527 research outputs found
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 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
Weak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem
Schur duality decomposes many copies of a quantum state into subspaces
labeled by partitions, a decomposition with applications throughout quantum
information theory. Here we consider applying Schur duality to the problem of
distinguishing coset states in the standard approach to the hidden subgroup
problem. We observe that simply measuring the partition (a procedure we call
weak Schur sampling) provides very little information about the hidden
subgroup. Furthermore, we show that under quite general assumptions, even a
combination of weak Fourier sampling and weak Schur sampling fails to identify
the hidden subgroup. We also prove tight bounds on how many coset states are
required to solve the hidden subgroup problem by weak Schur sampling, and we
relate this question to a quantum version of the collision problem.Comment: 21 page
Quantum algorithm for a generalized hidden shift problem
Consider the following generalized hidden shift problem:
given a function f on {0,...,M − 1} × ZN promised to be
injective for fixed b and satisfying f(b, x) = f(b + 1, x + s)
for b = 0, 1,...,M − 2, find the unknown shift s ∈ ZN.
For M = N, this problem is an instance of the abelian
hidden subgroup problem, which can be solved efficiently on
a quantum computer, whereas for M = 2, it is equivalent
to the dihedral hidden subgroup problem, for which no
efficient algorithm is known. For any fixed positive �, we give
an efficient (i.e., poly(logN)) quantum algorithm for this
problem provided M ≥ N^∈. The algorithm is based on the
“pretty good measurement” and uses H. Lenstra’s (classical)
algorithm for integer programming as a subroutine
Hidden Translation and Translating Coset in Quantum Computing
We give efficient quantum algorithms for the problems of Hidden Translation
and Hidden Subgroup in a large class of non-abelian solvable groups including
solvable groups of constant exponent and of constant length derived series. Our
algorithms are recursive. For the base case, we solve efficiently Hidden
Translation in , whenever is a fixed prime. For the induction
step, we introduce the problem Translating Coset generalizing both Hidden
Translation and Hidden Subgroup, and prove a powerful self-reducibility result:
Translating Coset in a finite solvable group is reducible to instances of
Translating Coset in and , for appropriate normal subgroups of
. Our self-reducibility framework combined with Kuperberg's subexponential
quantum algorithm for solving Hidden Translation in any abelian group, leads to
subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in
any solvable group.Comment: Journal version: change of title and several minor update
Exponential algorithmic speedup by quantum walk
We construct an oracular (i.e., black box) problem that can be solved
exponentially faster on a quantum computer than on a classical computer. The
quantum algorithm is based on a continuous time quantum walk, and thus employs
a different technique from previous quantum algorithms based on quantum Fourier
transforms. We show how to implement the quantum walk efficiently in our
oracular setting. We then show how this quantum walk can be used to solve our
problem by rapidly traversing a graph. Finally, we prove that no classical
algorithm can solve this problem with high probability in subexponential time.Comment: 24 pages, 7 figures; minor corrections and clarification
Finding hidden Borel subgroups of the general linear group
We present a quantum algorithm for solving the hidden subgroup problem in the
general linear group over a finite field where the hidden subgroup is promised
to be a conjugate of the group of the invertible lower triangular matrices. The
complexity of the algorithm is polynomial when size of the base field is not
much smaller than the degree.Comment: 12pt, 10 page
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