@article {1240,
title = {From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups
},
year = {2005},
month = {2005/04/11},
abstract = { 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.
},
doi = {10.1109/SFCS.2005.38},
url = {http://arxiv.org/abs/quant-ph/0504083v2},
author = {Dave Bacon and Andrew M. Childs and Wim van Dam}
}
@article {1239,
title = {Optimal measurements for the dihedral hidden subgroup problem},
year = {2005},
month = {2005/01/10},
abstract = { 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.
},
url = {http://arxiv.org/abs/quant-ph/0501044v2},
author = {Dave Bacon and Andrew M. Childs and Wim van Dam}
}
@article {1260,
title = {Universal simulation of Markovian quantum dynamics},
journal = {Physical Review A},
volume = {64},
year = {2001},
month = {2001/11/9},
abstract = { 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.
},
doi = {10.1103/PhysRevA.64.062302},
url = {http://arxiv.org/abs/quant-ph/0008070v2},
author = {Dave Bacon and Andrew M. Childs and Isaac L. Chuang and Julia Kempe and Debbie W. Leung and Xinlan Zhou}
}