Directed st-connectivity with few paths is in quantum logspace

TL;DR

This paper demonstrates that counting $st$-paths in directed graphs with few paths can be done efficiently in quantum logspace, revealing new relationships between quantum and classical complexity classes and identifying potential separations.

Contribution

It introduces a quantum logspace procedure for counting $st$-paths with few paths and shows how to recognize such graphs, advancing understanding of quantum complexity class separations.

Findings

Quantum logspace procedure for counting $st$-paths with few paths.

Recognition of graphs with polynomially many $st$-paths in quantum logspace.

First natural candidate for separating BQL from L and BPL.

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.