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}
}
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… Expand
View PDF on arXiv
70 Citations
Highly Influential Citations
3
Background Citations
20
Methods Citations
16

70 Citations

Citation Type

Citation Type

Quantum walk algorithm for element distinctness
  • A. Ambainis
  • Mathematics, Physics
  • 45th Annual IEEE Symposium on Foundations of Computer Science
  • 2004
TLDR
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. Expand
Quantum Walk Algorithm for Element Distinctness
TLDR
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. Expand
An Improved Claw Finding Algorithm Using Quantum Walk
  • S. Tani
  • Mathematics, Computer Science
  • MFCS
  • 2007
TLDR
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. Expand
Learning-Graph-Based Quantum Algorithm for k-Distinctness
  • Aleksandrs Belovs
  • Mathematics, Physics
  • 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
  • 2012
TLDR
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. Expand
Applications of Adversary Method in Quantum Query Algorithms
In the thesis, we use a recently developed tight characterisation of quantum query complexity, the adversary bound, to develop new quantum algorithms and lower bounds. Our results are as follows: *… Expand
Promised and Distributed Quantum Search
TLDR
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. Expand
Claw finding algorithms using quantum walk
  • S. Tani
  • Computer Science, Physics
  • Theor. Comput. Sci.
  • 2009
TLDR
This paper describes an optimal algorithm that uses quantum walk to solve the claw finding problem given two functions, f and g, with domain sizes N and M, respectively, and the goal of the problem is to find x and y such that f(x)=g(y). Expand
A Quantum Approach to Subset-Sum and Similar Problems
TLDR
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. Expand
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… Expand
Improved Quantum Information Set Decoding
TLDR
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. Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 22 REFERENCES
Quantum Walk Algorithm for Element Distinctness
TLDR
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. Expand
Quantum algorithms for element distinctness
  • H. Buhrman, C. Dürr, +4 authors R. D. Wolf
  • Mathematics, Computer Science
  • Proceedings 16th Annual IEEE Conference on Computational Complexity
  • 2001
TLDR
An O(N/sup 3/4/ log N) quantum upper bound for the element distinctness problem in the comparison complexity model is implied and a lower bound of /spl Omega/(/spl radic/N) comparisons for this problem is proved. Expand
Quantum lower bound for the collision problem
TLDR
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
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… Expand
Quantum lower bounds for the collision and the element distinctness problems
  • Yaoyun Shi
  • Mathematics, Physics
  • The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
  • 2002
TLDR
It is proved that any quantum algorithm for finding a collision in an r-to-one function must evaluate the function /spl Omega/ ((n/r)/sup 1/3/) times, where n is the size of the domain and r|n. Expand
Quantum Lower Bound for the Collision Problem with Small Range
  • S. Kutin
  • Computer Science, Mathematics
  • Theory Comput.
  • 2005
TLDR
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. Expand
On the Quantum Query Complexity of Detecting Triangles in Graphs
We show that in the quantum query model the complexity of detecting a triangle in an undirected graph on $n$ nodes can be done using $O(n^{1+{3\over 7}}\log^{2}n)$ quantum queries. The same… Expand
Spatial search by quantum walk
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 a… Expand
Exponential algorithmic speedup by a quantum walk
TLDR
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
Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range
TLDR
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. Expand
...
1
2
3
...