A subexponential-time quantum algorithm for the dihedral hidden subgroup problem

We present a quantum algorithm for the dihedral hidden subgroup problem with time and query complexity $O(\exp(C\sqrt{\log N}))$. In this problem an oracle computes a function $f$ on the dihedral group $D_N$ which is invariant under a hidden reflection in $D_N$. By contrast the classical query complexity of DHSP is $O(\sqrt{N})$. The algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group. The algorithm begins with the quantum character transform on the group, just as for other hidden subgroup problems. Then it tensors irreducible representations of $D_N$ and extracts summands to obtain target representations. Finally, state tomography on the target representations reveals the hidden subgroup.Comment: 11 pages. Revised in response to referee reports. Early sections are more accessible; expanded section on other hidden subgroup problem

Quantum algorithm for a generalized hidden shift problem

Consider the following generalized hidden shift problem: given a function f on {0,...,M − 1} × ZN promised to be injective for fixed b and satisfying f(b, x) = f(b + 1, x + s) for b = 0, 1,...,M − 2, find the unknown shift s ∈ ZN. For M = N, this problem is an instance of the abelian hidden subgroup problem, which can be solved efficiently on a quantum computer, whereas for M = 2, it is equivalent to the dihedral hidden subgroup problem, for which no efficient algorithm is known. For any fixed positive �, we give an efficient (i.e., poly(logN)) quantum algorithm for this problem provided M ≥ N^∈. The algorithm is based on the “pretty good measurement” and uses H. Lenstra’s (classical) algorithm for integer programming as a subroutine

Improved Low-qubit Hidden Shift Algorithms

Hidden shift problems are relevant to assess the quantum security of various cryptographic constructs. Multiple quantum subexponential time algorithms have been proposed. In this paper, we propose some improvements on a polynomial quantum memory algorithm proposed by Childs, Jao and Soukharev in 2010. We use subset-sum algorithms to significantly reduce its complexity. We also propose new tradeoffs between quantum queries, classical time and classical memory to solve this problem

From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups

We approach the hidden subgroup problem by performing the so-called pretty good measurement on hidden subgroup states. For various groups that can be expressed as the semidirect product of an abelian group and a cyclic group, we show that the pretty good measurement is optimal and that its probability of success and unitary implementation are closely related to an average-case algebraic problem. By solving this problem, we find efficient quantum algorithms for a number of nonabelian hidden subgroup problems, including some for which no efficient algorithm was previously known: certain metacyclic groups as well as all groups of the form (Z_p)^r X| Z_p for fixed r (including the Heisenberg group, r=2). In particular, our results show that entangled measurements across multiple copies of hidden subgroup states can be useful for efficiently solving the nonabelian HSP.Comment: 18 pages; v2: updated references on optimal measuremen