# Quantum algorithms for the triangle problem

@inproceedings{Magniez2005QuantumAF, title={Quantum algorithms for the triangle problem}, author={F. Magniez and M. Santha and M. Szegedy}, booktitle={SODA '05}, year={2005} }

We present two new quantum algorithms that either find a triangle (a copy of <i>K</i><inf>3</inf>) in an undirected graph <i>G</i> on <i>n</i> nodes, or reject if <i>G</i> is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes <i>Õ</i>(<i>n</i><sup>10/7</sup>) queries. The second algorithm uses <i>Õ</i>(<i>n</i><sup>13/10</sup>) queries, and it is based on a new design concept of Ambainis [6] that incorporates the benefits of quantum walks into Grover search… Expand

#### Topics from this paper

#### 313 Citations

Quantum verification of matrix products

- Mathematics, Computer Science
- SODA '06
- 2006

A quantum algorithm that verifies a product of two n × n matrices over any integral domain with bounded error in worst-case time O(n) and expected time ω, where ω is the number of wrong entries is presented. Expand

Coins make quantum walks faster

- Mathematics, Computer Science
- SODA '05
- 2005

The result improves on a previous bound for quantum local search by Aaronson and Ambainis and generalizes the result to 3 and more dimensions where the walk yields the optimal performance of <i>O</i>(√N) and gives several extensions of quantum walk search algorithms and generic expressions for its performance for general graphs. Expand

Subcubic Equivalences Between Path, Matrix, and Triangle Problems

- Mathematics, Computer Science
- JACM
- 2018

Generic equivalences between matrix products over a large class of algebraic structures used in optimization, verifying a matrix product over the same structure, and corresponding triangle detection problems over the structure are shown. Expand

Quantum Algorithm for Triangle Finding in Sparse Graphs

- Mathematics, Computer Science
- Algorithmica
- 2016

It is shown that triangle finding can be solved with O(n5/4-\epsilon) queries for some constant Epsilon >0$$ϵ>0 whenever the graph has at most O (n^{2-c})$$O(n 2-c) edges for some constants c>0$$c>0. Expand

Extended Learning Graphs for Triangle Finding

- Mathematics, Computer Science
- STACS
- 2017

New quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and spare instances are presented and a framework is presented in order to easily combine and analyze them. Expand

Extended Learning Graphs for Triangle Finding

- Mathematics, Computer Science
- Algorithmica
- 2019

New quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and sparse instances are presented and a framework is presented in order to easily combine and analyze them. Expand

Quantum Algorithm for Triangle Finding in Sparse Graphs

- Mathematics, Computer Science
- ISAAC
- 2015

It is shown that triangle finding can be solved with \(O(n^{5/4-\epsilon })\) queries for some constant \(\ep silon >0\) whenever the graph has at most \(O (n^{2-c})\) edges for some constants \(c>0\). Expand

Improved Quantum Query Algorithms for Triangle Detection and Associativity Testing

- Computer Science, Mathematics
- Algorithmica
- 2015

This work gives a family of algorithms for detecting constant-sized subgraphs, which can possibly be directed and colored, and shows how this high-level language can be compiled as a learning graph and gives the resulting complexity. Expand

Improved quantum query algorithms for triangle finding and associativity testing

- Physics, Computer Science
- SODA
- 2013

It is shown that the quantum query complexity of detecting if an n-vertex graph contains a triangle is O(n9/7), and the main theorem shows how this high-level language can be compiled as a learning graph and gives the resulting complexity. Expand

Quantum Distributed Algorithm for Triangle Finding in the CONGEST Model

- Mathematics, Physics
- STACS
- 2020

A quantum distributed algorithm that solves the triangle finding problem in $\tilde O(n^{1/4})$ rounds in the CONGEST model gives another example of quantum algorithm beating the best known classical algorithms in distributed computing. Expand

#### References

SHOWING 1-10 OF 51 REFERENCES

Quantum lower bounds for the collision and the element distinctness problems

- Mathematics, Computer Science
- JACM
- 2004

These lower bounds provide evidence for the existence of cryptographic primitives that are immune to quantum cryptanalysis, and implies a quantum lower bound of Ω(<i>n</i><sup>2/3</sup>) queries for the element distinctness problem, which is to determine whether <i*n> integers are all distinct. Expand

One-dimensional quantum walks

- Mathematics, Computer Science
- STOC '01
- 2001

A quantum analog of the symmetric random walk, which the authors call the Hadamard walk, is analyzed, which has position that is nearly uniformly distributed in the range after steps, in sharp contrast to the classical random walk. Expand

Quantum lower bound for the collision problem

- Mathematics, Physics
- STOC '02
- 2002

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

On the time required to recognize properties of graphs: a problem

- Computer Science
- SIGA
- 1973

In a recent paper [i], Holt and Reingold have proved the following results: any algorithm which, given an n-node graph, detects whether or not the graph enjoys property P must in the worst case probe 0(n 2) entries of the incidence matrix. Expand

Quantum Decision Trees and Semidefinite Programming.

- Mathematics
- 2001

We reformulate the notion of quantum query complexity in terms of inequalities and equations for a set of positive matrices, which we view as a quantum analogue of a decision tree. Using the new… Expand

Quantum lower bounds by polynomials

- Mathematics, Computer Science
- JACM
- 2001

This work examines the number of queries to input variables that a quantum algorithm requires to compute Boolean functions on {0,1}N in the black-box model and gives asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Expand

On the Quantum Query Complexity of Detecting Triangles in Graphs

- Mathematics, Physics
- 2003

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

Quantum walk algorithm for element distinctness

- Mathematics, Physics
- 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. Expand

An Ω(n5/4) lower bound on the randomized complexity of graph properties

- Mathematics, Computer Science
- Comb.
- 1991

Yao's lower bound on the randomized complexity of any nontrivial monotone graph property from Ω (n log1/12n) to Ω(n5/4) is improved. Expand

Lower bounds for randomized and quantum query complexity using Kolmogorov arguments

- Mathematics
- 2004

We prove a very general lower bound technique for quantum and randomized query complexity, that is easy to prove as well as to apply. To achieve this, we introduce the use of Kolmogorov complexity to… Expand