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… 
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References

SHOWING 1-10 OF 23 REFERENCES
Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem
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
The Hidden Subgroup Problem and Quantum Computation Using Group Representations
TLDR
A natural generalization of the algorithm for the abelian case to the nonabelian case is analyzed and it is shown that the algorithm determines the normal core of a hidden subgroup: in particular, normal subgroups can be determined.
Normal subgroup reconstruction and quantum computation using group representations
TLDR
It is shown that an immediate generalization of the Abelian case solution to the non-Abelian case does not efficiently solve Graph Isomorphism.
From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups
TLDR
The results show that entangled measurements across multiple copies of hidden subgroup states can be useful for efficiently solving the nonabelian HSP.
On the Power of Random Bases in Fourier Sampling: Hidden Subgroup Problem in the Heisenberg Group
TLDR
It is shown that the random strong method can be quite powerful under certain conditions on the group G and that the HSP can be solved using polynomially many random strong Fourier samplings followed by a possibly exponential classical post-processing without further queries.
Fast Quantum Fourier Transforms for a Class of Non-Abelian Groups
An algorithm is presented allowing the construction of fast Fourier transforms for any solvable group on a classical computer. The special structure of the recursion formula being the core of this
Quantum measurements and the Abelian Stabilizer Problem
  • A. Kitaev
  • Mathematics
    Electron. Colloquium Comput. Complex.
  • 1996
TLDR
A polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm is presented, based on a procedure for measuring an eigenvalue of a unitary operator.
Quantum error correction via codes over GF(4)
TLDR
In the present paper the problem of finding quantum-error-correcting codes is transformed into one of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product.
The power of basis selection in fourier sampling: hidden subgroup problems in affine groups
TLDR
It is shown that hidden subgroups of semidirect products of the form ℤ<inf><i>q</i></inf> ×ℤ <i>p</i</i>, can be efficiently determined by the strong standard method and a simple closure property is proved for the class of groups over which the hidden subgroup problem can be solved efficiently.
Quantum mechanical algorithms for the nonabelian hidden subgroup problem
We provide positive and negative results concerning the “standard method” of identifying a hidden subgroup of a nonabelian group using a quantum computer.
...
...