15 research outputs found
The Optimal Single Copy Measurement for the Hidden Subgroup Problem
The optimization of measurements for the state distinction problem has
recently been applied to the theory of quantum algorithms with considerable
successes, including efficient new quantum algorithms for the non-abelian
hidden subgroup problem. Previous work has identified the optimal single copy
measurement for the hidden subgroup problem over abelian groups as well as for
the non-abelian problem in the setting where the subgroups are restricted to be
all conjugate to each other. Here we describe the optimal single copy
measurement for the hidden subgroup problem when all of the subgroups of the
group are given with equal a priori probability. The optimal measurement is
seen to be a hybrid of the two previously discovered single copy optimal
measurements for the hidden subgroup problem.Comment: 8 pages. Error in main proof fixe
From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups
We approach the hidden subgroup problem by performing the so-called pretty
good measurement on hidden subgroup states. For various groups that can be
expressed as the semidirect product of an abelian group and a cyclic group, we
show that the pretty good measurement is optimal and that its probability of
success and unitary implementation are closely related to an average-case
algebraic problem. By solving this problem, we find efficient quantum
algorithms for a number of nonabelian hidden subgroup problems, including some
for which no efficient algorithm was previously known: certain metacyclic
groups as well as all groups of the form (Z_p)^r X| Z_p for fixed r (including
the Heisenberg group, r=2). In particular, our results show that entangled
measurements across multiple copies of hidden subgroup states can be useful for
efficiently solving the nonabelian HSP.Comment: 18 pages; v2: updated references on optimal measuremen
The Symmetric Group Defies Strong Fourier Sampling
The dramatic exponential speedups of quantum algorithms over their best existing classical counterparts were ushered in by the technique of Fourier sampling, introduced by Bernstein and Vazirani and developed by Simon and Shor into an approach to the hidden subgroup problem. This approach has proved successful for abelian groups, leading to efficient algorithms for factoring, extracting discrete logarithms, and other number-theoretic problems. We show, however, that this method cannot resolve the hidden subgroup problem in the symmetric groups, even in the weakest, information-theoretic sense. In particular, we show that the Graph Isomorphism problem cannot be solved by this approach. Our work implies that any quantum approach based upon the measurement of coset states must depart from the original framework by using entangled measurements on multiple coset states