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Quantum algorithms for subset finding

Andrew M. Childs, Eisenberg, J M

Published Nov 6, 2003• 53 citations
OpenAlex

Abstract

Recently, Ambainis gave an O(N^(2/3))-query quantum walk algorithm for element distinctness, and more generally, an O(N^(L/(L+1)))-query algorithm for finding L equal numbers. We point out that this algorithm actually solves a much more general problem, the problem of finding a subset of size L that satisfies any given property. We review the algorithm and give a considerably simplified analysis of its query complexity. We present several applications, including two algorithms for the problem of finding an L-clique in an N-vertex graph. One of these algorithms uses O(N^(2L/(L+1))) edge queries, and the other uses \\tilde{O}(N^((5L-2)/(2L+4))), which is an improvement for L <= 5. The latter algorithm generalizes a recent result of Magniez, Santha, and Szegedy, who considered the case L=3 (finding a triangle). We also pose two open problems regarding continuous time quantum walk and lower bounds.

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Symposium on Discrete Algorithms•Jan 23, 2005•83 citations
Recently, Ambainis gave an O(N^(2/3))-query quantum walk algorithm for element distinctness, and more generally, an O(N^(L/(L+1)))-query algorithm for finding L equal numbers. We point out that this algorithm actually solves a much more general problem, the problem of finding a subset of size L that satisfies any given property. We review the algorithm and give a considerably simplified analysis of its query complexity. We present several applications, including two algorithms for the problem of finding an L-clique in an N-vertex graph. One of these algorithms uses O(N^(2L/(L+1))) edge queries, and the other uses \tilde{O}(N^((5L-2)/(2L+4))), which is an improvement for L <= 5. The latter algorithm generalizes a recent result of Magniez, Santha, and Szegedy, who considered the case L=3 (finding a triangle). We also pose two open problems regarding continuous time quantum walk and lower bounds.
Recently, Ambainis gave an O(N2/3)O(N^{2/3})O(N2/3)-query discrete-time quantum walk algorithm for the element distinctness problem, and more generally, an O(NL/(L+1))O(N^{L/(L+1)})O(NL/(L+1))-query algorithm for finding LLL equal numbers. We review this algorithm and give a simplified and tightened analysis of its query complexity using techniques previously applied to the analysis of continuous-time quantum walk. We also briefly discuss applications of the algorithm and pose two open problems regarding continuous-time quantum walk and lower bounds.
An O(n1.3)O(n^{1.3})O(n1.3) Quantum Algorithm for the Triangle Problem
We present a new quantum algorithm that either finds a triangle (a copy of K3) in an undirected graph G on n nodes, or it outputs “reject ” if G is triangle free. The algorithm uses O(n 1.3) queries, and it is based on a new design concept of Ambainis [Amb03] that incorporates the benefits of quantum walks into Grover search [Gro96]. The algorithm both improves on, and is simpler than a recent algorithm of Szegedy [Sze03] which has Õ(n10/7) query complexity. The Triangle Problem was first treated in [BDH + 01], where an algorithm with O ( √ n|E|) query complexity was presented (here |E | is the number of edges of G). 1
We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N^{2/3}) query quantum algorithm. This improves the previous O(N^{3/4}) query quantum algorithm of Buhrman et.al. (quant-ph/0007016) and matches the lower bound by Shi (quant-ph/0112086). The algorithm also solves the generalization of element distinctness in which we have to find k equal items among N items. For this problem, we get an O(N^{k/(k+1)}) query quantum algorithm.
Suppose we have k matrices of size n by n. We are given an oracle that knows all the entries of k matrices, that is, we can query the oracle an (i,j) entry of the l-th matrix. The goal is to test if each pair of k matrices commute with each other or not with as few queries to the oracle as possible. In order to solve this problem, we use a theorem of Mario Szegedy (quant-ph/0401053) that relates a hitting time of a classical random walk to that of a quantum walk. We also take a look at another method of quantum walk by Andris Ambainis (quant-ph/0311001). We apply both walks into triangle finding problem (quant-ph/0310134) and matrix verification problem (quant-ph/0409035) to compare the powers of the two different walks. We also present Ambainis's method of lower bounding technique (quant-ph/0002066) to obtain a lower bound for this problem. It turns out Szegedy's algorithm can be generalized to solve similar problems. Therefore we use Szegedy's theorem to analyze the problem of matrix set commutativity. We give an O(k^{4/5}n^{9/5}) algorithm as well as a lower bound of Omega(k^{1/2}n). We generalize the technique used in coming up with the upper bound to solve a broader range of similar problems. This is probably the first problem to be studied on the quantum query complexity using quantum walks that involves more than one parameter, here, k and n.
The quantum walk is a powerful tool to develop quantum algorithms, which usually are based on searching for a vertex in a graph with multiple marked vertices, with Ambainis's quantum algorithm for solving the element distinctness problem being the most shining example. In this work, we address the problem of calculating analytical expressions of the time complexity of finding a marked vertex using quantum-walk-based search algorithms with multiple marked vertices on arbitrary graphs, extending previous analytical methods based on Szegedy's quantum walk, which can be applied only to bipartite graphs. Two examples based on the coined quantum walk on two-dimensional lattices and hypercubes show the details of our method.
We present two new quantum algorithms that either find a triangle (a copy of K3) in an undirected graph G on n nodes, or reject if G is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes Õ(n10/7) queries. The second algorithm uses Õ(n13/10) queries, and it is based on a design concept of Ambainis [6] that incorporates the benefits of quantum walks into Grover search [18]. The first algorithm uses only O(log n) qubits in its quantum subroutines, whereas the second one uses O(n) qubits. The Triangle Problem was first treated in [12], where an algorithm with O(n + √ nm) query complexity was presented, where m is the number of edges of G.