# Quantum algorithms for solvable groups

@inproceedings{Watrous2001QuantumAF, title={Quantum algorithms for solvable groups}, author={John Watrous}, booktitle={STOC '01}, year={2001} }

In this paper we give a polynomial-time quantum algorithm for computing orders of solvable groups. Several other problems, such as testing membership in solvable groups, testing equality of subgroups in a given solvable group, and testing normality of a subgroup in a given solvable group, reduce to computing orders of solvable groups and therefore admit polynomial-time quantum algorithms as well. Our algorithm works in the setting of black-box groups, wherein none of these problems have…

## 94 Citations

Quantum Algorithms for a Set of Group Theoretic Problems

- MathematicsInt. J. Found. Comput. Sci.
- 2015

This work introduces two decision problems, StabilizerDand Orbit CosetD, and gives quantum reductions from them to the problem Orbit Superposition (Friedl et al., 2003), as well as quantum reductions…

An Efficient Quantum Algorithm for Some Instances of the Group Isomorphism Problem

- MathematicsSTACS
- 2010

This paper presents a quantum algorithm solving instances of the nonabelian group isomorphism problem exponentially faster than the best known classical algorithms.

Interactive Proofs with Polynomial-Time Quantum Prover for Computing the Order of Solvable Groups

- Computer Science, MathematicsMFCS
- 2018

The result shows that computing the order of a solvable group, which is one of the most general problems for which quantum computing exhibits an exponential speed-up with respect to classical computing, can be realized in this model.

Efficient quantum algorithms for the hidden subgroup problem over semi-direct product groups

- Mathematics, Computer ScienceQuantum Inf. Comput.
- 2007

The hidden subgroup problem (HSP) over the class of semidirect product groups Zpr ⋊ Zq, for p and q prime is considered, and a polynomial-time quantum algorithm solving the HSP over all the groups of one of these classes is described.

Quantum Complexity of Testing Group Commutativity

- Mathematics, Computer ScienceAlgorithmica
- 2007

A quite optimal quantum algorithm is constructed for the problem of testing the commutativity of a black-box group specified by its k generators, whose complexity is in $\tilde{O}(k^{2/3})$.

Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem

- MathematicsSPAA '01
- 2001

In this paper we show that certain special cases of the hidden subgroup problem can be solved in polynomial time by a quantum algorithm. These special cases involve finding hidden normal subgroups of…

Quantum Property Testing for Solvable Groups

- Mathematics
- 2006

Property testing has been extensively studied and its target is to determine whether a given object satisfies a certain property or it is far from the property. In this paper, we construct an…

A ug 2 00 5 Quantum Complexity of Testing Group Commutativity ∗

- Mathematics, Computer Science
- 2005

A quite optimal quantum algorithm is constructed for the problem of testing the commutativity of a black-box group specified by its k generators, and it is proved the optimality of the algorithm of Pak for the randomized model.

Quantum Property Testing of Group Solvability

- Mathematics, Computer ScienceAlgorithmica
- 2009

This paper constructs an efficient quantum algorithm that tests whether (Γ,⋅) is a solvable group, or is far from any solable group, and the number of queries used by the algorithm is polylogarithmic in the size of the set Γ.

Quantum algorithms for problems in number theory, algebraic geometry, and group theory

- Computer Science
- 2012

This article will review the current state of quantum algorithms, focusing on algorithms for problems with an algebraic flavor that achieve an apparent superpolynomial speedup over classical computation.

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