3,537 research outputs found
Remote Sampling with Applications to General Entanglement Simulation
We show how to sample exactly discrete probability distributions whose
defining parameters are distributed among remote parties. For this purpose, von
Neumann's rejection algorithm is turned into a distributed sampling
communication protocol. We study the expected number of bits communicated among
the parties and also exhibit a trade-off between the number of rounds of the
rejection algorithm and the number of bits transmitted in the initial phase.
Finally, we apply remote sampling to the simulation of quantum entanglement in
its most general form possible, when an arbitrary number of parties share
systems of arbitrary dimensions on which they apply arbitrary measurements (not
restricted to being projective measurements). In case the dimension of the
systems and the number of possible outcomes per party is bounded by a constant,
it suffices to communicate an expected O(m^2) bits in order to simulate exactly
the outcomes that these measurements would have produced on those systems,
where m is the number of participants.Comment: 17 pages, 1 figure, 4 algorithms (protocols); Complete generalization
of previous paper arXiv:1303.5942 [cs.IT] -- Exact simulation of the GHZ
distribution -- by the same author
Exact Classical Simulation of the GHZ Distribution
John Bell has shown that the correlations entailed by quantum
mechanics cannot be reproduced by a classical process involving
non-communicating parties. But can they be simulated with the help
of bounded communication? This problem has been studied for more
than twenty years and it is now well understood in the case of
bipartite entanglement. However, the issue was still widely open for
multipartite entanglement, even for the simplest case, which is
the tripartite Greenberger-Horne-Zeilinger (GHZ) state.
We give an exact simulation of arbitrary independent von Neumann
measurements on general n-partite GHZ states. Our protocol
requires O(n^2) bits of expected communication between the
parties, and O(n*log(n)) expected time is sufficient to carry it
out in parallel. Furthermore, we need only an expectation of
O(n) independent unbiased random bits, with no need for the
generation of continuous real random variables nor prior shared
random variables. In the case of equatorial measurements, we
improve earlier results with a protocol that needs only O(n*log(n)) bits of communication and O(log^2(n)) parallel time. At the
cost of a slight increase in the number of bits communicated, these
tasks can be accomplished with a constant expected number of rounds
Optimal Discrete Uniform Generation from Coin Flips, and Applications
This article introduces an algorithm to draw random discrete uniform
variables within a given range of size n from a source of random bits. The
algorithm aims to be simple to implement and optimal both with regards to the
amount of random bits consumed, and from a computational perspective---allowing
for faster and more efficient Monte-Carlo simulations in computational physics
and biology. I also provide a detailed analysis of the number of bits that are
spent per variate, and offer some extensions and applications, in particular to
the optimal random generation of permutations.Comment: first draft, 22 pages, 5 figures, C code implementation of algorith
Quantum rejection sampling
Rejection sampling is a well-known method to sample from a target
distribution, given the ability to sample from a given distribution. The method
has been first formalized by von Neumann (1951) and has many applications in
classical computing. We define a quantum analogue of rejection sampling: given
a black box producing a coherent superposition of (possibly unknown) quantum
states with some amplitudes, the problem is to prepare a coherent superposition
of the same states, albeit with different target amplitudes. The main result of
this paper is a tight characterization of the query complexity of this quantum
state generation problem. We exhibit an algorithm, which we call quantum
rejection sampling, and analyze its cost using semidefinite programming. Our
proof of a matching lower bound is based on the automorphism principle which
allows to symmetrize any algorithm over the automorphism group of the problem.
Our main technical innovation is an extension of the automorphism principle to
continuous groups that arise for quantum state generation problems where the
oracle encodes unknown quantum states, instead of just classical data.
Furthermore, we illustrate how quantum rejection sampling may be used as a
primitive in designing quantum algorithms, by providing three different
applications. We first show that it was implicitly used in the quantum
algorithm for linear systems of equations by Harrow, Hassidim and Lloyd.
Secondly, we show that it can be used to speed up the main step in the quantum
Metropolis sampling algorithm by Temme et al.. Finally, we derive a new quantum
algorithm for the hidden shift problem of an arbitrary Boolean function and
relate its query complexity to "water-filling" of the Fourier spectrum.Comment: 19 pages, 5 figures, minor changes and a more compact style (to
appear in proceedings of ITCS 2012
The universe as quantum computer
This article reviews the history of digital computation, and investigates
just how far the concept of computation can be taken. In particular, I address
the question of whether the universe itself is in fact a giant computer, and if
so, just what kind of computer it is. I will show that the universe can be
regarded as a giant quantum computer. The quantum computational model of the
universe explains a variety of observed phenomena not encompassed by the
ordinary laws of physics. In particular, the model shows that the the quantum
computational universe automatically gives rise to a mix of randomness and
order, and to both simple and complex systems.Comment: 16 pages, LaTe
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