01370nas a2200133 4500008004100000245012700041210006900168260001500237520088400252100001601136700002301152700001701175856004401192 2005 eng d00aFrom optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups
0 aFrom optimal measurement to efficient quantum algorithms for the c2005/04/113 a 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.
1 aBacon, Dave1 aChilds, Andrew, M.1 avan Dam, Wim uhttp://arxiv.org/abs/quant-ph/0504083v201943nas a2200133 4500008004100000245006600041210006600107260001500173520152100188100001601709700002301725700001701748856004401765 2005 eng d00aOptimal measurements for the dihedral hidden subgroup problem0 aOptimal measurements for the dihedral hidden subgroup problem c2005/01/103 a 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.
1 aBacon, Dave1 aChilds, Andrew, M.1 avan Dam, Wim uhttp://arxiv.org/abs/quant-ph/0501044v201258nas a2200181 4500008004100000245005500041210005500096260001400151490000700165520074300172100001600915700002300931700002200954700001700976700002200993700001701015856004401032 2001 eng d00aUniversal simulation of Markovian quantum dynamics0 aUniversal simulation of Markovian quantum dynamics c2001/11/90 v643 a Although the conditions for performing arbitrary unitary operations to
simulate the dynamics of a closed quantum system are well understood, the same
is not true of the more general class of quantum operations (also known as
superoperators) corresponding to the dynamics of open quantum systems. We
propose a framework for the generation of Markovian quantum dynamics and study
the resources needed for universality. For the case of a single qubit, we show
that a single nonunitary process is necessary and sufficient to generate all
unital Markovian quantum dynamics, whereas a set of processes parametrized by
one continuous parameter is needed in general. We also obtain preliminary
results for the unital case in higher dimensions.
1 aBacon, Dave1 aChilds, Andrew, M.1 aChuang, Isaac, L.1 aKempe, Julia1 aLeung, Debbie, W.1 aZhou, Xinlan uhttp://arxiv.org/abs/quant-ph/0008070v2