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Exponential Quantum Advantage for Pathfinding in Regular Sunflower Graphs

Jianqiang Li, Yu Tong

TL;DR

This paper finds a class of graphs that allows for exponential quantum-classical separation for the pathfinding problem with the adjacency list oracle, and this class of graphs is named regular sunflower graphs, and it is proved that, with high probability, a regular sunflower graph of degree at least $7 is a mild expander graph.

Abstract

Finding problems that allow for superpolynomial quantum speedup is one of the most important tasks in quantum computation. A key challenge is identifying problem structures that can only be exploited by quantum mechanics. In this paper, we find a class of graphs that allows for exponential quantum-classical separation for the pathfinding problem with the adjacency list oracle, and this class of graphs is named regular sunflower graphs. We prove that, with high probability, a regular sunflower graph of degree at least $7$ is a mild expander graph, that is, the spectral gap of the graph Laplacian is at least inverse polylogarithmic in the graph size. We provide an efficient quantum algorithm to find an $s$-$t$ path in the regular sunflower graph while any classical algorithm takes exponential time. This quantum advantage is achieved by efficiently preparing a $0$-eigenstate of the adjacency matrix of the regular sunflower graph as a quantum superposition state over the vertices, and this quantum state contains enough information to help us efficiently find an $s$-$t$ path in the regular sunflower graph. Because the security of an isogeny-based cryptosystem depends on the hardness of finding an $s$-$t$ path in an expander graph \cite{Charles2009}, a quantum speedup of the pathfinding problem on an expander graph is of significance. Our result represents a step towards this goal as the first provable exponential speedup for pathfinding in a mild expander graph.

Exponential Quantum Advantage for Pathfinding in Regular Sunflower Graphs

TL;DR

This paper finds a class of graphs that allows for exponential quantum-classical separation for the pathfinding problem with the adjacency list oracle, and this class of graphs is named regular sunflower graphs, and it is proved that, with high probability, a regular sunflower graph of degree at least $7 is a mild expander graph.

Abstract

Finding problems that allow for superpolynomial quantum speedup is one of the most important tasks in quantum computation. A key challenge is identifying problem structures that can only be exploited by quantum mechanics. In this paper, we find a class of graphs that allows for exponential quantum-classical separation for the pathfinding problem with the adjacency list oracle, and this class of graphs is named regular sunflower graphs. We prove that, with high probability, a regular sunflower graph of degree at least is a mild expander graph, that is, the spectral gap of the graph Laplacian is at least inverse polylogarithmic in the graph size. We provide an efficient quantum algorithm to find an - path in the regular sunflower graph while any classical algorithm takes exponential time. This quantum advantage is achieved by efficiently preparing a -eigenstate of the adjacency matrix of the regular sunflower graph as a quantum superposition state over the vertices, and this quantum state contains enough information to help us efficiently find an - path in the regular sunflower graph. Because the security of an isogeny-based cryptosystem depends on the hardness of finding an - path in an expander graph \cite{Charles2009}, a quantum speedup of the pathfinding problem on an expander graph is of significance. Our result represents a step towards this goal as the first provable exponential speedup for pathfinding in a mild expander graph.
Paper Structure (24 sections, 34 theorems, 130 equations, 3 figures, 1 algorithm)

This paper contains 24 sections, 34 theorems, 130 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Summary of the Main Results
  3. Overview of the algorithm
  4. From adjacency matrix $A$ to effective Hamiltonian $H$
  5. Prepare a $0$-eigenstate of the adjacency matrix $A$ as a quantum state
  6. Pathfinding in regular sunflower graph and $0$-eigenstate of its adjacency matrix
  7. Future outlook
  8. Organization:
  9. Background
  10. Graph expansion
  11. The oracle models
  12. The 0-eigenvector of the adjacency matrix
  13. The regular sunflower graphs $\mathcal{G}$
  14. Graph definition
  15. Expansion properties
  16. ...and 9 more sections

Key Result

Lemma 1

The 0-eigenstate $\ket{\eta^{\mathrm{odd}}}$ of the effective Hamiltonian $H$ (and also of the adjacency matrix $A$) defined in cor:spectral_properties can be prepared to $1/\operatorname{poly}(n)$ precision with $\operatorname{poly}(n)$ queries to the adjacency list oracle.

Figures (3)

  • Figure 1: An example of the regular sunflower graph $\mathcal{G}$ with $d=3, m=5, n=8$. The $s$ and $t$ vertices are marked out. The tree within the dashed rectangle is the subtree $\mathcal{T}_i$ (in this instance $i=3$). The leaves of the trees $\mathcal{T}_i$ are connected via $(d-1)/2$ random perfect matchings. We note that we only prove the expansion property for $d\geq 7$, and $d=3$ is chosen here for visual clarity.
  • Figure 2: Exponential Speedup Landscape of Pathfinding Problem
  • Figure 3: The supergraph consisting of the supervertices defined in \ref{['defn:supervertex']}. Each pair of supervertices are linked by an edge if there exists an edge in the sunflower graph $\mathcal{G}$ between two vertices contained in these two supervertices respectively.

Theorems & Definitions (76)

  • Lemma : 0-eigenstate preparation (informal)
  • Theorem : Finding the $s$-$t$ path (informal)
  • Definition 2.1: Neighborhood in graph
  • Definition 2.2: Edge boundary
  • Definition 2.3: $(K,\epsilon)$-vertex expander graph
  • Definition 2.4: Mild expander graph
  • Definition 2.5: The adjacency list oracle
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8: Block encoding gilyen2018QSingValTransf
  • ...and 66 more