(No) Quantum space-time tradeoff for USTCON

TL;DR

The paper advances quantum algorithms for undirected st-connectivity (USTCON) by showing a quantum walk-based method that attains time $\tilde{O}(n)$ and space $O(\log n)$ for ustcon$_{qw}$, thus eliminating a nontrivial quantum time-space tradeoff in that model. It introduces a Metropolis-Hastings–driven quantum walk (on an expanded graph) to achieve optimal resource usage, and proves a matching lower bound via a parity reduction. Additionally, it establishes a spectral-gap–promised tradeoff: for any $S\ge\Omega(\log n)$ and gap $\delta>0$, there is a quantum algorithm with $T=\tilde{O}\left(\frac{S}{\delta}+\sqrt{\frac{n}{\delta S}}\right)$, highlighting how mixing-time properties can enable tunable quantum resource allocation. The work also discusses QCRAM-based memory, the relationship between quantum walk access and array access, and the implications for quantum query complexity in logspace-related problems. Overall, the results demonstrate near-optimal quantum time-space performance for USTCON and clarify when a tradeoff is feasible under additional spectral assumptions.

Abstract

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 $T=\tilde{O}(n^2/S)$ 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 $\tilde{O}(n)$ and space $O(\log (n))$ simultaneously. This improves on previous results, which required either $O(\log (n))$ space and $\tilde{O}(n^{1.5})$ time, or $\tilde{O}(n)$ 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.

Figures (2)

• Figure 1: Quantum space-time tradeoffs for USTCON, with axes representing the time complexity and spectral gap promise (up to polylog-factors). The grey area represents the regime in which a non-trivial tradeoff is achieved. \ref{['thm:intro1']} (upper line) corresponds to the regime with space $S = O(\log n)$ and time $T = \widetilde{O}(n)$. \ref{['thm:intro2']} (grey area) corresponds to the regime with a promise on $\delta$, and interpolates between $S = O(\log n)$ and $T = \widetilde{O}(n)$, and $S = O((n\delta)^{1/3})$ and $T = \widetilde{O}(n^{1/3}/\delta^{2/3})$.
• Figure 2: The parity graph. We include an edge labelled by "$x_i$" (in red) if and only if $x_i=1$, and an edge labelled "$\bar{x}_i$" (in blue) if and only if $x_i=0$, meaning that for each vertex we include exactly one of the two incoming edges, and exactly one of the two outgoing edges. The resulting graph has $s$ and $t$ connected if and only if $\textsc{parity}(x)=1$.

Theorems & Definitions (29)

• Theorem 1.1: Informal
• Theorem 1.2: Informal
• Theorem 2.1: levin2017markov
• Theorem 2.2
• Lemma 2.3
• Lemma 3.3
• proof
• Lemma 4.2: beals2001QLowerBoundPolyfarhi1998parity
• Theorem 4.3
• proof