Sufficient conditions for convergence of the Sum-Product Algorithm

We derive novel conditions that guarantee convergence of the Sum-Product algorithm (also known as Loopy Belief Propagation or simply Belief Propagation) to a unique fixed point, irrespective of the initial messages. The computational complexity of the conditions is polynomial in the number of variables. In contrast with previously existing conditions, our results are directly applicable to arbitrary factor graphs (with discrete variables) and are shown to be valid also in the case of factors containing zeros, under some additional conditions. We compare our bounds with existing ones, numerically and, if possible, analytically. For binary variables with pairwise interactions, we derive sufficient conditions that take into account local evidence (i.e., single variable factors) and the type of pair interactions (attractive or repulsive). It is shown empirically that this bound outperforms existing bounds.Comment: 15 pages, 5 figures. Major changes and new results in this revised version. Submitted to IEEE Transactions on Information Theor

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

Quantum state detection of a superconducting flux qubit using a DC-SQUID in the inductive mode

We present a readout method for superconducting flux qubits. The qubit quantum flux state can be measured by determining the Josephson inductance of an inductively coupled DC superconducting quantum interference device (DC-SQUID). We determine the response function of the DC-SQUID and its back-action on the qubit during measurement. Due to driving, the qubit energy relaxation rate depends on the spectral density of the measurement circuit noise at sum and difference frequencies of the qubit Larmor frequency and SQUID driving frequency. The qubit dephasing rate is proportional to the spectral density of circuit noise at the SQUID driving frequency. These features of the backaction are qualitatively different from the case when the SQUID is used in the usual switching mode. For a particular type of readout circuit with feasible parameters we find that single shot readout of a superconducting flux qubit is possible.Comment: 11 pages, 3 figures; submitted to Phys. Rev.

Parametric coupling for superconducting qubits

We propose a scheme to couple two superconducting charge or flux qubits biased at their symmetry points with unequal energy splittings. Modulating the coupling constant between two qubits at the sum or difference of their two frequencies allows to bring them into resonance in the rotating frame. Switching on and off the modulation amounts to switching on and off the coupling which can be realized at nanosecond speed. We discuss various physical implementations of this idea, and find that our scheme can lead to rapid operation of a two-qubit gate.Comment: 6 page

Coherent Quantum Dynamics of a Superconducting Flux Qubit

We have observed coherent time evolution between two quantum states of a superconducting flux qubit comprising three Josephson junctions in a loop. The superposition of the two states carrying opposite macroscopic persistent currents is manipulated by resonant microwave pulses. Readout by means of switching-event measurement with an attached superconducting quantum interference device revealed quantum-state oscillations with high fidelity. Under strong microwave driving it was possible to induce hundreds of coherent oscillations. Pulsed operations on this first sample yielded a relaxation time of 900 nanoseconds and a free-induction dephasing time of 20 nanoseconds. These results are promising for future solid-state quantum computing.Comment: submitted 2 December 2002; accepted 4 February 200

Characterising exo-ringsystems around fast-rotating stars using the Rossiter-McLaughlin effect

Planetary rings produce a distinct shape distortion in transit lightcurves. However, to accurately model such lightcurves the observations need to cover the entire transit, especially ingress and egress, as well as an out-of-transit baseline. Such observations can be challenging for long period planets, where the transits may last for over a day. Planetary rings will also impact the shape of absorption lines in the stellar spectrum, as the planet and rings cover different parts of the rotating star (the Rossiter-McLaughlin effect). These line-profile distortions depend on the size, structure, opacity, obliquity and sky projected angle of the ring system. For slow rotating stars, this mainly impacts the amplitude of the induced velocity shift, however, for fast rotating stars the large velocity gradient across the star allows the line distortion to be resolved, enabling direct determination of the ring parameters. We demonstrate that by modeling these distortions we can recover ring system parameters (sky-projected angle, obliquity and size) using only a small part of the transit. Substructure in the rings, e.g. gaps, can be recovered if the width of the features ($\delta W$) relative to the size of the star is similar to the intrinsic velocity resolution (set by the width of the local stellar profile, $\gamma$) relative to the stellar rotation velocity ($v$ sin$i$, i.e. $\delta W / R_* \gtrsim v$sin$i$/$\gamma$). This opens up a new way to study the ring systems around planets with long orbital periods, where observations of the full transit, covering the ingress and egress, are not always feasible.Comment: Accepted for publication in MNRA

Nondestructive readout for a superconducting flux qubit

We present a new readout method for a superconducting flux qubit, based on the measurement of the Josephson inductance of a superconducting quantum interference device that is inductively coupled to the qubit. The intrinsic flux detection efficiency and back-action are suitable for a fast and nondestructive determination of the quantum state of the qubit, as needed for readout of multiple qubits in a quantum computer. We performed spectroscopy of a flux qubit and we measured relaxation times of the order of 80 $\mu s$.Comment: 4 pages, 4 figures; modified content, figures and references; accepted for publication in Phys. Rev. Let

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