Topologically non-trivial quantum layers

Given a complete non-compact surface embedded in R^3, we consider the Dirichlet Laplacian in a layer of constant width about the surface. Using an intrinsic approach to the layer geometry, we generalise the spectral results of an original paper by Duclos et al. to the situation when the surface does not possess poles. This enables us to consider topologically more complicated layers and state new spectral results. In particular, we are interested in layers built over surfaces with handles or several cylindrically symmetric ends. We also discuss more general regions obtained by compact deformations of certain layers.Comment: 15 pages, 6 figure

Weakly regular Floquet Hamiltonians with pure point spectrum

We study the Floquet Hamiltonian: -i omega d/dt + H + V(t) as depending on the parameter omega. We assume that the spectrum of H is discrete, {h_m (m = 1..infinity)}, with h_m of multiplicity M_m. and that V is an Hermitian operator, 2pi-periodic in t. Let J > 0 and set Omega_0 = [8J/9,9J/8]. Suppose that for some sigma > 0: sum_{m,n such that h_m > h_n} mu_{mn}(h_m - h_n)^(-sigma) < infinity where mu_{mn} = sqrt(min{M_m,M_n)) M_m M_n. We show that in that case there exist a suitable norm to measure the regularity of V, denoted epsilon, and positive constants, epsilon_* & delta_*, such that: if epsilon |Omega_0| - delta_* epsilon and the Floquet Hamiltonian has a pure point spectrum for all omega in Omega_infinity.Comment: 35 pages, Latex with AmsAr

Fast Decoders for Topological Quantum Codes

We present a family of algorithms, combining real-space renormalization methods and belief propagation, to estimate the free energy of a topologically ordered system in the presence of defects. Such an algorithm is needed to preserve the quantum information stored in the ground space of a topologically ordered system and to decode topological error-correcting codes. For a system of linear size L, our algorithm runs in time log L compared to L^6 needed for the minimum-weight perfect matching algorithm previously used in this context and achieves a higher depolarizing error threshold.Comment: 4 pages, 4 figure

Quantum waveguides with a lateral semitransparent barrier: spectral and scattering properties

We consider a quantum particle in a waveguide which consists of an infinite straight Dirichlet strip divided by a thin semitransparent barrier on a line parallel to the walls which is modeled by a $\delta$ potential. We show that if the coupling strength of the latter is modified locally, i.e. it reaches the same asymptotic value in both directions along the line, there is always a bound state below the bottom of the essential spectrum provided the effective coupling function is attractive in the mean. The eigenvalues and eigenfunctions, as well as the scattering matrix for energies above the threshold, are found numerically by the mode-matching technique. In particular, we discuss the rate at which the ground-state energy emerges from the continuum and properties of the nodal lines. Finally, we investigate a system with a modified geometry: an infinite cylindrical surface threaded by a homogeneous magnetic field parallel to the cylinder axis. The motion on the cylinder is again constrained by a semitransparent barrier imposed on a ``seam'' parallel to the axis.Comment: a LaTeX source file with 12 figures (11 of them eps); to appear in J. Phys. A: Math. Gen. Figures 3, 5, 8, 9, 11 are given at 300 dpi; higher resolution originals are available from the author

Bound States in Mildly Curved Layers

It has been shown recently that a nonrelativistic quantum particle constrained to a hard-wall layer of constant width built over a geodesically complete simply connected noncompact curved surface can have bound states provided the surface is not a plane. In this paper we study the weak-coupling asymptotics of these bound states, i.e. the situation when the surface is a mildly curved plane. Under suitable assumptions about regularity and decay of surface curvatures we derive the leading order in the ground-state eigenvalue expansion. The argument is based on Birman-Schwinger analysis of Schroedinger operators in a planar hard-wall layer.Comment: LaTeX 2e, 23 page

Topological code Autotune

Many quantum systems are being investigated in the hope of building a large-scale quantum computer. All of these systems suffer from decoherence, resulting in errors during the execution of quantum gates. Quantum error correction enables reliable quantum computation given unreliable hardware. Unoptimized topological quantum error correction (TQEC), while still effective, performs very suboptimally, especially at low error rates. Hand optimizing the classical processing associated with a TQEC scheme for a specific system to achieve better error tolerance can be extremely laborious. We describe a tool Autotune capable of performing this optimization automatically, and give two highly distinct examples of its use and extreme outperformance of unoptimized TQEC. Autotune is designed to facilitate the precise study of real hardware running TQEC with every quantum gate having a realistic, physics-based error model.Comment: 13 pages, 17 figures, version accepted for publicatio

Differential classification of states of consciousness using envelope- and phase-based functional connectivity

The development of sophisticated computational tools to quantify changes in the brain\u27s oscillatory dynamics across states of consciousness have included both envelope- and phase-based measures of functional connectivity (FC), but there are very few direct comparisons of these techniques using the same dataset. The goal of this study was to compare an envelope-based (i.e. Amplitude Envelope Correlation, AEC) and a phase-based (i.e. weighted Phase Lag Index, wPLI) measure of FC in their classification of states of consciousness. Nine healthy participants underwent a three-hour experimental anesthetic protocol with propofol induction and isoflurane maintenance, in which five minutes of 128-channel electroencephalography were recorded before, during, and after anesthetic-induced unconsciousness, at the following time points: Baseline; light sedation with propofol (Light Sedation); deep unconsciousness following three hours of surgical levels of anesthesia with isoflurane (Unconscious); five minutes prior to the recovery of consciousness (Pre-ROC); and three hours following the recovery of consciousness (Recovery). Support vector machine classification was applied to the source-localized EEG in the alpha (8-13 Hz) frequency band in order to investigate the ability of AEC and wPLI (separately and together) to discriminate i) the four states from Baseline; ii) Unconscious ( deep unconsciousness) vs. Pre-ROC ( light unconsciousness); and iii) responsiveness (Baseline, Light Sedation, Recovery) vs. unresponsiveness (Unconscious, Pre-ROC). AEC and wPLI yielded different patterns of global connectivity across states of consciousness, with AEC showing the strongest network connectivity during the Unconscious epoch, and wPLI showing the strongest connectivity during full consciousness (i.e., Baseline and Recovery). Both measures also demonstrated differential predictive contributions across participants and used different brain regions for classification. AEC showed higher classification accuracy overall, particularly for distinguishing anesthetic-induced unconsciousness from Baseline (83.7 ± 0.8%). AEC also showed stronger classification accuracy than wPLI when distinguishing Unconscious from Pre-ROC (i.e., deep from light unconsciousness) (AEC: 66.3 ± 1.2%; wPLI: 56.2 ± 1.3%), and when distinguishing between responsiveness and unresponsiveness (AEC: 76.0 ± 1.3%; wPLI: 63.6 ± 1.8%). Classification accuracy was not improved compared to AEC when both AEC and wPLI were combined. This analysis of source-localized EEG data demonstrates that envelope- and phase-based FC provide different information about states of consciousness but that, on a group level, AEC is better able to detect relative alterations in brain FC across levels of anesthetic-induced unconsciousness compared to wPLI

The Interpretation of Surface Easements in Severance Deeds as a Limit on Hydraulic Fracturing Practices

Hydraulic fracturing has driven a boom in natural gas production in the Marcellus Shale. While providing a growing source of domestic energy, this boom also raises signicant environmental concerns. Many of the impacts of hy draulic fracturing predominantly affect the inhabitants of the property where the drilling occurs. Yet when those inhabitants own only the surface estate, they have relatively little influence over whether and how the drilling occurs and will not profit from the gas extraction. This article provides a jurisdictional case stud4 set in West Virginia to assist in understanding the nuances relevant to an interpretation of the scope of express and implied surface easements pertinent to mineral extraction. West Virginia takes a unique approach to the accommodation doctrine. It permits a surface ovner to argue that certain overly burdensome practices may not have been contemplated by the parties to the original severance deed and easement, thus weakening the likelihood of their propriety and giving surface ovners leverage. Depending on the tlpe of easement at issue, the analysis can include a review of the burden to the surface, the nature of surface uses, the necessity of a practice, the compatibility of a practice, and/or contractual intention

Universal topological phase of 2D stabilizer codes

Two topological phases are equivalent if they are connected by a local unitary transformation. In this sense, classifying topological phases amounts to classifying long-range entanglement patterns. We show that all 2D topological stabilizer codes are equivalent to several copies of one universal phase: Kitaev's topological code. Error correction benefits from the corresponding local mappings.Comment: 4 pages, 3 figure