For distinguishing conjugate hidden subgroups, the pretty good measurement is as good as it gets

Recently Bacon, Childs and van Dam showed that the "pretty good measurement"(PGM) is optimal for the Hidden Subgroup Problem on the dihedral group Dn in the casewhere 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 inthe case where the hidden subgroup is a uniformly random conjugate of H. We go on toshow that when H forms a Gel'fand pair with its parent group, the PGM is the optimalmeasurement for any number of registers. In both cases we bound the probability thatthe optimal measurement succeeds. This generalizes the case of the dihedral group, andincludes a number of other examples of interest.

[1]  Julia Kempe,et al.  The hidden subgroup problem and permutation group theory , 2004, SODA '05.

[2]  Oded Regev,et al.  Quantum computation and lattice problems , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[3]  Dave Bacon,et al.  From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[4]  G. Pólya,et al.  Inequalities (Cambridge Mathematical Library) , 1934 .

[5]  Robert S. Kennedy,et al.  Optimum testing of multiple hypotheses in quantum detection theory , 1975, IEEE Trans. Inf. Theory.

[6]  Alexander Russell,et al.  Explicit Multiregister Measurements for Hidden Subgroup Problems; or, Fourier Sampling Strikes Back , 2005 .

[7]  Alexander Russell,et al.  The Symmetric Group Defies Strong Fourier Sampling: Part I , 2005, ArXiv.

[8]  Mark Ettinger,et al.  On Quantum Algorithms for Noncommutative Hidden Subgroups , 1998, STACS.

[9]  Sean Hallgren,et al.  Quantum algorithms for some hidden shift problems , 2003, SODA '03.

[10]  Lawrence Ip.  Shor ’ s Algorithm is Optimal , 2003 .

[11]  Jaikumar Radhakrishnan,et al.  On the Power of Random Bases in Fourier Sampling: Hidden Subgroup Problem in the Heisenberg Group , 2005, ICALP.

[12]  A. Terras.  Fourier Analysis on Finite Groups and Applications: Index , 1999 .

[13]  William K. Wootters,et al.  A ‘Pretty Good’ Measurement for Distinguishing Quantum States , 1994 .

[14]  Dave Bacon,et al.  Optimal measurements for the dihedral hidden subgroup problem , 2005, Chic. J. Theor. Comput. Sci..

[15]  Jean-Pierre Serre,et al.  Linear representations of finite groups , 1977, Graduate texts in mathematics.

[16]  Alexander Russell,et al.  The power of basis selection in fourier sampling: hidden subgroup problems in affine groups , 2004, SODA '04.

[17]  Joe Harris,et al.  Representation Theory: A First Course , 1991 .

[18]  Yonina C. Eldar,et al.  Optimal detection of symmetric mixed quantum states , 2004, IEEE Transactions on Information Theory.

[19]  Alexander Russell,et al.  The Symmetric Group Defies Strong Fourier Sampling , 2008, SIAM J. Comput..

[20]  Yuval Roichman,et al.  Upper bound on the characters of the symmetric groups , 1996 .

[21]  Martin Rötteler,et al.  Limitations of quantum coset states for graph isomorphism , 2006, STOC '06.

[22]  Greg Kuperberg.  A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem , 2005, SIAM J. Comput..

[23]  Emanuel Knill,et al.  The quantum query complexity of the hidden subgroup problem is polynomial , 2004, Inf. Process. Lett..