828 research outputs found
Efficient Quantum Transforms
Quantum mechanics requires the operation of quantum computers to be unitary,
and thus makes it important to have general techniques for developing fast
quantum algorithms for computing unitary transforms. A quantum routine for
computing a generalized Kronecker product is given. Applications include
re-development of the networks for computing the Walsh-Hadamard and the quantum
Fourier transform. New networks for two wavelet transforms are given. Quantum
computation of Fourier transforms for non-Abelian groups is defined. A slightly
relaxed definition is shown to simplify the analysis and the networks that
computes the transforms. Efficient networks for computing such transforms for a
class of metacyclic groups are introduced. A novel network for computing a
Fourier transform for a group used in quantum error-correction is also given.Comment: 30 pages, LaTeX2e, 7 figures include
On The Power of Exact Quantum Polynomial Time
We investigate the power of quantum computers when they are required to
return an answer that is guaranteed correct after a time that is upper-bounded
by a polynomial in the worst case. In an oracle setting, it is shown that such
machines can solve problems that would take exponential time on any classical
bounded-error probabilistic computer.Comment: 10 pages, LaTeX2e, no figure
On Quantum Algorithms for Noncommutative Hidden Subgroups
Quantum algorithms for factoring and discrete logarithm have previously been
generalized to finding hidden subgroups of finite Abelian groups. This paper
explores the possibility of extending this general viewpoint to finding hidden
subgroups of noncommutative groups. We present a quantum algorithm for the
special case of dihedral groups which determines the hidden subgroup in a
linear number of calls to the input function. We also explore the difficulties
of developing an algorithm to process the data to explicitly calculate a
generating set for the subgroup. A general framework for the noncommutative
hidden subgroup problem is discussed and we indicate future research
directions.Comment: 13 pages, no figures, LaTeX2
Lower Bounds on Quantum Query Complexity
Shor's and Grover's famous quantum algorithms for factoring and searching
show that quantum computers can solve certain computational problems
significantly faster than any classical computer. We discuss here what quantum
computers_cannot_ do, and specifically how to prove limits on their
computational power. We cover the main known techniques for proving lower
bounds, and exemplify and compare the methods.Comment: survey, 23 page
An Exact Quantum Polynomial-Time Algorithm for Simon's Problem
We investigate the power of quantum computers when they are required to
return an answer that is guaranteed to be correct after a time that is
upper-bounded by a polynomial in the worst case. We show that a natural
generalization of Simon's problem can be solved in this way, whereas previous
algorithms required quantum polynomial time in the expected sense only, without
upper bounds on the worst-case running time. This is achieved by generalizing
both Simon's and Grover's algorithms and combining them in a novel way. It
follows that there is a decision problem that can be solved in exact quantum
polynomial time, which would require expected exponential time on any classical
bounded-error probabilistic computer if the data is supplied as a black box.Comment: 12 pages, LaTeX2e, no figures. To appear in Proceedings of the Fifth
Israeli Symposium on Theory of Computing and Systems (ISTCS'97
Optimal Protocols for Nonlocality Distillation
Forster, Winkler, and Wolf recently showed that weak nonlocality can be
amplified by giving the first protocol that distills a class of nonlocal boxes
(NLBs) [Phys. Rev. Lett. 102, 120401 (2009)]. We first show that their protocol
is optimal among all non-adaptive protocols. We next consider adaptive
protocols. We show that the depth 2 protocol of Allcock et al. [Phys. Rev. A
80, 062107, (2009)] performs better than previously known adaptive depth 2
protocols for all symmetric NLBs. We present a new depth 3 protocol that
extends the known region of distillable NLBs. We give examples of NLBs for
which each of Forster et al.'s, Allcock et al.'s, and our protocol performs
best. The new understanding we develop is that there is no single optimal
protocol for NLB distillation. The choice of which protocol to use depends on
the noise parameters for the NLB.Comment: RevTeX4, 6 pages with 4 figure
A Quantum Observable for the Graph Isomorphism Problem
Suppose we are given two graphs on vertices. We define an observable in
the Hilbert space \Co[(S_n \wr S_2)^m] which returns the answer ``yes'' with
certainty if the graphs are isomorphic and ``no'' with probability at least
if the graphs are not isomorphic. We do not know if this observable
is efficiently implementable.Comment: 5 pages, no figure
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