Quantum walk algorithm for element distinctness2003 · Andris Ambainis
Primary reference for element distinctness by quantum walk; the source of this record's cost claim.
arxiv.org/abs/quant-ph/0311001 ↗Attested & literatureAlgorithmsQuantum walk
Element distinctness by quantum walk
Given oracle access to N items carrying no promise about the underlying function, find two items that are equal, or establish that all N are distinct.
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Given oracle access to N items carrying no promise about the underlying function, find two items that are equal, or establish that all N are distinct. Ambainis builds the algorithm out of a quantum walk instead of a direct search, which is what improves on the earlier O(N^(3/4))-query quantum algorithm the abstract credits to Buhrman et al. and matches the query lower bound of Shi. The same construction also solves the generalization in which k equal items must be found among N. The easier promised version is older: when f is r-to-one, Brassard, Hoyer and Tapp give an algorithm that finds a collision by using Grover's search in a novel way, and their technique also yields a claw-finding algorithm for a pair of functions and a space-time tradeoff. The Zoo groups the two under one entry because dropping the two-to-one promise is exactly what turns collision finding into element distinctness.
Circuit & simulation
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What this takes and returns
TakesNothingWhat joins here
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ReturnsNothingWhat joins here
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Where this sits
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- Walk over subsets that remember their queries Method
Takes A rule naming the neighbours of any vertex, a rule saying whether a vertex is marked, somewhere to start — one named vertex, or a distribution over them — and the size bounds the schedule is computed from: the number of vertices, the depth, the maximum degree, or the size of the subsets a vertex stands for. Which of those a method needs is a property of the walk, not of the slot. Returns A marked vertex, with the evidence it is marked already in hand from the work stored on it — or the report that the graph holds none, at a stated failure probability.
How it works
Ambainis builds the algorithm out of a quantum walk instead of a direct search, which is what improves on the earlier O(N^(3/4))-query quantum algorithm the abstract credits to Buhrman et al. and matches the query lower bound of Shi. The same construction also solves the generalization in which k equal items must be found among N. The easier promised version is older: when f is r-to-one, Brassard, Hoyer and Tapp give an algorithm that finds a collision by using Grover's search in a novel way, and their technique also yields a claw-finding algorithm for a pair of functions and a space-time tradeoff. The Zoo groups the two under one entry because dropping the two-to-one promise is exactly what turns collision finding into element distinctness. This record's speedup class, "Polynomial", is a secondary source's classification of the oracular algorithms it files this under — not a claim its primary paper makes. Not checked against the primary source yet. Reported cost: O(N^(2/3)) queries for element distinctness on N items, matching the lower bound of Shi and improving the previous O(N^(3/4))-query quantum algorithm; the same algorithm uses O(N^(k/(k+1))) queries to find k equal items among N. The Zoo gives Θ(N) classical query complexity for element distinctness, and for the two-to-one collision problem O(N^(1/3)) quantum queries against Θ(√N) classical randomized queries, with Brassard, Hoyer and Tapp stating O((N/r)^(1/3)) expected function evaluations for r-to-one functions..
Implementation
Unsupported
element-distinctness.txt
ALGORITHM: Element distinctness by quantum walk
PROBLEM: Given oracle access to N items carrying no promise about the underlying function, find two items that are equal, or establish that all N are distinct.
IDEA: Ambainis builds the algorithm out of a quantum walk instead of a direct search, which is what improves on the earlier O(N^(3/4))-query quantum algorithm the abstract credits to Buhrman et al. and matches the query lower bound of Shi. The same construction also solves the generalization in which k equal items must be found among N. The easier promised version is older: when f is r-to-one, Brassard, Hoyer and Tapp give an algorithm that finds a collision by using Grover's search in a novel way, and their technique also yields a claw-finding algorithm for a pair of functions and a space-time tradeoff. The Zoo groups the two under one entry because dropping the two-to-one promise is exactly what turns collision finding into element distinctness.
REPORTED COST: O(N^(2/3)) queries for element distinctness on N items, matching the lower bound of Shi and improving the previous O(N^(3/4))-query quantum algorithm; the same algorithm uses O(N^(k/(k+1))) queries to find k equal items among N. The Zoo gives Θ(N) classical query complexity for element distinctness, and for the two-to-one collision problem O(N^(1/3)) quantum queries against Θ(√N) classical randomized queries, with Brassard, Hoyer and Tapp stating O((N/r)^(1/3)) expected function evaluations for r-to-one functions.
BASIS: abstract of arXiv:quant-ph/0311001: "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 [link mangled to 'this http URL' on the abs page] (quant-ph/0007016) and matches the lower bound by Shi (quant-ph/0112086)", and "For this problem, we get an O(N^{k/(k+1)}) query quantum algorithm"; Zoo entry "Collision Finding and Element Distinctness": element distinctness "has Θ(N) classical query complexity", and for the collision problem "The classical randomized query complexity of this problem is Θ(√N), whereas ... a quantum computer can achieve this using O(N^{1/3}) queries"; abstract of arXiv:quant-ph/9705002: "finds collisions in arbitrary r-to-one functions after only O((N/r)^(1/3)) expected evaluations of the function".
PRIMARY SOURCE: Andris Ambainis (2003), Quantum walk algorithm for element distinctness — https://arxiv.org/abs/quant-ph/0311001
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Literature & references
Quantum Algorithm for the Collision Problem1997 · Gilles Brassard, Peter Hoyer, Alain Tapp
The promised half of the same Zoo entry. For arbitrary r-to-one functions given as a black box the authors find collisions in O((N/r)^(1/3)) expected evaluations, more efficient than any classical algorithm even with randomization, and they extend the technique to claw finding in pairs of functions and exhibit a space-time tradeoff.
arxiv.org/abs/quant-ph/9705002 ↗