Directed st-connectivity with few paths is in quantum logspace

2024-08-22

arXiv.org

AI Summary

This paper investigates quantum logarithmic-space complexity for $s$-$t$ path counting and connectivity in directed graphs where the number of paths from $s$ and to $t$ is polynomially bounded (i.e., $leq mathrm{poly}(n)$). We present the first $ mathbf{BQSPACE}(O(log n)) $ algorithm that decides $s$-$t$ connectivity and exactly counts $s$-$t$ paths using only $O(log n)$ qubits—breaking the classical $ mathbf{DSPACE}(O(log^2 n / log log n)) $ upper bound. This establishes $s$-$t$ connectivity as a natural problem in $ mathbf{BQL} $, revealing deep connections with unambiguous logspace ($ mathbf{UL} $) and sparse subclasses of $ mathbf{NL} $. Moreover, we identify the first natural candidate language potentially separating $ mathbf{BQL} $ from $ mathbf{L} $ and $ mathbf{BPL} $. To support scalability, we design an efficient graph preprocessing scheme enabling online verification of polynomial path-boundedness for arbitrary node pairs.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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, some preprocessing in our approach also allows us to verify whether there are at most polynomially many paths between any two nodes in $mathsf{BQSPACE}(O(log n))$. This yields the first natural candidate for a language problem separating $mathsf{BQL}$ from $mathsf{L}$ and $mathsf{BPL}$. Until now, all candidates separating these classes were promise problems.

Problem

• Count st-paths in directed graphs quantumly
• Compare quantum and classical space complexity bounds
• Identify natural BQL-L separation candidates

Innovation

• Quantum logspace for directed st-connectivity
• Counting paths with polynomial bounds
• Recognizing graphs with few paths