For Distinguishing Conjugate Hidden Subgroups, the Pretty Good Measurement is as Good as it Gets

2005/01/31 by Cristopher Moore, Alexander Russell, Moore, Cristopher +1 · 1 citation
Physics and Astronomy · #FOS: Physical sciences #Quantum Physics (quant-ph) #quant-ph

paper · pdf · doi:10.48550/arxiv.quant-ph/0501177

proved additional conditions for optimality due to Yuen, Kennedy and Max; also bounded the probability of success in the one-register and multiregister cases

arxiv created 2005/05/20 · arxiv updated 2009/12/01

Abstract

Recently Bacon, Childs and van Dam showed that the ``pretty good measurement'' (PGM) is optimal for the Hidden Subgroup Problem on the dihedral group D_n in the case where the hidden subgroup is chosen uniformly from the n involutions. We show that, for any group and any subgroup H, the PGM is the optimal one-register experiment in the case where the hidden subgroup is a uniformly random conjugate of H. We go on to show that when H forms a Gel'fand pair with its parent group, the PGM is the optimal measurement for any number of registers. In both cases we bound the probability that the optimal measurement succeeds. This generalizes the case of the dihedral group, and includes a number of other examples of interest.

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