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Optimal measurements for the dihedral hidden subgroup problem. (English) Zbl 1117.81010

Summary: 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 \(\nu=k\log_2 N\), 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.

MSC:

81P68 Quantum computation
68Q05 Models of computation (Turing machines, etc.) (MSC2010)
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
81P15 Quantum measurement theory, state operations, state preparations
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