Alphabetic Minimax Trees of Degree at Most t*

Problems in circuit fan-out reduction motivate the study of constructing various types of weighted trees that are optimal with respect to maximum weighted path length. An upper bound on the maximum weighted path length and an efficient construction algorithm will be presented for trees of degree at most t, along with their implications for circuit fan-out reduction

Rounding and Chaining LLL: Finding Faster Small Roots of Univariate Polynomial Congruences

International audienceIn a seminal work at EUROCRYPT '96, Coppersmith showed how to find all small roots of a univariate polynomial congruence in polynomial time: this has found many applications in public-key cryptanalysis and in a few security proofs. However, the running time of the algorithm is a high-degree polynomial, which limits experiments: the bottleneck is an LLL reduction of a high-dimensional matrix with extra-large coefficients. We present in this paper the first significant speedups over Coppersmith's algorithm. The first speedup is based on a special property of the matrices used by Coppersmith's algorithm, which allows us to provably speed up the LLL reduction by rounding, and which can also be used to improve the complexity analysis of Coppersmith's original algorithm. The exact speedup depends on the LLL algorithm used: for instance, the speedup is asymptotically quadratic in the bit-size of the small-root bound if one uses the Nguyen-Stehlé L2 algorithm. The second speedup is heuristic and applies whenever one wants to enlarge the root size of Coppersmith's algorithm by exhaustive search. Instead of performing several LLL reductions independently, we exploit hidden relationships between these matrices so that the LLL reductions can be somewhat chained to decrease the global running time. When both speedups are combined, the new algorithm is in practice hundreds of times faster for typical parameters

Noise Stabilization of Self-Organized Memories

We investigate a nonlinear dynamical system which ``remembers'' preselected values of a system parameter. The deterministic version of the system can encode many parameter values during a transient period, but in the limit of long times, almost all of them are forgotten. Here we show that a certain type of stochastic noise can stabilize multiple memories, enabling many parameter values to be encoded permanently. We present analytic results that provide insight both into the memory formation and into the noise-induced memory stabilization. The relevance of our results to experiments on the charge-density wave material $NbSe_3$ is discussed.Comment: 29 pages, 6 figures, submitted to Physical Review

Theoretical characterization of a model of aragonite crystal orientation in red abalone nacre

Nacre, commonly known as mother-of-pearl, is a remarkable biomineral that in red abalone consists of layers of 400-nm thick aragonite crystalline tablets confined by organic matrix sheets, with the $(001)$ crystal axes of the aragonite tablets oriented to within $\pm$ 12 degrees from the normal to the layer planes. Recent experiments demonstrate that this orientational order develops over a distance of tens of layers from the prismatic boundary at which nacre formation begins. Our previous simulations of a model in which the order develops because of differential tablet growth rates (oriented tablets growing faster than misoriented ones) yield patterns of tablets that agree qualitatively and quantitatively with the experimental measurements. This paper presents an analytical treatment of this model, focusing on how the dynamical development and eventual degree of order depend on model parameters. Dynamical equations for the probability distributions governing tablet orientations are introduced whose form can be determined from symmetry considerations and for which substantial analytic progress can be made. Numerical simulations are performed to relate the parameters used in the analytic theory to those in the microscopic growth model. The analytic theory demonstrates that the dynamical mechanism is able to achieve a much higher degree of order than naive estimates would indicate.Comment: 20 pages, 3 figure

Second Harmonic Coherent Driving of a Spin Qubit in a Si/SiGe Quantum Dot

We demonstrate coherent driving of a single electron spin using second harmonic excitation in a Si/SiGe quantum dot. Our estimates suggest that the anharmonic dot confining potential combined with a gradient in the transverse magnetic field dominates the second harmonic response. As expected, the Rabi frequency depends quadratically on the driving amplitude and the periodicity with respect to the phase of the drive is twice that of the fundamental harmonic. The maximum Rabi frequency observed for the second harmonic is just a factor of two lower than that achieved for the first harmonic when driving at the same power. Combined with the lower demands on microwave circuitry when operating at half the qubit frequency, these observations indicate that second harmonic driving can be a useful technique for future quantum computation architectures.Comment: 9 pages, 9 figure