On total functions, existence theorems and computational complexity
Theoretical Computer Science
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
Quantum mechanical algorithms for the nonabelian hidden subgroup problem
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Quantum computation and quantum information
Quantum computation and quantum information
Orbit-stabilizer problems and computing normalizers for polycyclic groups
Journal of Symbolic Computation
Hidden translation and orbit coset in quantum computing
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
The power of basis selection in fourier sampling: hidden subgroup problems in affine groups
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Quantum Computation and Lattice Problems
SIAM Journal on Computing
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
On the impossibility of a quantum sieve algorithm for graph isomorphism
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Computing in solvable matrix groups
SFCS '92 Proceedings of the 33rd Annual Symposium on Foundations of Computer Science
Las Vegas algorithms for matrix groups
SFCS '93 Proceedings of the 1993 IEEE 34th Annual Foundations of Computer Science
An efficient quantum algorithm for the hidden subgroup problem in extraspecial groups
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
Finding conjugate stabilizer subgroups in PSL(2; q) and related groups
Quantum Information & Computation
Solutions to the hidden subgroup problem on some metacyclic groups
TQC'09 Proceedings of the 4th international conference on Theory of Quantum Computation, Communication, and Cryptography
Finding hidden Borel subgroups of the general linear group
Quantum Information & Computation
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In this paper we extend the algorithm for extraspecial groups in [12], and show 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. The algorithm presented here has several additional features. It contains a powerful classical reduction for the hidden subgroup problem in nilpotent groups of constant nilpotency class to the specific case where the group is a p-group of exponent p and the subgroup is either trivial or cyclic. This reduction might also be useful for dealing with groups of higher nilpotency class. The quantum part of the algorithm uses well chosen group actions based on some automorphisms of nil-2 groups. The right choice of the actions requires the solution of a system of quadratic and linear equations. The existence of a solution is guaranteed by the Chevalley-Warning theorem, and we prove that it can also be found efficiently.