# Learning-Graph-Based Quantum Algorithm for k-Distinctness

@article{Belovs2012LearningGraphBasedQA, title={Learning-Graph-Based Quantum Algorithm for k-Distinctness}, author={Aleksandrs Belovs}, journal={2012 IEEE 53rd Annual Symposium on Foundations of Computer Science}, year={2012}, pages={207-216} }

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.

## 74 Citations

A Time-Efficient Quantum Walk for 3-Distinctness Using Nested Updates

- Computer Science, MathematicsArXiv
- 2013

An extension to the quantum walk search framework that facilitates quantum walks with nested updates is presented, to give a quantum walk algorithm for 3-Distinctness with query complexity ~O(n^{5/7}), matching the best known upper bound up to log factors.

Time-Efficient Quantum Walks for 3-Distinctness

- Computer Science, MathematicsICALP
- 2013

Two quantum walk algorithms for 3-Distinctness are presented, improving the previous $\tilde{O}(n^{3/4})$ and matching the best known upper bound for query complexity (obtained via learning graphs) up to log factors.

Quantum Algorithms for Finding Constant-Sized Sub-hypergraphs

- Mathematics, Computer ScienceCOCOON
- 2014

We develop a general framework to construct quantum algorithms that detect if a 3-uniform hypergraph given as input contains a sub-hypergraph isomorphic to a prespecified constant-sized hypergraph.…

Quantum algorithms for finding constant-sized sub-hypergraphs

- Mathematics, PhysicsTheor. Comput. Sci.
- 2016

We develop a general framework to construct quantum algorithms that detect if a 3-uniform hypergraph given as input contains a sub-hypergraph isomorphic to a prespecified constant-sized hypergraph.…

Quantum Walks and Electric Networks

- Physics, Mathematics
- 2013

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…

Adversary lower bound for the k-sum problem

- Mathematics, PhysicsITCS '13
- 2013

We prove a tight quantum query lower bound Omega(nk/(k+1)) for the problem of deciding whether there exist k numbers among n that sum up to a prescribed number, provided that the alphabet size is…

On the Power of Non-adaptive Learning Graphs

- Mathematics, Physics2013 IEEE Conference on Computational Complexity
- 2013

We introduce a notion of the quantum query complexity of a certificate structure. This is a formalisation of a well-known observation that many quantum query algorithms only require the knowledge of…

On the Power of Non-adaptive Learning Graphs

- Computer Sciencecomputational complexity
- 2014

A notion of the quantum query complexity of a certificate structure is introduced and there exists a function possessing the certificate structure such that a learning graph gives an optimal quantum query algorithm for it.

Recent Developments in Quantum Algorithms and Complexity

- Computer ScienceDCFS
- 2014

This work surveys several recent developments in quantum algorithms and complexity, including Reichardt's characterization of quantum query algorithms via span programs and exact quantum algorithms with superlinear advantage over the best classical algorithm.

Applications of Adversary Method in Quantum Query Algorithms

- Mathematics, Physics
- 2014

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:
*…

## References

SHOWING 1-10 OF 29 REFERENCES

Quantum algorithms for subset finding

- Mathematics, PhysicsQuantum 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.

Quantum Algorithm for k-distinctness with Prior Knowledge on the Input

- Mathematics, Physics
- 2011

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…

Adversary lower bound for the k-sum problem

- Mathematics, PhysicsITCS '13
- 2013

We prove a tight quantum query lower bound Omega(nk/(k+1)) for the problem of deciding whether there exist k numbers among n that sum up to a prescribed number, provided that the alphabet size is…

Quantum walk algorithm for element distinctness

- Mathematics, Physics45th 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.

Span programs for functions with constant-sized 1-certificates: extended abstract

- Mathematics, PhysicsSTOC '12
- 2012

The power of the approach is proved by designing a quantum algorithm for the triangle problem with query complexity O(n35/27) that is better than O( n13/10) of the best previously known algorithm by Magniez et al.

The Quantum Query Complexity of Algebraic Properties

- Computer Science, PhysicsFCT
- 2007

The first application of the new quantum random walk technique by Magniez, Nayak, Roland, and Santha that improves the previous bounds by Ambainis [3] and Szegedy [23] is given.

Improving Quantum Query Complexity of Boolean Matrix Multiplication Using Graph Collision

- Mathematics, PhysicsICALP
- 2012

The quantum query complexity of Boolean matrix multiplication is typically studied as a function of the matrix dimension, n, as well as the number of 1s in the output, l. We prove an upper bound of…

Quantum lower bounds by quantum arguments

- Mathematics, Computer ScienceSTOC '00
- 2000

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.

Span Programs and Quantum Algorithms for st-Connectivity and Claw Detection

- Mathematics, Computer ScienceESA
- 2012

An algorithm is given that uses O(n) queries to the adjacency matrix of an n-vertex graph to decide if vertices s and t are connected, under the promise that they either are connected by a path of length at most d, or are disconnected.

Quantum Query Complexity of Subgraph Containment with Constant-sized Certificates

- Mathematics, Physics
- 2011

We study the quantum query complexity of constant-sized subgraph containment. Such problems include determining whether an $ n $-vertex graph contains a triangle, clique or star of some size. For a…