Quantum Measurements for Hidden Subgroup Problems with Optimal Sample Complexity

Abstract

One of the central issues in the hidden subgroup problem is to bound the sample complexity, i.e., the number of identical samples of coset states sufficient and necessary to solve the problem. In this paper, we present general bounds for the sample complexity of the identification and decision versions of the hidden subgroup problem. As a consequence of the bounds, we show that the sample complexity for both of the decision and identification versions is $Θ(\log|\HH|/\log p)$ for a candidate set $\HH$ of hidden subgroups in the case that the candidate subgroups have the same prime order $p$, which implies that the decision version is at least as hard as the identification version in this case. In particular, it does so for the important instances such as the dihedral and the symmetric hidden subgroup problems. Moreover, the upper bound of the identification is attained by the pretty good measurement. This shows that the pretty good measurement can identify any hidden subgroup of an arbitrary group with at most $O(\log|\HH|)$ samples.

11 pages. Adding an application to quantum encryption schemes and correcting editorial errors

References in corpus (4)

• From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups
• A Subexponential Time Algorithm for the Dihedral Hidden Subgroup Problem with Polynomial Space
• For Distinguishing Conjugate Hidden Subgroups, the Pretty Good Measurement is as Good as it Gets
• The Hidden Subgroup Problem in Affine Groups: Basis Selection in Fourier Sampling