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arXiv:2407.14398

Exponential Quantum Advantage for Pathfinding in Regular Sunflower Graphs

Jianqiang Li, Tong, Yu

Published Jul 19, 2024• 3 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2407.14398

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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 777 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 sss-ttt path in the regular sunflower graph while any classical algorithm takes exponential time. This quantum advantage is achieved by efficiently preparing a 000-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 sss-ttt path in the regular sunflower graph. Because the security of an isogeny-based cryptosystem depends on the hardness of finding an sss-ttt 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.
We demonstrate the possibility of (sub)exponential quantum speedup via a quantum algorithm that follows an adiabatic path of a gapped Hamiltonian with no sign problem. The Hamiltonian that exhibits this speed-up comes from the adjacency matrix of an undirected graph whose vertices are labeled by n-bit strings, and we can view the adiabatic evolution as an efficient O(poly(n))-time quantum algorithm for finding a specific “EXIT” vertex in the graph given the “ENTRANCE” vertex. On the other hand we show that if the graph is given via an adjacency-list oracle, there is no classical algorithm that finds the “EXIT” with probability greater than exp(−nδ) using at most exp(nδ) queries for δ= 1/5 − o(1). Our construction of the graph is somewhat similar to the “welded-trees” construction of Childs et al., but uses additional ideas of Hastings for achieving a spectral gap and a short adiabatic path.
Quantum computers can sometimes exponentially outperform classical ones, but only for problems with sufficient structure. While it is well known that query problems with full permutation symmetry can have at most polynomial quantum speedup -- even for partial functions -- it is unclear how far this condition must be relaxed to enable exponential speedup. In particular, it is natural to ask whether exponential speedup is possible for (partial) graph properties, in which the input describes a graph and the output can only depend on its isomorphism class. We show that the answer to this question depends strongly on the input model. In the adjacency matrix model, we prove that the bounded-error randomized query complexity RRR of any graph property P\mathcal{P}P has R(P)=O(Q(P)6)R(\mathcal{P}) = O(Q(\mathcal{P})^{6})R(P)=O(Q(P)6), where QQQ is the bounded-error quantum query complexity. This negatively resolves an open question of Montanaro and de Wolf in the adjacency matrix model. More generally, we prove R(P)=O(Q(P)3l)R(\mathcal{P}) = O(Q(\mathcal{P})^{3l})R(P)=O(Q(P)3l) for any lll-uniform hypergraph property P\mathcal{P}P in the adjacency matrix model. In direct contrast, in the adjacency list model for bounded-degree graphs, we exhibit a promise problem that shows an exponential separation between the randomized and quantum query complexities.
Establishing quantum speedup for computationally hard problems of practical relevance, particularly combinatorial optimization problems, remains a central challenge in quantum computation. In this work, we identify a structurally defined family of classically hard maximum independent set (MIS) instances, and design and analyze a non-stoquastic adiabatic quantum optimization algorithm that exploits this structure. The algorithm runs in polynomial time and achieves an exponential speedup over both transverse-field quantum annealing and state-of-the-art classical solvers on these instances, under assumptions supported by analytical and numerical evidence. We identify the essential quantum mechanism enabling the speedup as the use of a non-stoquastic XX-driver to access a larger sign-structured admissible subspace beyond the stoquastic regime, which allows sign-generating quantum interference to create smooth evolution paths that bypass tunneling. This identifies a distinctive quantum mechanism underlying the speedup and explains why no efficient classical analogue is likely to exist. In addition, our analysis produces scalable small-scale models, derived from our structural reduction, that capture the essential dynamics of the algorithm. These models provide a concrete opportunity for verification of the quantum advantage mechanism on currently available universal quantum computers.
In the welded tree problem, an algorithm is tasked with navigating a graph formed from two binary trees joined at the leaves through a ``weld'' of connecting edges. A quantum walk can navigate from root to root exponentially faster than any classical algorithm. However, known efficient quantum algorithms cannot find a path between the roots, as recording the path destroys constructive interference and thus the speedup. We prove that this is inherent: any quantum algorithm needs exponentially many queries to find a path between the roots of an independently matched welded tree graph. This provides an example of a problem that a quantum computer can solve exponentially faster than any classical algorithm by exploring exponentially many paths in superposition, but where it is provably intractable to find any such path. The proof uses compressed permutation oracles to record the progress of a quantum algorithm as it queries the graph. We show that the compressed database remains path-free up to a small error. By controlling such errors and bounding the progress of the algorithm with each compressed oracle query, we show that Ω(2n/12)Ω(2^{n/12})Ω(2n/12) queries are required to find a path in a height-nnn tree with constant success probability.
We provide a rigorous theoretical analysis of quantum algorithms for shortest path problems in weighted graphs, establishing complexity-theoretic foundations for quantum acceleration in this fundamental computational domain. Building upon recent quantum adaptations of classical graph algorithms, we develop a unified framework that characterizes when and why quantum approaches offer computational advantages for single-source and all-pairs shortest path computations. Our analysis encompasses both positive-weight and negative-weight scenarios, deriving conditional speedup theorems that depend on graph structure, weight distribution, and oracle access models. We prove that quantum amplitude amplification and quantum walks can provide polynomial to exponential speedups under specific structural assumptions, while establishing hardness results that identify problem instances where quantum advantage is provably absent. The theoretical framework introduces novel complexity measures that capture graph-theoretic properties relevant to quantum algorithm performance, including quantum diameter, entanglement capacity, and interference efficiency. We extend our analysis to dynamic graph scenarios where edge weights evolve over time, establishing bounds on quantum advantage preservation under graph modifications. This work contributes fundamental insights into quantum algorithmic graph theory, clarifying the boundaries of quantum computational advantage and providing theoretical guidance for algorithm design in network optimization, routing, and graph analytics applications.
Starting from the entrance of a welded tree, a quantum walk algorithm can find its exit vertex exponentially faster than any classical algorithm. However, it has been an open question whether any quantum algorithm is able to efficiently find a path from the entrance to the exit. We answer this by proving an exponential quantum query lower bound for finding such path. Specifically, for trees of height nnn, any quantum query algorithm requires at least Ω(2n/24)Ω(2^{n/24})Ω(2n/24) queries in order to succeed with constant probability. Our proof uses the compressed permutation oracle technique in order to construct databases which track the graph information an algorithm has learned and forgotten, and show that no efficient quantum algorithm can build an entrance-to-exit path in these records.