# Quantum Query Complexity of Some Graph Problems

@article{Drr2006QuantumQC, title={Quantum Query Complexity of Some Graph Problems}, author={C. D{\"u}rr and M. Heiligman and P. H{\o}yer and M. Mhalla}, journal={SIAM J. Comput.}, year={2006}, volume={35}, pages={1310-1328} }

Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example, we show that the query complexity of Minimum Spanning Tree is in $\Theta(n^{3/2})$ in the matrix model and in $\Theta(\sqrt{nm})$ in the array model, while the… Expand

#### Topics from this paper

#### 36 Citations

Quantum Algorithms for Matching and Network Flows

- Mathematics, Computer Science
- STACS
- 2006

We present quantum algorithms for some graph problems: finding a maximal bipartite matching in time $O(n\sqrt{m}logn)$, finding a maximal non-bipartite matching in time $O(n^2(\sqrt{m/n}+log n)log… Expand

Quantum Query Complexity of Minor-Closed Graph Properties

- Mathematics, Computer Science
- SIAM J. Comput.
- 2012

It is shown that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity $\Theta(n^{3/2})$, and an adversary lower bound is proved. Expand

Quantum Algorithm for Shortest Path Search in Directed Acyclic Graph

- Mathematics
- 2019

In this work, we consider a well-known “Single Source Shortest Path Search” problems for weighted directed acyclic graphs (DAGs). We suggest a quantum algorithm with time complexity $O(\sqrt {nm}… Expand

Quantum and classical query complexities of local search are polynomially related

- Mathematics, Computer Science
- STOC
- 2004

It is shown that the deterministic, randomized and quantum query complexities of the problem are polynomially related and this generalizes earlier results of Aldous and Aar and solves the main open problem in Aar. Expand

Quantum Algorithms for Connectivity and Related Problems

- Computer Science, Mathematics
- ESA
- 2018

It is shown that the negative witness size in an st-connectivity span program is exactly characterized by the capacitance of the input graph, which gives a tight analysis for algorithms based on st-Connectivity span programs on any set of inputs, and an algorithm for estimating the algebraic connectivity of a graph is given. Expand

Claw finding algorithms using quantum walk

- Computer Science, Physics
- Theor. Comput. Sci.
- 2009

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

Quantum Algorithm for Dynamic Programming Approach for DAGs. Applications for Zhegalkin Polynomial Evaluation and Some Problems on DAGs

- Mathematics, Computer Science
- UCNC
- 2019

A quantum algorithm for dynamic programming approach for problems on directed acyclic graphs (DAGs) that can solve problems that use OR, AND, NAND, MAX and MIN functions as the main transition steps and is useful for a couple of problems. Expand

Quantum SDP-Solvers: Better upper and lower bounds

- Computer Science, Physics
- Quantum
- 2020

A general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure for quantum algorithms, and some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver has to scale linearly with $mn$ when $m\approx n$, which is the same as classical. Expand

Quantum SDP-Solvers: Better Upper and Lower Bounds

- Computer Science, Mathematics
- 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
- 2017

New techniques for quantum algorithms are developed, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure, and some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver has to scale linearly with mn, which is the same as classical. Expand

Quantum Speedup Based on Classical Decision Trees

- Computer Science, Physics
- Quantum
- 2020

A simple proof of and generalization of quantum query upper bounds for non-binary input as well as output alphabets is given, which shows that topological sorting of vertices of a directed graph $\mathcal{G}$ can be done with O(n^{3/2})$ quantum queries in the adjacency matrix model. Expand

#### References

SHOWING 1-10 OF 22 REFERENCES

Lower Bounds for Fully Dynamic Connectivity Problems in Graphs

- Mathematics, Computer Science
- Algorithmica
- 1998

An amortized lower bound of $\Omega$ (log n / {k (log log n} + log b)) per edge insertion, deletion, or query operation in the cell probe model is shown, which are the first lower bounds for fully dynamic connectivity problems. Expand

Quantum Query Complexity for Some Graph Problems

- Computer Science
- SOFSEM
- 2004

It is proved in these cases that it is impossible to provide a better application of Ambainis’ technique for these problems, because some of the new lower bounds do not close the gap between the best upper and lower bounds. Expand

Bounds for small-error and zero-error quantum algorithms

- Mathematics, Computer Science
- 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)
- 1999

We present a number of results related to quantum algorithms with small error probability and quantum algorithms that are zero-error. First, we give a tight analysis of the trade-offs between the… Expand

Quantum Simulations of Classical Random Walks and Undirected Graph Connectivity

- Computer Science, Physics
- J. Comput. Syst. Sci.
- 2001

This paper shows that space-bounded quantum Turing machines can efficiently simulate a limited class of random processes-random walks on undirected graphs-without relying on measurements during the computation, and demonstrates that the Undirected graph connectivity problem for regular graphs can be solved by one-sided error quantum Turing Machines that run in logspace and require a single measurement at the end of their computations. Expand

A Quantum Algorithm for Finding the Minimum

- Mathematics, Physics
- ArXiv
- 1996

A simple quantum algorithm whicholves the minimum searching problem using O(√N) probes using the mainsubroutine of Grover’s recent quantum searching algorithm. Expand

Lower bounds for quantum computation and communication

- Mathematics
- 1999

The description of the state of an n-bit quantum system requires 2n-1 complex numbers. This exponentially large information capacity of quantum states has been exploited in recent results showing… Expand

The quantum query complexity of approximating the median and related statistics

- Mathematics, Physics
- STOC '99
- 1999

The main ingredient in the proof is a polynomial degree lower bound for real multilinear polynomials that ``approximate'' symmetric partial boolean functions, which immediately yields lower bounds for the problems of approximating the kth-smallest element, approximates the mean of a sequence of numbers, and that of approximately counting the number of ones of a boolean function. Expand

Tight bounds on quantum searching

- Computer Science, Physics
- 1996

A lower bound on the efficiency of any possible quantum database searching algorithm is provided and it is shown that Grover''s algorithm nearly comes within a factor 2 of being optimal in terms of the number of probes required in the table. Expand

Quantum lower bounds by quantum arguments

- Mathematics, Computer Science
- STOC '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. Expand

An exact quantum polynomial-time algorithm for Simon's problem

- Mathematics, Physics
- Proceedings of the Fifth Israeli Symposium on Theory of Computing and Systems
- 1997

It is shown that there is a decision problem that can be solved in exact quantum polynomial time, which would require expected exponential time on any classical bounded-error probabilistic computer if the data is supplied as a black box. Expand