Constructions and Noise Threshold of Hyperbolic Surface Codes

We show how to obtain concrete constructions of homological quantum codes based on tilings of 2D surfaces with constant negative curvature (hyperbolic surfaces). This construction results in two-dimensional quantum codes whose tradeoff of encoding rate versus protection is more favorable than for the surface code. These surface codes would require variable length connections between qubits, as determined by the hyperbolic geometry. We provide numerical estimates of the value of the noise threshold and logical error probability of these codes against independent X or Z noise, assuming noise-free error correction

Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology

We introduce some algebraic geometric models in cosmology related to the "boundaries" of space-time: Big Bang, Mixmaster Universe, Penrose's crossovers between aeons. We suggest to model the kinematics of Big Bang using the algebraic geometric (or analytic) blow up of a point $x$. This creates a boundary which consists of the projective space of tangent directions to $x$ and possibly of the light cone of $x$. We argue that time on the boundary undergoes the Wick rotation and becomes purely imaginary. The Mixmaster (Bianchi IX) model of the early history of the universe is neatly explained in this picture by postulating that the reverse Wick rotation follows a hyperbolic geodesic connecting imaginary time axis to the real one. Penrose's idea to see the Big Bang as a sign of crossover from "the end of previous aeon" of the expanding and cooling Universe to the "beginning of the next aeon" is interpreted as an identification of a natural boundary of Minkowski space at infinity with the Big Bang boundary

A speech envelope landmark for syllable encoding in human superior temporal gyrus.

The most salient acoustic features in speech are the modulations in its intensity, captured by the amplitude envelope. Perceptually, the envelope is necessary for speech comprehension. Yet, the neural computations that represent the envelope and their linguistic implications are heavily debated. We used high-density intracranial recordings, while participants listened to speech, to determine how the envelope is represented in human speech cortical areas on the superior temporal gyrus (STG). We found that a well-defined zone in middle STG detects acoustic onset edges (local maxima in the envelope rate of change). Acoustic analyses demonstrated that timing of acoustic onset edges cues syllabic nucleus onsets, while their slope cues syllabic stress. Synthesized amplitude-modulated tone stimuli showed that steeper slopes elicited greater responses, confirming cortical encoding of amplitude change, not absolute amplitude. Overall, STG encoding of the timing and magnitude of acoustic onset edges underlies the perception of speech temporal structure

Formal Verification of Neural Network Controlled Autonomous Systems

In this paper, we consider the problem of formally verifying the safety of an autonomous robot equipped with a Neural Network (NN) controller that processes LiDAR images to produce control actions. Given a workspace that is characterized by a set of polytopic obstacles, our objective is to compute the set of safe initial conditions such that a robot trajectory starting from these initial conditions is guaranteed to avoid the obstacles. Our approach is to construct a finite state abstraction of the system and use standard reachability analysis over the finite state abstraction to compute the set of the safe initial states. The first technical problem in computing the finite state abstraction is to mathematically model the imaging function that maps the robot position to the LiDAR image. To that end, we introduce the notion of imaging-adapted sets as partitions of the workspace in which the imaging function is guaranteed to be affine. We develop a polynomial-time algorithm to partition the workspace into imaging-adapted sets along with computing the corresponding affine imaging functions. Given this workspace partitioning, a discrete-time linear dynamics of the robot, and a pre-trained NN controller with Rectified Linear Unit (ReLU) nonlinearity, the second technical challenge is to analyze the behavior of the neural network. To that end, we utilize a Satisfiability Modulo Convex (SMC) encoding to enumerate all the possible segments of different ReLUs. SMC solvers then use a Boolean satisfiability solver and a convex programming solver and decompose the problem into smaller subproblems. To accelerate this process, we develop a pre-processing algorithm that could rapidly prune the space feasible ReLU segments. Finally, we demonstrate the efficiency of the proposed algorithms using numerical simulations with increasing complexity of the neural network controller

Closed-Loop Targeted Memory Reactivation during Sleep Improves Spatial Navigation

Sounds associated with newly learned information that are replayed during non-rapid eye movement (NREM) sleep can improve recall in simple tasks. The mechanism for this improvement is presumed to be reactivation of the newly learned memory during sleep when consolidation takes place. We have developed an EEG-based closed-loop system to precisely deliver sensory stimulation at the time of down-state to up-state transitions during NREM sleep. Here, we demonstrate that applying this technology to participants performing a realistic navigation task in virtual reality results in a significant improvement in navigation efficiency after sleep that is accompanied by increases in the spectral power especially in the fast (12\u201315 Hz) sleep spindle band. Our results show promise for the application of sleep-based interventions to drive improvement in real-world tasks

Center vortex model for the infrared sector of SU(3) Yang-Mills theory: Topological susceptibility

The topological susceptibility of the SU(3) random vortex world-surface ensemble, an effective model of infrared Yang-Mills dynamics, is investigated. The model is implemented by composing vortex world-surfaces of elementary squares on a hypercubic lattice, supplemented by an appropriate specification of vortex color structure on the world-surfaces. Topological charge is generated in this picture by writhe and self-intersection of the vortex world-surfaces. Systematic uncertainties in the evaluation of the topological charge, engendered by the hypercubic construction, are discussed. Results for the topological susceptibility are reported as a function of temperature and compared to corresponding measurements in SU(3) lattice Yang-Mills theory. In the confined phase, the topological susceptibility of the random vortex world-surface ensemble appears quantitatively consistent with Yang-Mills theory. As the temperature is raised into the deconfined regime, the topological susceptibility falls off rapidly, but significantly less so than in SU(3) lattice Yang-Mills theory. Possible causes of this deviation, ranging from artefacts of the hypercubic description to more physical sources, such as the adopted vortex dynamics, are discussed.Comment: 30 pages, 6 figure

Spectral Theory of Discrete Processes

We offer a spectral analysis for a class of transfer operators. These transfer operators arise for a wide range of stochastic processes, ranging from random walks on infinite graphs to the processes that govern signals and recursive wavelet algorithms; even spectral theory for fractal measures. In each case, there is an associated class of harmonic functions which we study. And in addition, we study three questions in depth: In specific applications, and for a specific stochastic process, how do we realize the transfer operator $T$ as an operator in a suitable Hilbert space? And how to spectral analyze $T$ once the right Hilbert space $\mathcal{H}$ has been selected? Finally we characterize the stochastic processes that are governed by a single transfer operator. In our applications, the particular stochastic process will live on an infinite path-space which is realized in turn on a state space $S$. In the case of random walk on graphs $G$, $S$ will be the set of vertices of $G$. The Hilbert space $\mathcal{H}$ on which the transfer operator $T$ acts will then be an $L^{2}$ space on $S$, or a Hilbert space defined from an energy-quadratic form. This circle of problems is both interesting and non-trivial as it turns out that $T$ may often be an unbounded linear operator in $\mathcal{H}$; but even if it is bounded, it is a non-normal operator, so its spectral theory is not amenable to an analysis with the use of von Neumann's spectral theorem. While we offer a number of applications, we believe that our spectral analysis will have intrinsic interest for the theory of operators in Hilbert space.Comment: 34 pages with figures removed, for the full version with all the figures please go to http://www.siue.edu/~msong/Research/spectrum.pd