Local Decoders for the 2D and 4D Toric Code

We analyze the performance of decoders for the 2D and 4D toric code which are local by construction. The 2D decoder is a cellular automaton decoder formulated by Harrington which explicitly has a finite speed of communication and computation. For a model of independent $X$ and $Z$ errors and faulty syndrome measurements with identical probability we report a threshold of $0.133\%$ for this Harrington decoder. We implement a decoder for the 4D toric code which is based on a decoder by Hastings arXiv:1312.2546 . Incorporating a method for handling faulty syndromes we estimate a threshold of $1.59\%$ for the same noise model as in the 2D case. We compare the performance of this decoder with a decoder based on a 4D version of Toom's cellular automaton rule as well as the decoding method suggested by Dennis et al. arXiv:quant-ph/0110143 .Comment: 22 pages, 21 figures; fixed typos, updated Figures 6,7,8,

A semi-analytical approach to molecular dynamics

Despite numerous computational advances over the last few decades, molecular dynamics still favors explicit (and thus easily-parallelizable) time integrators for large scale numerical simulation. As a consequence, computational efficiency in solving its typically stiff oscillatory equations of motion is hampered by stringent stability requirements on the time step size. In this paper, we present a semi-analytical integration scheme that offers a total speedup of a factor 30 compared to the Verlet method on typical MD simulation by allowing over three orders of magnitude larger step sizes. By efficiently approximating the exact integration of the strong (harmonic) forces of covalent bonds through matrix functions, far improved stability with respect to time step size is achieved without sacrificing the explicit, symplectic, time-reversible, or fine-grained parallelizable nature of the integration scheme. We demonstrate the efficiency and scalability of our integrator on simulations ranging from DNA strand unbinding and protein folding to nanotube resonators

Lie Symmetry Analysis for Cosserat Rods

We consider a subsystem of the Special Cosserat Theory of Rods and construct an explicit form of its solution that depends on three arbitrary functions in (s,t) and three arbitrary functions in t. Assuming analyticity of the arbitrary functions in a domain under consideration, we prove that the obtained solution is analytic and general. The Special Cosserat Theory of Rods describes the dynamic equilibrium of 1-dimensional continua, i.e. slender structures like fibers, by means of a system of partial differential equations.Comment: 12 Pages, 1 Figur

Zero-Level-Set Encoder for Neural Distance Fields

Neural shape representation generally refers to representing 3D geometry using neural networks, e.g., to compute a signed distance or occupancy value at a specific spatial position. Previous methods tend to rely on the auto-decoder paradigm, which often requires densely-sampled and accurate signed distances to be known during training and testing, as well as an additional optimization loop during inference. This introduces a lot of computational overhead, in addition to having to compute signed distances analytically, even during testing. In this paper, we present a novel encoder-decoder neural network for embedding 3D shapes in a single forward pass. Our architecture is based on a multi-scale hybrid system incorporating graph-based and voxel-based components, as well as a continuously differentiable decoder. Furthermore, the network is trained to solve the Eikonal equation and only requires knowledge of the zero-level set for training and inference. Additional volumetric samples can be generated on-the-fly, and incorporated in an unsupervised manner. This means that in contrast to most previous work, our network is able to output valid signed distance fields without explicit prior knowledge of non-zero distance values or shape occupancy. In other words, our network computes approximate solutions to the boundary-valued Eikonal equation. It also requires only a single forward pass during inference, instead of the common latent code optimization. We further propose a modification of the loss function in case that surface normals are not well defined, e.g., in the context of non-watertight surface-meshes and non-manifold geometry. We finally demonstrate the efficacy, generalizability and scalability of our method on datasets consisting of deforming 3D shapes, single class encoding and multiclass encoding, showcasing a wide range of possible applications