Quantum Measurements for Hidden Subgroup Problems with Optimal Sample Complexity [article]

Masahito Hayashi, Akinori Kawachi, Hirotada Kobayashi
2007 arXiv   pre-print
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(||/ p) for a candidate set of hidden
more » ... bgroups 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(||) samples.
arXiv:quant-ph/0604174v3 fatcat:o5yz5vlv5newfpvszzvh4xrqdq