27,614 research outputs found
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
Quantum Rejection Sampling
Let H be a finite dimensional Hilbert space and ρ, σ ∈ D(H) be quantum states in H
such that S(ρ||σ) is finite. In this thesis, we consider the following communication task involving two parties Alice and Bob. Suppose that Alice is given a classical description of the states ρ and σ. Given an unlimited number of copies of an entangled state whose marginal on Bob’s side is σ, Alice’s goal is to help Bob output a single copy of the state ρ by sending a single message to Bob in a one-way LOCC (short for local operation and classical communication) protocol. We propose a class of one-way LOCC protocols for this task which we refer to as quantum rejection sampling protocols. Inspired by the classical rejection sampling protocol of Harsha, Jain, McAllester, and Radhakrishnan [25] for a similar classical communication task, we introduce the Greedy Quantum Rejection Sampler. We characterize the expected communication cost of the protocol in terms of max-relative entropy of ρ with respect to σ, in the case where the state ρ is a pure state and prove that the Greedy Quantum Rejection Sampler is an optimal quantum rejection sampling protocol in this case. We describe an optimal quantum rejection sampling protocol in terms of a semidefinite program and we find general lower bounds and upper bounds on the expected communication cost of the optimal protocol. We propose an LOCC compression protocol based on the Greedy Quantum Rejection Sampler protocol, for lossless compression of an arbitrary pure state quantum source in the visible compression model and we show an upper bound on the average length of this encoding. The upper bound is always less than or equal to the Shannon entropy of the quantum source and the gap between the two quantities can be arbitrary large. Finally, we introduce a high-entanglement deterministic exact protocol for remote preparation of an arbitrary ensemble of quantum states. Our protocol is based on a quantum rejection sampling protocol which uses a prefix-free encoding for communication of the messages. We establish an upper bound on the expected communication cost of this protocol for the worst case choice of the target state in terms of the max-information in Bob’s output register at the end of the protocol about Alice’s classical input register. Furthermore, this protocol can be used as a non-oblivious universal protocol for exact remote preparation of an arbitrary d-dimensional state at an expected communication cost of at most lg(d) +O (lg(lg(d))).4 month
Monte Carlo sampling from the quantum state space. I
High-quality random samples of quantum states are needed for a variety of
tasks in quantum information and quantum computation. Searching the
high-dimensional quantum state space for a global maximum of an objective
function with many local maxima or evaluating an integral over a region in the
quantum state space are but two exemplary applications of many. These tasks can
only be performed reliably and efficiently with Monte Carlo methods, which
involve good samplings of the parameter space in accordance with the relevant
target distribution. We show how the standard strategies of rejection sampling,
importance sampling, and Markov-chain sampling can be adapted to this context,
where the samples must obey the constraints imposed by the positivity of the
statistical operator. For a comparison of these sampling methods, we generate
sample points in the probability space for two-qubit states probed with a
tomographically incomplete measurement, and then use the sample for the
calculation of the size and credibility of the recently-introduced optimal
error regions [see New J. Phys. 15 (2013) 123026]. Another illustration is the
computation of the fractional volume of separable two-qubit states.Comment: 13 pages, 5 figures, 1 table, 26 reference
Metropolis Methods for Quantum Monte Carlo Simulations
Since its first description fifty years ago, the Metropolis Monte Carlo
method has been used in a variety of different ways for the simulation of
continuum quantum many-body systems. This paper will consider some of the
generalizations of the Metropolis algorithm employed in quantum Monte Carlo:
Variational Monte Carlo, dynamical methods for projector monte carlo ({\it
i.e.} diffusion Monte Carlo with rejection), multilevel sampling in path
integral Monte Carlo, the sampling of permutations, cluster methods for lattice
models, the penalty method for coupled electron-ionic systems and the Bayesian
analysis of imaginary time correlation functions.Comment: Proceedings of "Monte Carlo Methods in the Physical Sciences"
Celebrating the 50th Anniversary of the Metropolis Algorith
The chaotic chameleon
Various local hidden variables models for the singlet correlations exploit
the detection loophole, or other loopholes connected with post-selection on
coincident arrival times. I consider the connection with a probabilistic
simulation technique called rejection-sampling, and pose some natural questions
concerning what can be achieved and what cannot be achieved with local (or
distributed) rejection sampling. In particular a new and more serious loophole,
which we call the coincidence loophole, is introduced.Comment: v.2: 7pp; conjecture 1 disproved by Gisin & Gisin (1999) but which
leads to new open problems and conjectures. v.3: minor correction and
addition. To appear in proceedings of "Quantum Probability and Infinite
Dimensional Analysis", Greifswald, 2003; World Scientific. v.4: major
revision in view of solution of another conjecture by Larsson & Gill (2003
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