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Hiding, Shuffling, and Cycle Finding: Quantum Algorithms on Edge Lists

Amin Shiraz Gilani, Daochen Wang, Pei Wu, Xingyu Zhou

TL;DR

This work analyzes triangle and cycle problems on the edge-list input model under quantum query complexity. It introduces hiding and shuffling transforms to study how input organization affects complexity, and develops Mirroring and Exclusion lemmas to handle rare events within the recording query framework. The authors obtain tight or near-tight bounds: Triangle finding on sparse graphs achieves $Q( ext{Triangle})= ilde{ heta}(m^{5/7})$ with matching upper bounds via Belovs's learning graph, and extend these ideas to $k$-cycle finding with $Q(k ext{-CYCLE})= ilde{ heta}(m^{3/4-1/(2^{k+2}-4)})$; they also connect Triangle to classical problems like $3 ext{-DIST}$ and $3 ext{-SUM}$, showing a rich interplay between input hiding, input shuffling, and cycle structure. The techniques pave the way for lifting lower bounds to broader problems in quantum query complexity and offer a unified perspective on graph problems in minimalist input models with potential implications for average-case and cryptographic settings.

Abstract

The edge list model is arguably the simplest input model for graphs, where the graph is specified by a list of its edges. In this model, we study the quantum query complexity of three variants of the triangle finding problem. The first asks whether there exists a triangle containing a target edge and raises general questions about the hiding of a problem's input among irrelevant data. The second asks whether there exists a triangle containing a target vertex and raises general questions about the shuffling of a problem's input. The third asks whether there exists a triangle; this problem bridges the $3$-distinctness and $3$-sum problems, which have been extensively studied by both cryptographers and complexity theorists. We provide tight or nearly tight results for these problems as well as some first answers to the general questions they raise. Furthermore, given any graph with low maximum degree, such as a typical random sparse graph, we prove that the quantum query complexity of finding a length-$k$ cycle in its length-$m$ edge list is $m^{3/4-1/(2^{k+2}-4)\pm o(1)}$, which matches the best-known upper bound for the quantum query complexity of $k$-distinctness on length-$m$ inputs up to an $m^{o(1)}$ factor. We prove the lower bound by developing new techniques within Zhandry's recording query framework [CRYPTO '19] as generalized by Hamoudi and Magniez [ToCT '23]. These techniques extend the framework to treat any non-product distribution that results from conditioning a product distribution on the absence of rare events. We prove the upper bound by adapting Belovs's learning graph algorithm for $k$-distinctness [FOCS '12]. Finally, assuming a plausible conjecture concerning only cycle finding, we show that the lower bound can be lifted to an essentially tight lower bound on the quantum query complexity of $k$-distinctness, which is a long-standing open question.

Hiding, Shuffling, and Cycle Finding: Quantum Algorithms on Edge Lists

TL;DR

This work analyzes triangle and cycle problems on the edge-list input model under quantum query complexity. It introduces hiding and shuffling transforms to study how input organization affects complexity, and develops Mirroring and Exclusion lemmas to handle rare events within the recording query framework. The authors obtain tight or near-tight bounds: Triangle finding on sparse graphs achieves with matching upper bounds via Belovs's learning graph, and extend these ideas to -cycle finding with ; they also connect Triangle to classical problems like and , showing a rich interplay between input hiding, input shuffling, and cycle structure. The techniques pave the way for lifting lower bounds to broader problems in quantum query complexity and offer a unified perspective on graph problems in minimalist input models with potential implications for average-case and cryptographic settings.

Abstract

The edge list model is arguably the simplest input model for graphs, where the graph is specified by a list of its edges. In this model, we study the quantum query complexity of three variants of the triangle finding problem. The first asks whether there exists a triangle containing a target edge and raises general questions about the hiding of a problem's input among irrelevant data. The second asks whether there exists a triangle containing a target vertex and raises general questions about the shuffling of a problem's input. The third asks whether there exists a triangle; this problem bridges the -distinctness and -sum problems, which have been extensively studied by both cryptographers and complexity theorists. We provide tight or nearly tight results for these problems as well as some first answers to the general questions they raise. Furthermore, given any graph with low maximum degree, such as a typical random sparse graph, we prove that the quantum query complexity of finding a length- cycle in its length- edge list is , which matches the best-known upper bound for the quantum query complexity of -distinctness on length- inputs up to an factor. We prove the lower bound by developing new techniques within Zhandry's recording query framework [CRYPTO '19] as generalized by Hamoudi and Magniez [ToCT '23]. These techniques extend the framework to treat any non-product distribution that results from conditioning a product distribution on the absence of rare events. We prove the upper bound by adapting Belovs's learning graph algorithm for -distinctness [FOCS '12]. Finally, assuming a plausible conjecture concerning only cycle finding, we show that the lower bound can be lifted to an essentially tight lower bound on the quantum query complexity of -distinctness, which is a long-standing open question.

Paper Structure

This paper contains 20 sections, 12 theorems, 21 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Our results
  3. $\mathop{\mathrm{\mathsf{TriangleEdge}}}\nolimits$ and Hiding.
  4. $\mathop{\mathrm{\mathsf{TriangleVertex}}}\nolimits$ and Shuffling.
  5. Triangle Finding.
  6. Cycle Finding.
  7. Technical overview
  8. $\mathop{\mathrm{\mathsf{TriangleEdge}}}\nolimits$ and Hiding.
  9. $\mathop{\mathrm{\mathsf{TriangleVertex}}}\nolimits$ and Shuffling.
  10. Triangle Finding (lower bound).
  11. Triangle Finding (upper bound).
  12. Cycle Finding.
  13. Preliminaries
  14. Notation.
  15. Function definitions.
  16. ...and 5 more sections

Key Result

Theorem 1.2

$Q(\mathop{\mathrm{\mathsf{HIDE}}}\nolimits_m[\mathop{\mathrm{\mathrm{ED}}}\nolimits_d]) = \widetilde{\Theta}(\sqrt{m}d^{1/6})$. Consequently, where the input is a length-$m$ edge list and the target edge has $d$ neighboring edges.

Theorems & Definitions (26)

  • Definition 1.1: Hiding transform
  • Theorem 1.2: Informal version of \ref{['prop:hided_ED', 'prop:triangle_edge']}
  • Definition 1.3: Shuffling transform
  • Theorem 1.4: Informal version of \ref{['thm:shuffling-adj-list', 'prop:shuffling_adj_matrix']}
  • Theorem 1.4: Informal version of \ref{['thm:shuffling-adj-list', 'prop:shuffling_adj_matrix']}
  • Definition 1.5: Shuffled direct sum
  • Theorem 1.6: Informal version of \ref{['prop:triangle_search', 'cor:worst-case-search']}
  • Theorem 1.7: Informal version of \ref{['thm:worst-case-decision', 'cor:worst-case-decision']}
  • Theorem 1.8: Informal version of \ref{['cor:kcycle_lower_bound', 'cor:worst-case-decision-k']}
  • Definition 1.9
  • ...and 16 more