Truncating the loop series expansion for Belief Propagation

Recently, M. Chertkov and V.Y. Chernyak derived an exact expression for the partition sum (normalization constant) corresponding to a graphical model, which is an expansion around the Belief Propagation solution. By adding correction terms to the BP free energy, one for each "generalized loop" in the factor graph, the exact partition sum is obtained. However, the usually enormous number of generalized loops generally prohibits summation over all correction terms. In this article we introduce Truncated Loop Series BP (TLSBP), a particular way of truncating the loop series of M. Chertkov and V.Y. Chernyak by considering generalized loops as compositions of simple loops. We analyze the performance of TLSBP in different scenarios, including the Ising model, regular random graphs and on Promedas, a large probabilistic medical diagnostic system. We show that TLSBP often improves upon the accuracy of the BP solution, at the expense of increased computation time. We also show that the performance of TLSBP strongly depends on the degree of interaction between the variables. For weak interactions, truncating the series leads to significant improvements, whereas for strong interactions it can be ineffective, even if a high number of terms is considered.Comment: 31 pages, 12 figures, submitted to Journal of Machine Learning Researc

Superconductor-insulator transition in nanowires and nanowire arrays

Superconducting nanowires are the dual elements to Josephson junctions, with quantum phase-slip processes replacing the tunneling of Cooper pairs. When the quantum phase-slip amplitude ES is much smaller than the inductive energy EL, the nanowire responds as a superconducting inductor. When the inductive energy is small, the response is capacitive. The crossover at low temperatures as a function of ES/EL is discussed and compared with earlier experimental results. For one-dimensional and two-dimensional arrays of nanowires quantum phase transitions are expected as a function of ES/EL. They can be tuned by a homogeneous magnetic frustration.Comment: 15 pages, 10 figure

The merger of vertically offset quasi-geostrophic vortices

We examine the critical merging distance between two equal-volume, equal-potential-vorticity quasi-geostrophic vortices. We focus on how this distance depends on the vertical offset between the two vortices, each having a unit mean height-to-width aspect ratio. The vertical direction is special in the quasi-geostrophic model (used to capture the leading-order dynamical features of stably stratified and rapidly rotating geophysical flows) since vertical advection is absent. Nevertheless vortex merger may still occur by horizontal advection. In this paper, we first investigate the equilibrium states for the two vortices as a function of their vertical and horizontal separation. We examine their basic properties together with their linear stability. These findings are next compared to numerical simulations of the nonlinear evolution of two spheres of potential vorticity. Three different regimes of interaction are identified, depending on the vertical offset. For a small offset, the interaction differs little from the case when the two vortices are horizontally aligned. On the other hand, when the vertical offset is comparable to the mean vortex radius, strong interaction occurs for greater horizontal gaps than in the horizontally aligned case, and therefore at significantly greater full separation distances. This perhaps surprising result is consistent with the linear stability analysis and appears to be a consequence of the anisotropy of the quasi-geostrophic equations. Finally, for large vertical offsets, vortex merger results in the formation of a metastable tilted dumbbell vortex.Publisher PDFPeer reviewe

Phase-slip flux qubits

In thin superconducting wires, phase-slip by thermal activation near the critical temperature is a well-known effect. It has recently become clear that phase-slip by quantum tunnelling through the energy barrier can also have a significant rate at low temperatures. In this paper it is suggested that quantum phase-slip can be used to realize a superconducting quantum bit without Josephson junctions. A loop containing a nanofabricated very thin wire is biased with an externally applied magnetic flux of half a flux quantum, resulting in two states with opposite circulating current and equal energy. Quantum phase-slip should provide coherent coupling between these two macroscopic states. Numbers are given for a wire of amorphous niobium-silicon that can be fabricated with advanced electron beam lithography.Comment: Submitted to New Journal of Physics, special issue solid state quantum informatio

Quantum transitions induced by the third cumulant of current fluctuations

We investigate the transitions induced by external current fluctuations on a small probe quantum system. The rates for the transitions between the energy states are calculated using the real-time Keldysh formalism for the density matrix evolution. We especially detail the effects of the third cumulant of current fluctuations inductively coupled to a quantum bit and propose a setup for detecting the frequency-dependent third cumulant through the transitions it induces.Comment: 4 pages, 3 figure

Statistical-mechanical iterative algorithms on complex networks

The Ising models have been applied for various problems on information sciences, social sciences, and so on. In many cases, solving these problems corresponds to minimizing the Bethe free energy. To minimize the Bethe free energy, a statistical-mechanical iterative algorithm is often used. We study the statistical-mechanical iterative algorithm on complex networks. To investigate effects of heterogeneous structures on the iterative algorithm, we introduce an iterative algorithm based on information of heterogeneity of complex networks, in which higher-degree nodes are likely to be updated more frequently than lower-degree ones. Numerical experiments clarified that the usage of the information of heterogeneity affects the algorithm in BA networks, but does not influence that in ER networks. It is revealed that information of the whole system propagates rapidly through such high-degree nodes in the case of Barab{\'a}si-Albert's scale-free networks.Comment: 7 pages, 6 figure