# Quantum algorithms for subset finding

@article{Childs2005QuantumAF,
title={Quantum algorithms for subset finding},
author={Andrew M. Childs and Jason M. Eisenberg},
journal={Quantum Inf. Comput.},
year={2005},
volume={5},
pages={593-604}
}
• Published 6 November 2003
• Computer Science
• Quantum Inf. Comput.
Recently, Ambainis gave an O(N2/3)-query discrete-time quantum walk algorithm forthe element distinctness problem, and more generally, an O(NL/(L+1))-query algorithmfor finding L equal numbers. We review this algorithm and give a simplified and tightenedanalysis of its query complexity using techniques previously applied to the analysis ofcontinuous-time quantum walk. We also briefly discuss applications of the algorithm andpose two open problenm regarding continuous-time quantum walk and lower…
75 Citations
Quantum walk algorithm for element distinctness
• A. Ambainis
• Computer Science
45th Annual IEEE Symposium on Foundations of Computer Science
• 2004
An O(N/sup k/(k+1)/) query quantum algorithm is given for the generalization of element distinctness in which the authors have to find k equal items among N items.
An Improved Claw Finding Algorithm Using Quantum Walk
• S. Tani
• Mathematics, Computer Science
MFCS
• 2007
A quantum-walk-based algorithm is described that can be generalized to find a claw of k functions for any constant integer k > 1, where the domains of the functions may have different size.
Learning-Graph-Based Quantum Algorithm for k-Distinctness
• Aleksandrs Belovs
• Computer Science
2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
• 2012
A quantum algorithm solving the k-distinctness problem in a less number of queries than the previous algorithm by Ambainis is presented, and the complexity analysis is much simpler.
Applications of Adversary Method in Quantum Query Algorithms
A recently developed tight characterisation of quantum query complexity, the adversary bound, is used to develop new quantum algorithms and lower bounds, and a generalisation ofquantum walks that connects electrical properties of a graph and its quantum hitting time is developed.
Promised and Distributed Quantum Search
A tight bound of the quantum query complexity for the Claw Finding problem is shown, improving previous upper and lower bounds by Buhrman, Durr, Heiligman, Hoyer, Magniez, Santha and de Wolf.
Claw finding algorithms using quantum walk
• S. Tani
• Computer Science, Mathematics
Theor. Comput. Sci.
• 2009
A Quantum Approach to Subset-Sum and Similar Problems
This paper studies the subset-sum problem by using a quantum heuristic approach similar to the verification circuit of quantum Arthur-Merlin games and shows that the exact solution of the subset sum problem my be obtained in polynomial time and the exponential speed up over the classical algorithms may be possible.
Quantum Walks and Electric Networks
We prove that a quantum walk can detect the presence of a marked element in a graph in $O(\sqrt{WR})$ steps for any initial probability distribution on vertices. Here, $W$ is the total weight of the
Improved Quantum Information Set Decoding
This paper gives an alternative view on the quantum walk based algorithms proposed by Kachigar and Tillich (PQCrypto'17) and translates May-Ozerov Near Neighbour technique (Eurocrypt'15) to an update-and-query' language more suitable for the quantumWalk framework.
Quantum Algorithm for Commutativity Testing of a Matrix Set
It turns out Szegedy's algorithm can be generalized to solve similar problems and is used to analyze the problem of matrix set commutativity, which is probably the first problem to be studied on the quantum query complexity using quantum walks that involves more than one parameter.

## References

SHOWING 1-10 OF 20 REFERENCES
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.
A quantum lower bound for the collision problem
We extend Shi's 2002 quantum lower bound for collision in $r$-to-one functions with $n$ inputs. Shi's bound of $\Omega((n/r)^{1/3})$ is tight, but his proof applies only in the case where the range
Quantum Lower Bound for the Collision Problem with Small Range
• S. Kutin
• Mathematics, Computer Science
Theory Comput.
• 2005
A modified version of Aaronson and Shi's quantum lower bound for the r-to-one collision problem that removes a restriction that applies only when the range has size at least 3n/2.
On the Quantum Query Complexity of Detecting Triangles in Graphs
In the quantum query model the complexity of detecting a triangle in an undirected graph on nodes can be done using O(n^{1+{3\over 7}}\log^{2}n) quantum queries using the same complexity bound for outputting the triangle if there is any.
Spatial search by quantum walk
• Computer Science
• 2004
This work considers an alternative search algorithm based on a continuous-time quantum walk on a graph and shows that full {radical}(N) speedup can be achieved on a d-dimensional periodic lattice for d>4.
Exponential algorithmic speedup by a quantum walk
• 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.
Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range
• A. Ambainis
• Computer Science, Mathematics
Theory Comput.
• 2005
A general method for proving quantum lower bounds for problems with small range is given, and a better lower bound is obtained on the polynomial degree of the two-level AND-OR tree.
Quantum Mechanics Helps in Searching for a Needle in a Haystack
Quantum mechanics can speed up a range of search applications over unsorted data. For example, imagine a phone directory containing $N$ names arranged in completely random order. To find someone's
Quantum computation and decision trees
• Computer Science, Mathematics
• 1998
This work devise a quantum-mechanical algorithm that evolves a state, initially localized at the root, through the tree, and proves that if the classical strategy succeeds in reaching level $n$ in time polynomial in $n,$ then so does the quantum algorithm.
An $O(n^{1.3})$ Quantum Algorithm for the Triangle Problem
• Computer Science
• 2003
A new quantum algorithm is presented that either finds a triangle in an undirected graph G on n nodes, or it outputs `reject'' if $G$ is triangle free, and it is based on a new design concept of Ambainis that incorporates the benefits of quantum walks into Grover search.