Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum Information and Computation

Quantum algorithms for subset finding

Andrew M. Childs, J.M. Eisenberg

Published Nov 1, 2005• 37 citations
Publisher

Abstract

Recently, Ambainis gave an $O(N^{2/3})$-query discrete-time quantum walk algorithm for the element distinctness problem, and more generally, an $O(N^{L/(L+1)})$-query algorithm for finding $L$ 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.

DOI
10.26421/qic5.7-7

Related papers

Quantum algorithms for subset finding

Andrew M. Childs, Eisenberg, J M

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.

•Nov 6, 2003•53 citations

Quantum walk algorithm for element distinctness

Andris Ambainis

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.

arXiv:quant-ph/0311001•Nov 1, 2003•20 citations

Quantum Walk Algorithm for Element Distinctness

Andris Ambainis

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 $O(N^{3/4})$ quantum algorithm of Buhrman et al. [SIAM J. Comput., 34 (2005), pp. 1324–1330] and matches the lower bound of Aaronson and Shi [J. ACM, 51 (2004), pp. 595–605]. We also give an $O(N^{k/(k+1)})$ query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among N items.

SIAM Journal on Computing•Jan 1, 2007•704 citations

Quantum Walk Algorithm for Element Distinctness

Andris Ambainis

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/sup 2/3/) query quantum algorithm. This improves the previous O(N/sup 3/4/) quantum algorithm of Buhrman et al. and matches the lower bound by Shi. We also give an O(N/sup k/(k+1)/) query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among N items.

Annual Symposium on Foundations of Computer Science•Dec 23, 2004•215 citations

Optimal parallel quantum query algorithms

Stacey Jeffery, Frédéric Magniez, Ronald de Wolf

We study the complexity of quantum query algorithms that make p queries in parallel in each timestep. This model is in part motivated by the fact that decoherence times of qubits are typically small, so it makes sense to parallelize quantum algorithms as much as possible. We show tight bounds for a number of problems, specifically Theta((n/p)^{2/3}) p-parallel queries for element distinctness and Theta((n/p)^{k/(k+1)} for k-sum. Our upper bounds are obtained by parallelized quantum walk algorithms, and our lower bounds are based on a relatively small modification of the adversary lower bound method, combined with recent results of Belovs et al. on learning graphs. We also prove some general bounds, in particular that quantum and classical p-parallel complexity are polynomially related for all total functions f when p is small compared to f's block sensitivity.

arXiv:1309.6116•Sep 24, 2013•7 citations

Multidimensional Quantum Walks, with Application to $k$-Distinctness

Stacey Jeffery, Sebastian Zur

While the quantum query complexity of $k$-distinctness is known to be $O\left(n^{3/4-1/4(2^k-1)}\right)$ for any constant $k \geq 4$, the best previous upper bound on the time complexity was $\widetilde{O}\left(n^{1-1/k}\right)$. We give a new upper bound of $\widetilde{O}\left(n^{3/4-1/4(2^k-1)}\right)$ 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 $O(n^2)$ time using this new technique, showing that the new quantum walk framework can achieve exponential speedups. Comment: Journal version for TheoretiCS

arXiv:2208.13492•TheoretiCS•Mar 4, 2025•3 citations
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.
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.
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.
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(N2/3)O(N^{2/3})O(N2/3) query quantum algorithm. This improves the previous O(N3/4)O(N^{3/4})O(N3/4) quantum algorithm of Buhrman et al. [SIAM J. Comput., 34 (2005), pp. 1324–1330] and matches the lower bound of Aaronson and Shi [J. ACM, 51 (2004), pp. 595–605]. We also give an O(Nk/(k+1))O(N^{k/(k+1)})O(Nk/(k+1)) query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among N items.
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/sup 2/3/) query quantum algorithm. This improves the previous O(N/sup 3/4/) quantum algorithm of Buhrman et al. and matches the lower bound by Shi. We also give an O(N/sup k/(k+1)/) query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among N items.
We study the complexity of quantum query algorithms that make p queries in parallel in each timestep. This model is in part motivated by the fact that decoherence times of qubits are typically small, so it makes sense to parallelize quantum algorithms as much as possible. We show tight bounds for a number of problems, specifically Theta((n/p)^{2/3}) p-parallel queries for element distinctness and Theta((n/p)^{k/(k+1)} for k-sum. Our upper bounds are obtained by parallelized quantum walk algorithms, and our lower bounds are based on a relatively small modification of the adversary lower bound method, combined with recent results of Belovs et al. on learning graphs. We also prove some general bounds, in particular that quantum and classical p-parallel complexity are polynomially related for all total functions f when p is small compared to f's block sensitivity.
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