# Quantum walk algorithm for element distinctness

@article{Ambainis2004QuantumWA,
title={Quantum walk algorithm for element distinctness},
author={Andris Ambainis},
journal={45th Annual IEEE Symposium on Foundations of Computer Science},
year={2004},
pages={22-31}
}
• A. Ambainis
• Published 2004
• Mathematics, Physics, Computer Science
• 45th Annual IEEE Symposium on Foundations of Computer Science
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… Expand
649 Citations

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#### References

SHOWING 1-10 OF 85 REFERENCES
Quantum algorithms for subset finding
• Mathematics, Physics
• Quantum Inf. Comput.
• 2005
This algorithm is reviewed and a simplified and tightened analysis of its query complexity is given using techniques previously applied to the analysis of continuous-time quantum walk. Expand
Quantum Algorithms for Element Distinctness
We present several applications of quantum amplitude amplification for deciding whether all elements in the image of a given function are distinct, for finding an intersection of two sorted tables,Expand
Quantum random-walk search algorithm
• Physics, Computer Science
• 2003
It will be shown that this algorithm performs an oracle search on a database of N items with $O(\sqrt{N})$ calls to the oracle, yielding a speedup similar to other quantum search algorithms. Expand
Quantum lower bounds by quantum arguments
Two new Ω(√N) lower bounds on computing AND of ORs and inverting a permutation and more uniform proofs for several known lower bounds which have been previously proven via a variety of different techniques are proved. Expand
Quantum lower bound for the collision problem
A lower bound of Ω(n1/5) is shown on the number of queries needed by a quantum computer to solve the problem of deciding whether two sets are equal or disjoint on a constant fraction of elements. Expand
Quantum cryptanalysis of hash and claw-free functions
• Mathematics, Computer Science
• SIGA
• 1997
A quantum algorithm that finds collisions in arbitrary functions after only O(3&radic;N/&tau;) expected evaluations of the function, more efficient than the best possible classical algorithm, even allowing probabilism. Expand
Quantum computation and decision trees
• Physics
• 1998
Many interesting computational problems can be reformulated in terms of decision trees. A natural classical algorithm is to then run a random walk on the tree, starting at the root, to see if theExpand
Spatial search by quantum walk
• Physics
• 2004
Grover's quantum search algorithm provides a way to speed up combinatorial search, but is not directly applicable to searching a physical database. Nevertheless, Aaronson and Ambainis showed that aExpand
Quantum speed-up of Markov chain based algorithms
• M. Szegedy
• Mathematics, Computer Science
• 45th Annual IEEE Symposium on Foundations of Computer Science
• 2004
It is shown that under certain conditions, the quantum version of the Markov chain gives rise to a quadratic speed-up, and that the quantum escape time, just like its classical version, depends on the spectral properties of the transition matrix with the marked rows and columns deleted. Expand
Exponential algorithmic speedup by a quantum walk
• Mathematics, Computer Science
• STOC '03
• 2003
A black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer is constructed and it is proved that no classical algorithm can solve the problem in subexponential time. Expand