Bounding experimental quantum error rates relative to fault-tolerant thresholds

Rigorously establishing that the error in an experimental quantum operation is beneath the threshold for fault-tolerant quantum computation currently requires considering the worst-case error, which can be orders of magnitude smaller than the average gate infidelities routinely reported in experiments. We show that an improved bound on the worst-case error can be obtained by also considering the recently-introduced unitarity of the noise where the upper and lower bounds differ by a factor of $\approx 2.45$ for unital qubit channels. We prove that the contribution from the nonunital part of any noise map is at most on the order of the average gate infidelity and so is negligible relative to any coherent contribution. We also show that the "average" error rate when measurements are not restricted to an eigenbasis containing the state of the system exhibits the same scaling as the worst-case error, which, for coherent noise, is the square-root of the infidelity. We also obtain improved bounds for the diamond distance when the noise map is known (or approximately known).Comment: 12 pages, comments welcome. Improves on bounds involving the unitarity presented in arxiv.org/abs/1510.05653. v2: added discussion of the average gate error, simplified proofs and expositions and improved bound

Explosive Percolation in Social and Physical Networks

We discuss several interesting random network models which exhibit (provable) explosive transitions and their applications.Comment: As first written as a breviu

The Fields of a Charged Particle in Hyperbolic Motion

A particle in hyperbolic motion produces electric fields that appear to terminate in mid-air, violating Gauss's law. The resolution to this paradox has been known for sixty years, but exactly why the naive approach fails is not so clear.Comment: To appear, American Journal of Physic

Probabilistic Convergence Guarantees for Type II Pulse Coupled Oscillators

We show that a large class of pulse coupled oscillators converge with high probability from random initial conditions on a large class of graphs with time delays. Our analysis combines previous local convergence results, probabilistic network analysis, and a new classification scheme for Type II phase response curves to produce rigorous lower bounds for convergence probabilities based on network density. These bounds are then used to develop a simple, fast and rigorous computational analytic technique. These results suggest new methods for the analysis of pulse coupled oscillators, and provide new insights into the operation of biological Type II phase response curves and also the design of decentralized and minimal clock synchronization schemes in sensor nets.Comment: 5 pages, 3 figures, for submission to PR

A Novel Variational Autoencoder with Applications to Generative Modelling, Classification, and Ordinal Regression

We develop a novel probabilistic generative model based on the variational autoencoder approach. Notable aspects of our architecture are: a novel way of specifying the latent variables prior, and the introduction of an ordinality enforcing unit. We describe how to do supervised, unsupervised and semi-supervised learning, and nominal and ordinal classification, with the model. We analyze generative properties of the approach, and the classification effectiveness under nominal and ordinal classification, using two benchmark datasets. Our results show that our model can achieve comparable results with relevant baselines in both of the classification tasks.Comment: The first version [v1] contains our paper submitted (on 9 February, 2018) to and later rejected from the Thirty-Fifth International Conference on Machine Learning (ICML 2018); earlier version of the paper was submitted (on 13 October, 2017 [UTC]) to and later rejected from the Twenty-First International Conference on Artificial Intelligence and Statistics (AISTATS 2018

Berry phase mechanism for optical gyrotropy in stripe-ordered cuprates

Optical gyrotropy, the lifting of degeneracy between left and right circularly polarized light, can be generated by either time-reversal or chiral symmetry breaking. In the high-$T_c$ superconductor La$_{2-x}$Ba$_x$CuO$_4$ (LBCO), gyrotropy onsets at the same temperature as charge stripe order, suggesting that the rotation of the stripe direction from one plane to the next generates a helical pattern that breaks chiral symmetry. In order to further test this chiral stacking hypothesis it is necessary to develop an understanding of the physical mechanism by which chirality generates gyrotropy. In this paper we show that optical gyrotropy is a consequence of Berry curvature in the momentum space of chiral metals. We describe a physical picture showing that gyrotropy in chiral metals is closely related to the anomalous Hall effect in itinerant ferromagnets. We then calculate the magnitude of the gyrotropic response for a given Berry curvature using the semiclassical picture of anomalous velocity and Boltzmann transport theory. To connect this physical picture with experiment, we calculate the Berry curvature in two tight-binding models. The first model is motivated by the structure of LBCO and illustrates how the gyrotropy is created when the stripe perturbations are added to a simple cubic model. In the second model, we examine the dramatic enhancement of the gyrotropic coefficient when Rashba spin-orbit coupling is introduced. The magnitude of the rotation of polarization on reflection expected based these models is calculated and compared with experimental data

Noise tailoring for scalable quantum computation via randomized compiling

Quantum computers are poised to radically outperform their classical counterparts by manipulating coherent quantum systems. A realistic quantum computer will experience errors due to the environment and imperfect control. When these errors are even partially coherent, they present a major obstacle to achieving robust computation. Here, we propose a method for introducing independent random single-qubit gates into the logical circuit in such a way that the effective logical circuit remains unchanged. We prove that this randomization tailors the noise into stochastic Pauli errors, leading to dramatic reductions in worst-case and cumulative error rates, while introducing little or no experimental overhead. Moreover we prove that our technique is robust to variation in the errors over the gate sets and numerically illustrate the dramatic reductions in worst-case error that are achievable. Given such tailored noise, gates with significantly lower fidelity are sufficient to achieve fault-tolerant quantum computation, and, importantly, the worst case error rate of the tailored noise can be directly and efficiently measured through randomized benchmarking experiments. Remarkably, our method enables the realization of fault-tolerant quantum computation under the error rates observed in recent experiments.Comment: 7+6 pages, comments welcom

Smoothing Rotation Curves and Mass Profiles

We show that spiral activity can erase pronounced features in disk galaxy rotation curves. We present simulations of growing disks, in which the added material has a physically motivated distribution, as well as other examples of physically less realistic accretion. In all cases, attempts to create unrealistic rotation curves were unsuccessful because spiral activity rapidly smoothed away features in the disk mass profile. The added material was redistributed radially by the spiral activity, which was itself provoked by the density feature. In the case of a ridge-like feature in the surface density profile, we show that two unstable spiral modes develop, and the associated angular momentum changes in horseshoe orbits remove particles from the ridge and spread them both inwards and outwards. This process rapidly erases the density feature from the disk. We also find that the lack of a feature when transitioning from disk to halo dominance in the rotation curves of disk galaxies, the so called "disk-halo conspiracy", could also be accounted for by this mechanism. We do not create perfectly exponential mass profiles in the disk, but suggest that this mechanism contributes to their creation.Comment: Minor corrections in proofs and updates to references, 16 pages, 16 figure

Constraining the Milky Way's Hot Gas Halo with OVII and OVII Emission Lines

The Milky Way hosts a hot ($\approx 2 \times 10^6$ K), diffuse, gaseous halo based on detections of z = 0 OVII and OVIII absorption lines in quasar spectra and emission lines in blank-sky spectra. Here we improve constraints on the structure of the hot gas halo by fitting a radial model to a much larger sample of OVII and OVIII emission line measurements from XMM-Newton EPIC-MOS spectra compared to previous studies ($\approx$ 650 sightlines). We assume a modified $\beta$-model for the halo density distribution and a constant-density Local Bubble from which we calculate emission to compare with the observations. We find an acceptable fit to the OVIII emission line observations with $\chi^{2}_{red}$ (dof) = 1.08 (644) for best-fit parameters of $n_o r_c^{3\beta} = 1.35 \pm 0.24$ cm$^{-3}$ kpc$^{3\beta}$ and $\beta = 0.50 \pm 0.03$ for the hot gas halo and negligible Local Bubble contribution. The OVII observations yield an unacceptable $\chi^{2}_{red}$ (dof) = 4.69 (645) for similar best-fit parameters, which is likely due to temperature or density variations in the Local Bubble. The OVIII fitting results imply hot gas masses of $M$(< 50 kpc) = $3.8_{-0.3}^{+0.3} \times 10^{9} M_{\odot}$ and $M$(< 250 kpc) = $4.3_{-0.8}^{+0.9} \times 10^{10} M_{\odot}$, accounting for $\lesssim$ 50% of the Milky Way's missing baryons. We also explore our results in the context of optical depth effects in the halo gas, the halo gas cooling properties, temperature and entropy gradients in the halo gas, and the gas metallicity distribution. The combination of absorption and emission line analyses implies a sub-solar gas metallicity that decreases with radius, but that also must be $\geq 0.3 Z_{\odot}$ to be consistent with the pulsar dispersion measure toward the LMC.Comment: 26 pages, 13 figures, Accepted to Ap

Randomized Benchmarking with Confidence

Randomized benchmarking is a promising tool for characterizing the noise in experimental implementations of quantum systems. In this paper, we prove that the estimates produced by randomized benchmarking (both standard and interleaved) for arbitrary Markovian noise sources are remarkably precise by showing that the variance due to sampling random gate sequences is small. We discuss how to choose experimental parameters, in particular the number and lengths of random sequences, in order to characterize average gate errors with rigorous confidence bounds. We also show that randomized benchmarking can be used to reliably characterize time-dependent Markovian noise (e.g., when noise is due to a magnetic field with fluctuating strength). Moreover, we identify a necessary property for time-dependent noise that is violated by some sources of non-Markovian noise, which provides a test for non-Markovianity.Comment: 31 pages, 1 figure. v2: clarifications, refs added. v3: typos fixed, results unchange