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ACM Symposium on Theory of Computing (STOC)

Multidimensional Quantum Walks

Stacey Jeffery, Sebastian Zur

Published May 16, 2023• 7 citations
Publisher

Abstract

While the quantum query complexity of k-distinctness is known to be O(n3/4 − 1/4(2k−1)) for any constant k≥ 4 [Belovs, FOCS 2012], the best previous upper bound on the time complexity was O(n1−1/k). We give a new upper bound of O(n3/4 − 1/4(2k−1)) on the time complexity, matching the query complexity up to polylogarithmic factors. In order to achieve this upper bound, we give a new technique for designing quantum walk search algorithms, which is an extension of the electric network framework. We also show how to solve the welded trees problem in O(n) queries and O(n2) time using this new technique, showing that the new quantum walk framework can achieve exponential speedups.

DOI
10.1145/3564246.3585158

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While the quantum query complexity of k-distinctness is known to be O(n3/4 − 1/4(2k−1)) for any constant k≥ 4 [Belovs, FOCS 2012], the best previous upper bound on the time complexity was O(n1−1/k). We give a new upper bound of O(n3/4 − 1/4(2k−1)) on the time complexity, matching the query complexity up to polylogarithmic factors. In order to achieve this upper bound, we give a new technique for designing quantum walk search algorithms, which is an extension of the electric network framework. We also show how to solve the welded trees problem in O(n) queries and O(n2) time using this new technique, showing that the new quantum walk framework can achieve exponential speedups.
Multidimensional Quantum Walks, with Application to kkk-Distinctness
While the quantum query complexity of kkk-distinctness is known to be O(n3/4−1/4(2k−1))O\left(n^{3/4-1/4(2^k-1)}\right)O(n3/4−1/4(2k−1)) for any constant k≥4k \geq 4k≥4, the best previous upper bound on the time complexity was O~(n1−1/k)\widetilde{O}\left(n^{1-1/k}\right)O(n1−1/k). We give a new upper bound of O~(n3/4−1/4(2k−1))\widetilde{O}\left(n^{3/4-1/4(2^k-1)}\right)O(n3/4−1/4(2k−1)) on the time complexity, matching the query complexity up to polylogarithmic factors. In order to achieve this upper bound, we give a new technique for designing quantum walk search algorithms, which is an extension of the electric network framework. We also show how to solve the welded trees problem in O(n)O(n)O(n) queries and O(n2)O(n^2)O(n2) time using this new technique, showing that the new quantum walk framework can achieve exponential speedups. Comment: Journal version for TheoretiCS
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.
Quantum computing offers the potential to solve problems more efficiently than classical methods, yet understanding its capabilities requires new conceptual frameworks. This dissertation explores quantum algorithms with a focus on maintaining connections to classical intuition, clarifying both their strengths and limitations. In Part I, we explore the capabilities of a class of quantum algorithms known as quantum walks. These quantum walks, the quantum analogs of classical random walks, serve as powerful yet accessible tools for solving a variety of computational problems. Our key contribution is the development of a novel way to construct quantum walks, resulting in multidimensional quantum walks. By applying these walks to the kkk-distinctness problem, we achieve a time-efficient algorithm matching the best-known query upper bound up to polylogarithmic factors. Additionally, applying them to the welded tree problem results in exponential speedups, marking the first instance of a discrete quantum walk to do so. Furthermore, we extend the well-known link between quantum walks and electrical networks to the multidimensional quantum walks. In Part II, we turn to the limitations of quantum algorithms by examining techniques for establishing quantum query lower bounds, representing the minimum computational cost required for any quantum algorithm to solve a given computational problem, regardless of the quantum algorithm used. To this goal, we introduce the multiplicative ladder adversary method, a simplified version of the multiplicative adversary method which unifies the compressed oracle technique within the broader adversary framework while extending its applicability, offering a more intuitive approach to establishing quantum lower bounds.
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.
Query complexity is one of the several notions of complexity de ned to measure the cost \nof algorithms. It plays an important role in demonstrating the quantum advantage, that \nquantum computing is faster than classical computation in solving some problems. Kempe \nshowed that a discrete time quantum walk on hypercube hits the antipodal point with \nexponentially fewer queries than a simple random walk [K05]. Childs et al. showed that a \ncontinuous time quantum walk on \\Glued Trees" detects the label of a special vertex with \nexponentially fewer queries than any classical algorithm [CCD03], and the result translates \nto discrete time quantum walk by an e cient simulation. \nBuilding on these works, we examine the query complexity of variations of the hyper- \ncube and Glued Tree problems. We rst show the gap between quantum and classical query \nalgorithms for a modi ed hypercube problem is at most polynomial. We then strengthen \nthe query complexity gap for the Glued Tree label detection problem by improving a classical lower bound technique; and we prove such a lower bound is nearly tight by giving a \nclassical query algorithm whose query complexity matches the lower bound, up to a polylog \nfactor.
Inspired by the Elitzur-Vaidman bomb testing problem [19], we introduce a new query complexity model, which we call bomb query complexity B(f). We investigate its relationship with the usual quantum query complexity Q(f), and show that B(f) = Θ(Q(f)2).This result gives a new method to upper bound the quantum query complexity: we give a method of finding bomb query algorithms from classical algorithms, which then provide nonconstructive upper bounds on [EQUATION]. We subsequently were able to give explicit quantum algorithms matching our upper bound method. We apply this method on the single-source shortest paths problem on unweighted graphs, obtaining an algorithm with O(n1.5) quantum query complexity, improving the best known algorithm of [EQUATION] [21]. Applying this method to the maximum bipartite matching problem gives an O(n1.75) algorithm, improving the best known trivial O(n2) upper bound.