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Annual Symposium on Foundations of Computer Science

Learning-Graph-Based Quantum Algorithm for k-Distinctness

Aleksandrs Belovs

Published Oct 1, 2012• 62 citations
Publisher

Abstract

We present a quantum algorithm solving the k-distinctness problem in a less number of queries than the previous algorithm by Ambainis. The construction uses a modified learning graph approach. Compared to the recent paper by Belovs and Lee, the algorithm doesn't require any prior information on the input, and the complexity analysis is much simpler.

DOI
10.1109/focs.2012.18

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We present a quantum algorithm solving the k-distinctness problem in a less number of queries than the previous algorithm by Ambainis. The construction uses a modified learning graph approach. Compared to the recent paper by Belovs and Lee, the algorithm doesn't require any prior information on the input, and the complexity analysis is much simpler.
We present a quantum algorithm solving the kkk-distinctness problem in O(n1−2k−2/(2k−1))O(n^{1-2^{k-2}/(2^k-1)})O(n1−2k−2/(2k−1)) queries with a bounded error. This improves the previous O(nk/(k+1))O(n^{k/(k+1)})O(nk/(k+1))-query algorithm by Ambainis. The construction uses a modified learning graph approach. Compared to the recent paper by Belovs and Lee arXiv:1108.3022, the algorithm doesn't require any prior information on the input, and the complexity analysis is much simpler. Additionally, we introduce an O(nα1/6)O(\sqrt{n}α^{1/6})O(n​α1/6) algorithm for the graph collision problem where ααα is the independence number of the graph.
We study the problem of learning an unknown graph provided via an oracle using a quantum algorithm. We consider three query models. In the first model (OR queries), the oracle returns whether a given subset of the vertices contains any edges. In the second (parity queries), the oracle returns the parity of the number of edges in a subset. In the third model, we are given copies of the graph state corresponding to the graph. We give quantum algorithms that achieve speedups over the best possible classical algorithms in the OR and parity query models, for some families of graphs, and give quantum algorithms in the graph state model whose complexity is similar to the parity query model. For some parameter regimes, the speedups can be exponential in the parity query model. On the other hand, without any promise on the graph, no speedup is possible in the OR query model. A main technique we use is the quantum algorithm for solving the combinatorial group testing problem, for which a query-efficient quantum algorithm was given by Belovs. Here we additionally give a time-efficient quantum algorithm for this problem, based on the algorithm of Ambainis et al. for a gapped version of the group testing problem. We also give simple time-efficient quantum algorithms based on Fourier sampling and amplitude amplification for learning the exact-half and majority functions, which almost match the optimal complexity of Belovs' algorithms.
It is known that the dual of the general adversary bound can be used to build quantum query algorithms with optimal complexity. Despite this result, not many quantum algorithms have been designed this way. This paper shows another example of such algorithm. We use the learning graph technique from arXiv:1105.4024 to give a quantum algorithm for kkk-distinctness problem that runs in o(n3/4)o(n^{3/4})o(n3/4) queries, for a fixed kkk, given some prior knowledge on the structure of the input. The best known quantum algorithm for the unconditional problem uses O(nk/(k+1))O(n^{k/(k+1)})O(nk/(k+1)) queries.
We study the problem of learning an unknown graph provided via an oracle using a quantum algorithm. We consider three query models. In the first model ("OR queries"), the oracle returns whether a given subset of the vertices contains any edges. In the second ("parity queries"), the oracle returns the parity of the number of edges in a subset. In the third model, we are given copies of the graph state corresponding to the graph. We give quantum algorithms that achieve speedups over the best possible classical algorithms in the OR and parity query models, for some families of graphs, and give quantum algorithms in the graph state model whose complexity is similar to the parity query model. For some parameter regimes, the speedups can be exponential in the parity query model. On the other hand, without any promise on the graph, no speedup is possible in the OR query model. A main technique we use is the quantum algorithm for solving the combinatorial group testing problem, for which a query-efficient quantum algorithm was given by Belovs. Here we additionally give a time-efficient quantum algorithm for this problem, based on the algorithm of Ambainis et al.\ for a "gapped" version of the group testing problem. We also give simple time-efficient quantum algorithms based on Fourier sampling and amplitude amplification for learning the exact-half and majority functions, which almost match the optimal complexity of Belovs' algorithms.
Let HHH be a fixed kkk-vertex graph with mmm edges and minimum degree d>0d >0d>0. We use the learning graph framework of Belovs to show that the bounded-error quantum query complexity of determining if an nnn-vertex graph contains HHH as a subgraph is O(n2−2/k−t)O(n^{2-2/k-t})O(n2−2/k−t), where $ t = \max{\frac{k^2- 2(m+1)}{k(k+1)(m+1)}, \frac{2k - d - 3}{k(d+1)(m-d+2)}}$. The previous best algorithm of Magniez et al. had complexity O~(n2−2/k)\widetilde O(n^{2-2/k})O(n2−2/k).
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.