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A Quantum Time-Space Tradeoff for Directed $st$-Connectivity

Stacey Jeffery, Galina Pass

TL;DR

This work proposes a quantum time-space tradeoff for directed st-connectivity (DSTCON). By separating quantum space from the dominant classical space, the authors design a quantum algorithm that, for any $S \ge \log^2(n)$, uses total space $S$ and achieves time $T \le 2^{\tfrac{1}{2}\log(n)\log(n/S)+O(\log n\log\log n)}$ with only $O(\log^2 n)$ quantum space, yielding a quadratic or near-quadratic speedup over the best known classical tradeoffs in the small-space regime. The core technical advance is a quantum short-path subroutine, Dist$_L$, built via recursively constructed switching networks, enabling a reversible, bounded-error quantum evaluation of reachability within path-length $L$. The results demonstrate that classical space can be traded for quantum time in a nontrivial way for a NL-complete problem and open directions toward tighter quantum-space bounds and broader directed-graph applications. The framework relies on span-program-inspired switching networks to compose quantum subroutines without incurring prohibitive overheads, marking a first quantum time-space tradeoff for DSTCON with practical quantum-space guarantees.

Abstract

Directed $st$-connectivity (DSTCON) is the problem of deciding if there exists a directed path between a pair of distinguished vertices $s$ and $t$ in an input directed graph. This problem appears in many algorithmic applications, and is also a fundamental problem in complexity theory, due to its ${\sf NL}$-completeness. We show that for any $S\geq \log^2(n)$, there is a quantum algorithm for DSTCON using space $S$ and time $T\leq 2^{\frac{1}{2}\log(n)\log(n/S)+o(\log^2(n))}$, which is an (up to quadratic) improvement over the best classical algorithm for any $S=o(\sqrt{n})$. Of the $S$ total space used by our algorithm, only $O(\log^2(n))$ is quantum space - the rest is classical. This effectively means that we can trade off classical space for quantum time.

A Quantum Time-Space Tradeoff for Directed $st$-Connectivity

TL;DR

This work proposes a quantum time-space tradeoff for directed st-connectivity (DSTCON). By separating quantum space from the dominant classical space, the authors design a quantum algorithm that, for any , uses total space and achieves time with only quantum space, yielding a quadratic or near-quadratic speedup over the best known classical tradeoffs in the small-space regime. The core technical advance is a quantum short-path subroutine, Dist, built via recursively constructed switching networks, enabling a reversible, bounded-error quantum evaluation of reachability within path-length . The results demonstrate that classical space can be traded for quantum time in a nontrivial way for a NL-complete problem and open directions toward tighter quantum-space bounds and broader directed-graph applications. The framework relies on span-program-inspired switching networks to compose quantum subroutines without incurring prohibitive overheads, marking a first quantum time-space tradeoff for DSTCON with practical quantum-space guarantees.

Abstract

Directed -connectivity (DSTCON) is the problem of deciding if there exists a directed path between a pair of distinguished vertices and in an input directed graph. This problem appears in many algorithmic applications, and is also a fundamental problem in complexity theory, due to its -completeness. We show that for any , there is a quantum algorithm for DSTCON using space and time , which is an (up to quadratic) improvement over the best classical algorithm for any . Of the total space used by our algorithm, only is quantum space - the rest is classical. This effectively means that we can trade off classical space for quantum time.

Paper Structure

This paper contains 28 sections, 37 theorems, 121 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution:
  3. Our Techniques:
  4. Open Problems:
  5. Organization:
  6. Preliminaries
  7. Generating superpositions
  8. Graphs
  9. Switching Networks
  10. Flows, Circulations, and the Cut Space
  11. Quantum Algorithm for DSTCON
  12. BFS algorithm that calls the quantum short path subroutine
  13. Complexity comparison: classical vs. quantum
  14. Quantum short path subroutine
  15. Switching network
  16. ...and 13 more sections

Key Result

Lemma 2.1

Fix $\alpha \in \mathbb{R}^{\{0,1\}^m}$, and suppose there is a subroutine that can compute, given any prefix $p\in \{0,1\}^{\leq m}$, the partial sum of entries of $\alpha$ with prefix $p$, $S(p) = \sum_{s\in\{0,1\}^m:s \text{ has prefix } p}\alpha_s^2$, in time $T$. Then the state $\frac{1}{\sqrt{

Figures (5)

  • Figure 1: (L) An example of a switching network. (R) The same switching network, with edges switched on (thick) or off (dashed) by the assignment $x=10101$. In this example, $u$ and $v$ are connected by a path of edges labelled by variables that are true under $x$, and so the switching network accepts $x$.
  • Figure 2: The graph $G'$ constructed from $G$. It is clear that there is a $uv$-path of length at most $L$ in $G$ if and only if there is a $s_1v$-path of length at most $L+a$ in $G'$.
  • Figure 3: Graph construction for the switching network ${\cal N}_1(u)$ that decides whether each vertex of a graph $G = (V,E)$ is reachable from a vertex $u \in V$ by a path of length $1$. Each edge $([u],[u,v_i])$ of ${\cal N}_1(u)$ has query label $(u, v_i)$ and is "on" in ${\cal N}_1(u)(G)$ if and only if $(u, v_i) \in E$. Therefore, it holds that $v_i$ is reachable from $u$ by a path of length at most $1$ in $G$ if and only if $[u]$ and $[u, v_i]$ are connected in ${\cal N}_1(u)(G)$.
  • Figure 4: Graph construction for the switching network ${\cal N}_{2^\ell}(u)$ that decides whether each vertex of a graph $G = (V,E)$ is reachable from a vertex $u \in V$ by a path of length $2^\ell$.
  • Figure 5: Visualization of the state ${\lvert}p_{ij}\rangle = {\lvert}\bar{\theta}^0_i\rangle + {\lvert}\bar{\theta}^{1i}_{j}\rangle - {\lvert}\bar{\theta}^{2j}_{i}\rangle={\lvert}0,\bar{0}\rangle{\lvert}\bar{\theta}_i\rangle + {\lvert}1,i\rangle{\lvert}\bar{\theta}_{j}\rangle + {\lvert}2,j\rangle{\lvert}\bar{\theta}_{i}\rangle$ in ${\cal N}_{2^\ell}$, where $j\in \{0,1\}^{\log n}$ and $\ell\in [\log(L)]$. Here $\theta_j$ denotes the optimal unit $[u], [u,v_j]$-flow in ${\cal N}_{2^{\ell - 1}}$, and ${\lvert}\bar{\theta}_j\rangle$ is as in \ref{['eq:cropped']}. The superscript encodes the copy of ${\cal N}_{2^{\ell - 1}}$ in ${\cal N}_{2^\ell}$.

Theorems & Definitions (73)

  • Lemma 2.1: grover2002creating
  • Remark 2.2: Larger alphabets
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7: Basis Generation
  • Lemma 2.8
  • Theorem 2.9
  • Lemma 2.10
  • ...and 63 more