5) time, or Õ(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.
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