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arXiv · 2408.12473

Directed st-connectivity with few paths is in quantum logspace

Simon Apers·Roman Edenhofer
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Abstract

We present a $\mathsf{BQSPACE}(O(\log n))$-procedure to count $st$-paths on directed graphs for which we are promised that there are at most polynomially many paths starting in $s$ and polynomially many paths ending in $t$. For comparison, the best known classical upper bound in this case just to decide $st$-connectivity is $\mathsf{DSPACE}(O(\log^2 n/ \log \log n))$. The result establishes a new relationship between~$\mathsf{BQL}$ and unambiguity and fewness subclasses of $\mathsf{NL}$. Further, we also show how to \emph{recognize} directed graphs with at most polynomially many paths between any two nodes in $\mathsf{BQSPACE}(O(\log n))$. This yields the first natural candidate for a language separating $\mathsf{BQL}$ from $\mathsf{L}$ and~$\mathsf{BPL}$. Until now, all candidates potentially separating these classes were inherently promise problems.

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Simon Apers, Roman Edenhofer. 2024-08-22. Directed st-connectivity with few paths is in quantum logspace. https://arxiv.org/abs/2408.12473

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