How to Couple from the Past Using a Read-Once Source of Randomness

We give a new method for generating perfectly random samples from the stationary distribution of a Markov chain. The method is related to coupling from the past (CFTP), but only runs the Markov chain forwards in time, and never restarts it at previous times in the past. The method is also related to an idea known as PASTA (Poisson arrivals see time averages) in the operations research literature. Because the new algorithm can be run using a read-once stream of randomness, we call it read-once CFTP. The memory and time requirements of read-once CFTP are on par with the requirements of the usual form of CFTP, and for a variety of applications the requirements may be noticeably less. Some perfect sampling algorithms for point processes are based on an extension of CFTP known as coupling into and from the past; for completeness, we give a read-once version of coupling into and from the past, but it remains unpractical. For these point process applications, we give an alternative coupling method with which read-once CFTP may be efficiently used.Comment: 28 pages, 2 figure

One-shot CFTP; Application to a class of truncated Gaussian densities

In this paper, we introduce a new method for perfect simulation of multivariate densities. We use One-Shot CFTP (G. Roberts and J. Rosenthal, “One-shot coupling for certain stochastic recursive sequences,” Stochastic Processes and their Applications vol. 99 pp. 195–208, 2002) together with a monotone coupler for the Gibbs sampler, and implement the algorithm within the Read-Once CFTP protocol (D. B. Wilson, “How to couple form the past using a read-once source of randomness,” Random Structures and Algorithms vol. 16(1) pp. 85–113, 2000b). We illustrate our method by simulating efficiently from high-dimensional truncated normal distributions using the Gibbs sampler

Measurement-based quantum computation beyond the one-way model

We introduce novel schemes for quantum computing based on local measurements on entangled resource states. This work elaborates on the framework established in [Phys. Rev. Lett. 98, 220503 (2007), quant-ph/0609149]. Our method makes use of tools from many-body physics - matrix product states, finitely correlated states or projected entangled pairs states - to show how measurements on entangled states can be viewed as processing quantum information. This work hence constitutes an instance where a quantum information problem - how to realize quantum computation - was approached using tools from many-body theory and not vice versa. We give a more detailed description of the setting, and present a large number of new examples. We find novel computational schemes, which differ from the original one-way computer for example in the way the randomness of measurement outcomes is handled. Also, schemes are presented where the logical qubits are no longer strictly localized on the resource state. Notably, we find a great flexibility in the properties of the universal resource states: They may for example exhibit non-vanishing long-range correlation functions or be locally arbitrarily close to a pure state. We discuss variants of Kitaev's toric code states as universal resources, and contrast this with situations where they can be efficiently classically simulated. This framework opens up a way of thinking of tailoring resource states to specific physical systems, such as cold atoms in optical lattices or linear optical systems.Comment: 21 pages, 7 figure

On Extractors and Exposure-Resilient Functions for Sublogarithmic Entropy

We study deterministic extractors for oblivious bit-fixing sources (a.k.a. resilient functions) and exposure-resilient functions with small min-entropy: of the function's n input bits, k << n bits are uniformly random and unknown to the adversary. We simplify and improve an explicit construction of extractors for bit-fixing sources with sublogarithmic k due to Kamp and Zuckerman (SICOMP 2006), achieving error exponentially small in k rather than polynomially small in k. Our main result is that when k is sublogarithmic in n, the short output length of this construction (O(log k) output bits) is optimal for extractors computable by a large class of space-bounded streaming algorithms. Next, we show that a random function is an extractor for oblivious bit-fixing sources with high probability if and only if k is superlogarithmic in n, suggesting that our main result may apply more generally. In contrast, we show that a random function is a static (resp. adaptive) exposure-resilient function with high probability even if k is as small as a constant (resp. log log n). No explicit exposure-resilient functions achieving these parameters are known

Time of Philosophers, Time of Physicists, Time of Mathematicians

Is presentism compatible with relativity ? This question has been much debated since the argument first proposed by Rietdijk and Putnam. The goal of this text is to study the implications of relativity and quantum mechanics on presentism, possibilism, and eternalism. We put the emphasis on the implicit metaphysical preconceptions underlying each of these different approaches to the question of time. We show that there exists a unique version of presentism which is both non-trivial, in the sense that it does not reduce the present to a unique event, and compatible with special relativity and quantum mechanics: the one in which the present of an observer at a point is identified with the backward light cone of that point. However, this compatibility is achieved at the cost of a renouncement to the notion of an objective, observer-independent reality. We also argue that no non-trivial version of presentism survives in general relativity, except if some mechanism forbids the existence of closed timelike curves, in which case precisely one version of possibilism does survive. We remark that the above physical theories force the presentist/possibilist's view of reality to shrink and break up, whereas the eternalist, on the contrary, is forced to grant the status of reality to more and more entities. Finally, we identify mathematics as the "deus ex machina" allowing the eternalist to unify his vision of reality into a coherent whole, and offer to him an "idealist deal": to accept a mathematical ontology in exchange for the assurance of surviving any physical theory.Comment: 24 pages, 10 figure

A New Formula for Predicting Solar Cycles

A new formula for predicting solar cycles based on the current theoretical understanding of the solar cycle from flux transport dynamo is presented. Two important processes---fluctuations in the Babcock-Leighton mechanism and variations in the meridional circulation, which are believed to be responsible for irregularities of the solar cycle---are constrained by using observational data. We take the polar field near minima of the cycle as a measure of the randomness in the Babcock-Leighton process, and the decay rate near the minima as a consequence of the change in meridional circulation. We couple these two observationally derived quantities into a single formula to predict the amplitude of the future solar cycle. Our new formula suggests that the cycle 25 would be a moderate cycle. Whether this formula for predicting the future solar cycle can be justified theoretically is also discussed using simulations from the flux transport dynamo model.Comment: 12 pages, 6 figures, Accepted for publication in Ap