79 research outputs found
Atomaire gassen en draadloze netwerken
Get PDFVeel wetenschappers zoeken naar verbanden tussen ogenschijnlijk verschillende gebieden. Jaron Sanders zocht naar verbanden tussen draadloze netwerken en gassen van atomen, in de hoop op een vruchtbare kruisbestuiving tussen de wiskunde en natuurkunde
Clustering in Block Markov Chains
Get PDFThis paper considers cluster detection in Block Markov Chains (BMCs). These Markov chains are characterized by a block structure in their transition matrix. More precisely, the possible states are divided into a finite number of groups or clusters, such that states in the same cluster exhibit the same transition rates to other states. One observes a trajectory of the Markov chain, and the objective is to recover, from this observation only, the (initially unknown) clusters. In this paper we devise a clustering procedure that accurately, efficiently, and provably detects the clusters. We first derive a fundamental information-theoretical lower bound on the detection error rate satisfied under any clustering algorithm. This bound identifies the parameters of the BMC, and trajectory lengths, for which it is possible to accurately detect the clusters. We next develop two clustering algorithms that can together accurately recover the cluster structure from the shortest possible trajectories, whenever the parameters allow detection. These algorithms thus reach the fundamental detectability limit, and are optimal in that sense
Atomaire gassen en draadloze netwerken
No full textVeel wetenschappers zoeken naar verbanden tussen ogenschijnlijk verschillende gebieden. Jaron Sanders zocht naar verbanden tussen draadloze netwerken en gassen van atomen, in de hoop op een vruchtbare kruisbestuiving tussen de wiskunde en natuurkunde
Scaling Limits and Generic Bounds for Exploration Processes
Get PDFWe consider exploration algorithms of the random sequential adsorption type both for homogeneous random graphs and random geometric graphs based on spatial Poisson processes. At each step, a vertex of the graph becomes active and its neighboring nodes become blocked. Given an initial number of vertices N growing to infinity, we study statistical properties of the proportion of explored (active or blocked) nodes in time using scaling limits. We obtain exact limits for homogeneous graphs and prove an explicit central limit theorem for the final proportion of active nodes, known as the jamming constant, through a diffusion approximation for the exploration process which can be described as a unidimensional process. We then focus on bounding the trajectories of such exploration processes on random geometric graphs, i.e., random sequential adsorption. As opposed to exploration processes on homogeneous random graphs, these do not allow for such a dimensional reduction. Instead we derive a fundamental relationship between the number of explored nodes and the discovered volume in the spatial process, and we obtain generic bounds for the fluid limit and jamming constant: bounds that are independent of the dimension of space and the detailed shape of the volume associated to the discovered node. Lastly, using coupling techinques, we give trajectorial interpretations of the generic bounds.Fil: Bermolen, Paola. Universidad de la Republica. Facultad de Ingeniería; UruguayFil: Jonckheere, Matthieu Thimothy Samson. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Sanders, Jaron. Eindhoven Technical University; Países Bajo
Markov Chains and Hitting Times for Error Accumulation in Quantum Circuits
No full textWe study a classical model for the accumulation of errors in multi-qubit quantum computations. By modeling the error process in a quantum computation using two coupled Markov chains, we are able to capture a weak form of time-dependency between errors in the past and future. By subsequently using techniques from the field of discrete probability theory, we calculate the probability that error quantities such as the fidelity and trace distance exceed a threshold analytically. The formulae cover fairly generic error distributions, cover multi-qubit scenarios, and are applicable to the randomized benchmarking protocol. To combat the numerical challenge that may occur when evaluating our expressions, we additionally provide an analytical bound on the error probabilities that is of lower numerical complexity. Besides this, we study a model describing continuous errors accumulating in a single qubit. Finally, taking inspiration from the field of operations research, we illustrate how our expressions can be used to decide how many gates one can apply before too many errors accumulate with high probability, and how one can lower the rate of error accumulation in existing circuits through simulated annealing.Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Network Architectures and Service
Big Data, Big Libraries, Big Problems?: the 2014 LibTech Anti-talk?
Get PDFThe desire to create automatons is a familiar theme in human history, and during the age of the Enlightenment mechanical automatons became not only an “emblem of the cosmos”, but a symbol of man’s confidence that he would unlock nature’s greatest mysteries and fully harness her power. And yet only a century later, automatons had begun to represent human repression and servitude, a theme later picked up by writers of science fiction. Man’s confidence undeterred, the endgame of the modern scientific and technological mindset, or MSTM, seems to be increasingly coming into view with the rise of “information technology” in general and “Big data” in particular. Along with those who wield them, these can be seen as functioning together as a “mechanical muse” of sorts – surprisingly alluring – and, like a physical automaton can serve as a symbol – a microcosm – of what the MSTM sees (at the very least in practice) as the cosmic machine, our “final frontier”. And yet, individuals who unreflectively participate in these things – giving themselves over to them and seeking the powers afforded by the technology apart from technology’s rightful purposes – in fact yield to the same pragmatism and reductionism those wielding them are captive to. Thus, they ultimately nullify themselves philosophically, politically, and economically – their value increasingly being only the data concerning their persons, and its perceived usefulness. Likewise libraries, the time-honored place of, and symbol for, the intellectual flowering of the individual, will, insofar as they spurn the classical liberal arts (with the idea that things are intrinsically good, and in the case of humans, special as well) in favor of the alluring embrace of MSTM-driven “information technology” and Big data - unwittingly contribute to their irrelevance and demise as they find themselves increasingly less needed, valued, wanted. Likewise for the liberal arts as a whole, and in fact history itself, if the acid of a “science” untethered from what is, in fact, good (intrinsically), continues to gain strengt
Markov chains for error accumulation in quantum circuits
Get PDFWe study a model for the accumulation of errors in multi-qubit quantum computations, as well as a model describing continuous errors accumulating in a single qubit. By modeling the error process in a quantum computation using two coupled Markov chains, we are able to capture a weak form of time-dependency between errors in the past and future. By subsequently using techniques from the field of discrete probability theory, we calculate the probability that error measures such as the fidelity and trace distance exceed a threshold analytically. The formulae cover fairly generic error distributions, cover multi-qubit scenarios, and are applicable to e.g. the randomized benchmarking protocol. To combat the numerical challenge that may occur when evaluating our expressions, we additionally provide an analytical bound on the error probabilities that is of lower numerical complexity, and we also discuss a state space reduction that occurs for stabilizer circuits. Finally, taking inspiration from the field of operations research, we illustrate how our expressions can be used to e.g. decide how many gates one can apply before too many errors accumulate with high probability, and how one can lower the rate of error accumulation in existing circuits through simulated annealing
Reinforcement Learning in Block Markov Chains
No full textNowadays, reinforcement learning algorithms on Markov decision processes (MDPs) face computational issues when the state space is large. To reduce this state space of a MDP several state aggregation, or clustering, methodologies have been applied. Recently, a new clustering algorithm has been proposed that is able to cluster states from a single block Markov chain. A block Markov chain is a Markov chain with blocks in its transition matrix that correspond to clusters. Our aim was to investigate the possible combination of state aggregation in reinforcement learning on MDPs with clustering of states on a block Markov chain. First, we investigated the clustering algorithm and its properties to see its performance with different parameters and trajectory length. We compared the statistical properties of a pure Markov chain and the mixed Markov chain generated by a MDP. Afterwards, we verified the performance of the clustering algorithm on this mixed Markov chain. We proposed the BMC-MDP model that is able to model cluster based MDPs. We proposed C-PSRL, an algorithm, that consists of a single clustering step, on this newly introduced model. We compared its performance with a naïve approach and concluded that this new combined approach of clustering and MDP solving on a reduced space is a viable approach that reduces the computational complexity significantly. This research opened up the possibilities of more complex algorithms with, for example, multiple clustering steps. Moreover, if we can extend this clustering algorithm to clustering based on a state and action trajectory, this may results in an increased clustering performance and thereby enhance the performance of this general approach of optimizing on a cluster based MDP.Electrical Engineerin
Scaling limits and generic bounds for exploration processes
Get PDFArtículo publicado en Journal of Statistical Physics, v.169, 2017, pp. 989–1018We consider exploration algorithms of the random sequential adsorption type both for homogeneous random graphs and random geometric graphs based on spatial Poisson processes. At each step, a vertex of the graph becomes active and its neighboring nodes become blocked. Given an initial number of vertices N growing to infinity, we study statistical properties of the proportion of explored (active or blocked) nodes in time using scaling limits. We obtain exact limits for homogeneous graphs and prove an explicit central limit theorem for the final proportion of active nodes, known as the jamming constant, through a diffusion approximation for the exploration process which can be described as a unidimensional process. We then focus on bounding the trajectories of such exploration processes on random geometric graphs, i.e., random sequential adsorption. As opposed to exploration processes on homogeneous random graphs, these do not allow for such a dimensional reduction. Instead we derive a fundamental relationship between the number of explored nodes and the discovered volume in the spatial process, and we obtain generic bounds for the fluid limit and jamming constant: bounds that are independent of the dimension of space and the detailed shape of the volume associated to the discovered node. Lastly, using coupling techinques, we give trajectorial interpretations of the generic bounds. Keywords : Random sequential adsorption, Scaling limits, Random graph
Modeling Rydberg gases using random sequential adsorption on random graphs
Get PDFThe statistics of strongly interacting, ultracold Rydberg gases are governed by the interplay of two factors: geometrical restrictions induced by blockade effects and quantum mechanical effects. To shed light on their relative roles in the statistics of Rydberg gases, we compare three models in this paper: a quantum mechanical model describing the excitation dynamics within a Rydberg gas, a random sequential adsorption (RSA) process on a random geometric graph (RGG), and a RSA process on a decomposed random intersection graph (DRIG). The last model refers to choosing a particular subgraph of a mixture of two other random graphs. Contrary to the first two models, it lends itself for a rigorous mathematical analysis, and it is built specifically to have particular structural properties of a RGG. We establish for it a fluid limit describing the time evolution of the number of Rydberg atoms and show numerically that the expression remains accurate across a wider range of particle densities than an earlier approach based on an RSA process on an Erdos-Rényi random graph (ERRG). Finally, we also develop a heuristic using random graphs that gives a recursion to describe a normalized pair-correlation function of a Rydberg gas. Our results suggest that even without dissipation, on long timescales the statistics are affected most by the geometrical restrictions induced by blockade effects, while on short timescales the statistics are affected most by quantum mechanical effects
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