Optimal measurements for the dihedral hidden subgroup problem

TL;DR

This paper investigates the optimal measurement strategy for the dihedral hidden subgroup problem, revealing a sharp success probability threshold related to the number of state copies and linking the problem to subset sum challenges.

Contribution

It demonstrates that the pretty good measurement is optimal for this problem and establishes a threshold phenomenon in success probability based on the number of copies.

Findings

Optimal measurement is the pretty good measurement.

Success probability exhibits a sharp threshold at density nu=1.

Connections between the problem and subset sum algorithms are established.

Abstract

We consider the dihedral hidden subgroup problem as the problem of distinguishing hidden subgroup states. We show that the optimal measurement for solving this problem is the so-called pretty good measurement. We then prove that the success probability of this measurement exhibits a sharp threshold as a function of the density nu=k/log N, where k is the number of copies of the hidden subgroup state and 2N is the order of the dihedral group. In particular, for nu<1 the optimal measurement (and hence any measurement) identifies the hidden subgroup with a probability that is exponentially small in log N, while for nu>1 the optimal measurement identifies the hidden subgroup with a probability of order unity. Thus the dihedral group provides an example of a group G for which Omega(log|G|) hidden subgroup states are necessary to solve the hidden subgroup problem. We also consider the optimal measurement for determining a single bit of the answer, and show that it exhibits the same threshold. Finally, we consider implementing the optimal measurement by a quantum circuit, and thereby establish further connections between the dihedral hidden subgroup problem and average case subset sum problems. In particular, we show that an efficient quantum algorithm for a restricted version of the optimal measurement would imply an efficient quantum algorithm for the subset sum problem, and conversely, that the ability to quantum sample from subset sum solutions allows one to implement the optimal measurement.