# An Efficient Quantum Algorithm for the Hidden Subgroup Problem in Extraspecial Groups

@inproceedings{Ivanyos2007AnEQ, title={An Efficient Quantum Algorithm for the Hidden Subgroup Problem in Extraspecial Groups}, author={G{\'a}bor Ivanyos and Luc Sanselme and Miklos Santha}, booktitle={STACS}, year={2007} }

Extraspecial groups form a remarkable subclass of p-groups. They are also present in quantum information theory, in particular in quantum error correction. We give here a polynomial time quantum algorithm for finding hidden subgroups in extraspecial groups. Our approach is quite different from the recent algorithms presented in [17] and [2] for the Heisenberg group, the extraspecial p-group of size p3 and exponent p. Exploiting certain nice automorphisms of the extraspecial groups we define…

## 26 Citations

An Efficient Quantum Algorithm for the Hidden Subgroup Problem in Nil-2 Groups

- MathematicsAlgorithmica
- 2010

It is shown that the hidden subgroup problem in nil-2 groups, that is in groups of nilpotency class at most 2, can be solved efficiently by a quantum procedure, and the existence of a solution is guaranteed by the Chevalley-Warning theorem.

An Efficient Quantum Algorithm for the Hidden Subgroup Problem over Weyl-Heisenberg Groups

- Computer ScienceMMICS
- 2008

This work shows that the HSP over Weyl-Heisenberg groups can be solved efficiently on a quantum computer and uses Clebsch-Gordan decompositions to combine and reduce tensor products of irreducible representations.

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The hidden subgroup problem is a fundamental problem in quantum computation. It has many interesting instances for which we do not yet have an efficient classical algorithm and want to find or have…

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A aquantum algorithm is presented that correctly identifies hidden m-variate polynomials for all but a finitenumber of values of d with constant probability and that has a running time that is onlypolylogarithmic in d.

The Hidden Subgroup Problem for Universal Algebras

- Mathematics, Computer ScienceLICS
- 2020

This work identifies a class of algebras for which the generalized HSP exhibits super-polynomial speedup on a quantum computer compared to a classical one and proves a complete classification of every such power as quantum tractable, quantum intractable, or classicallyintractable.

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- Computer Science, MathematicsQuantum Inf. Comput.
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The new approach yields an efficient quantum algorithm for the bivariate HPGP even when the input consists of several level set superpositions, a more difficult version of the problem than the one where the input is given by an oracle.

Abelian Hypergroups and Quantum Computation

- MathematicsArXiv
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A stabilizer formalism using abelian hypergroups and an associated classical simulation theorem is developed and a new, hypergroup-based algorithm for the HNSP on nilpotent groups is developed.

Efficient Quantum Algorithm for the Hidden Parabola Problem

- Mathematics, Computer Science
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The success probability and the implementation of this measurement to certain classical problems involving polynomial equations are related and an efficient algorithm for hidden parabola is presented by establishing that the success probability of the measurement is lower bounded by a constant and that it can be implemented efficiently.

The Hidden Subgroup Problem

- Mathematics, Computer Science
- 2010

An overview of the Hidden Subgroup Problem (HSP) as of July 2010, including new results discovered since the survey of arXiv:quant-ph/0411037v1, and analyse Regev's algorithm for the poly(n)-uniqueSVP, proving how the degree of the polynomial is related to the oracle complexity used and suggesting several variants.

The computational power of normalizer circuits over black-box groups

- Computer Science, MathematicsArXiv
- 2014

This work presents a precise connection between Clifford circuits, Shor's factoring algorithm and several other famous quantum algorithms with exponential quantum speed-ups for solving Abelian hidden subgroup problems, and finds that normalizer circuits are powerful enough to factorize and solve classically-hard problems in the black-box setting.

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