Atomaire gassen en draadloze netwerken

Veel 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

Markov chains for error accumulation in quantum circuits

We 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

Almost sure convergence of dropout algorithms for neural networks

We investigate the convergence and convergence rate of stochastic training algorithms for Neural Networks (NNs) that, over the years, have spawned from Dropout (Hinton et al., 2012). Modeling that neurons in the brain may not fire, dropout algorithms consist in practice of multiplying the weight matrices of a NN component-wise by independently drawn random matrices with $\{0,1\}$-valued entries during each iteration of the Feedforward-Backpropagation algorithm. This paper presents a probability theoretical proof that for any NN topology and differentiable polynomially bounded activation functions, if we project the NN's weights into a compact set and use a dropout algorithm, then the weights converge to a unique stationary set of a projected system of Ordinary Differential Equations (ODEs). We also establish an upper bound on the rate of convergence of Gradient Descent (GD) on the limiting ODEs of dropout algorithms for arborescences (a class of trees) of arbitrary depth and with linear activation functions

Clustering in Block Markov Chains

This 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 $n$ possible states are divided into a finite number of $K$ 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

Sub-Poissonian statistics of jamming limits in ultracold Rydberg gases

Several recent experiments have established by measuring the Mandel Q parameter that the number of Rydberg excitations in ultracold gases exhibits sub-Poissonian statistics. This effect is attributed to the Rydberg blockade that occurs due to the strong interatomic interactions between highly excited atoms. Because of this blockade effect, the system can end up in a state in which all particles are either excited or blocked: a jamming limit. We analyze appropriately constructed random-graph models that capture the blockade effect, and derive formulae for the mean and variance of the number of Rydberg excitations in jamming limits. This yields an explicit relationship between the Mandel Q parameter and the blockade effect, and comparison to measurement data shows strong agreement between theory and experiment