Quantum search algorithms

  title={Quantum search algorithms},
  author={Andris Ambainis},
  journal={SIGACT News},
We review some of quantum algorithms for search problems: Grover's search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks. 


An overview of quantum algorithms of some graph problems and the rate of speedup compared to the corresponding classical algorithms is discussed.

Grover's Algorithm with Errors

Grover’s algorithm is a quantum search algorithm solving the unstructured search problem of size n in \(O(\sqrt{n})\) queries, while any classical algorithm needs O(n) queries [3].

Impossibility of a Quantum Speed-Up with a Faulty Oracle

We consider Grover's unstructured search problem in the setting where each oracle call has some small probability of failing. We show that no quantum speed-up is possible in this case.

Quantum algorithms for fixed points and invariant subgroups

A quantum algorithm is applied to solve problems concerning fixed points and invariant subgroups of automorphisms by computes the intersection of multiple unsorted multisets whose elements originate from the same set.

Quantum Walk Based Search Algorithms

An intuitive treatment of the discrete time quantization of classical Markov chains and how quantum walks can be applied to the following search problems: Element Distinctness, Matrix Product Verification, Restricted Range Associativity, Triangle, and Group Commutativity.

Quantum Complexity Bounds of Independent Set Problems

This work gives quantum algorithms for computing a maximal and a maximum independent set in a graph and improves the best classical complexity bounds for the corresponding problems.

Quantum Algorithms for Graph Problems A Survey

In this survey we give an overview about important methods to construct quantum algorithms and quantum lower bounds for graph problems. We show how to use these methods, and we give a summary about

Efficient distributed quantum computing

  • R. BealsS. Brierley M. Stather
  • Computer Science
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2013
A parallel quantum search algorithm is presented that can be used by algorithm designers without worrying whether the underlying architecture supports the connectivity of the circuit and improves the time–space trade-off for the element distinctness and collision finding problems.

Quantum walks and ground state problems

Since the appearance of Shor's factoring algorithm in 1994, the search for novel quantum computer algorithms has proved surprisingly difficult. Two design approaches that have yielded some progress

Computational speedups using small quantum devices

A hybrid quantum-classical algorithm is presented to solve 3-satisfiability problems involving n≫M variables that significantly speeds up its fully classical counterpart.



Quantum Counting

This work generalizes the Grover iteration in the light of a concept called amplitude amplification, and shows that the quadratic speedup obtained by the quantum searching algorithm over classical brute force can still be obtained for a large family of search problems for which good classical heuristics exist.


An overview of quantum walks is given, with emphasis on their algorithmic applications, which are quantum counterparts of Markov chains.

A framework for fast quantum mechanical algorithms

The sqrt(N) step quantum search algorithm is an immediate consequence of a framework for the design and analysis of quantum mechanical algorithms, and several other search-type applications are presented.

Exact quantum query complexity for total Boolean functions

We will show that if there exists a quantum query algorithm that exactly computes some total Boolean function f by making T queries, then there is a classical deterministic algorithm A that exactly

A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem

For the small examples that the authors could simulate, the quantum adiabatic algorithm worked well, providing evidence that quantum computers (if large ones can be built) may be able to outperform ordinary computers on hard sets of instances of NP-complete problems.

Quantum algorithms for element distinctness

We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Hoyer, and Tapp

Quantum algorithms for subset finding

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 random walks: An introductory overview

This article will introduce quantum random walks, review some of their properties and outline their striking differences to classical walks, introducing some of the main concepts and language of present day quantum information science in this context.

How significant are the known collision and element distinctness quantum algorithms?

The criterion that an algorithm width requires O(S) hardware to be considered significant if it produces a speedup of better than O(√S) over asimple quantum search algorithm is proposed.

Tight bounds on quantum searching

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.