Simulation of Nonequilibrium Dynamics on a Quantum Computer

We present a hybrid quantum-classical algorithm for the time evolution of out-of-equilibrium thermal states. The method depends upon classically computing a sparse approximation to the density matrix, and then time-evolving each matrix element via the quantum computer. For this exploratory study, we investigate the time-dependent Heisenberg model with five spins on the Rigetti Forest quantum virtual machine and a one spin system on the Rigetti 8Q-Agave quantum processor.Comment: 4 pages, 2 Figures, fixed typ

General Methods for Digital Quantum Simulation of Gauge Theories

A general scheme is presented for simulating gauge theories, with matter fields, on a digital quantum computer. A Trotterized time-evolution operator that respects gauge symmetry is constructed, and a procedure for obtaining time-separated, gauge-invariant operators is detailed. We demonstrate the procedure on small lattices, including the simulation of a 2+1D non-Abelian gauge theory.Comment: 13 pages, 7 figures, v3 includes clarifying comments, additional data and additional references. Matched published versio

Spatially Localized Unstable Periodic Orbits

Using an innovative damped-Newton method, we report the first calculation of many distinct unstable periodic orbits (UPOs) of a large high-dimensional extensively chaotic partial differential equation. A majority of the UPOs turn out to be spatially localized in that time dependence occurs only on portions of the spatial domain. With a particular weighting of 127 UPOs, the Lyapunov fractal dimension D=8.8 can be estimated with a relative error of 2%. We discuss the implications of these spatially localized UPOs for understanding and controlling spatiotemporal chaos.Comment: 16 pages (total), 3 eps figures (Includes two new references and a new footnote) Submitted to Physical Review Letter

The Reactivation of Main-Belt Comet 324P/La Sagra (P/2010 R2)

We present observations using the Baade Magellan and Canada-France-Hawaii telescopes showing that main-belt comet 324P/La Sagra, formerly known as P/2010 R2, has become active again for the first time since originally observed to be active in 2010-2011. The object appears point-source-like in March and April 2015 as it approached perihelion (true anomaly of ~300 deg), but was ~1 mag brighter than expected if inactive, suggesting the presence of unresolved dust emission. Activity was confirmed by observations of a cometary dust tail in May and June 2015. We find an apparent net dust production rate of <0.1 kg/s during these observations. 324P is now the fourth main-belt comet confirmed to be recurrently active, a strong indication that its activity is driven by sublimation. It now has the largest confirmed active range of all likely main-belt comets, and also the most distant confirmed inbound activation point at R~2.8 AU. Further observations during the current active period will allow direct comparisons of activity strength with 324P's 2010 activity.Comment: 5 pages, 3 figures, accepted for publication in MNRAS Letter

Comment on "Optimal Periodic Orbits of Chaotic Systems"

In a recent Letter, Hunt and Ott argued that SHORT-period unstable periodic orbits (UPOs) would be the invariant sets associated with a chaotic attractor that are most likely to optimize the time average of some smooth scalar performance function. In this Comment, we show that their conclusion does not hold generally and that optimal time averages may specifically require long-period UPOs. This situation can arise when long-period UPOs are able to spend substantial amounts of time in a region of phase space that is close to large values of the performance function.Comment: One Page, 1 Figure, Double Column format. Submitted to Physical Review Letter

Suppressing Coherent Gauge Drift in Quantum Simulations

Simulations of field theories on noisy quantum computers must contend with errors introduced by that noise. For gauge theories, a large class of errors violate gauge symmetry, and thus may result in unphysical processes occurring in the simulation. We present a method, applicable to non-Abelian gauge theories, for suppressing coherent gauge drift errors through the repeated application of pseudorandom gauge transformation. In cases where the dominant errors are gauge-violating, we expect this method to be a practical way to improve the accuracy of NISQ-era simulations.Comment: 5 pages, 1 figur

Stationarity and Redundancy of Multichannel EEG Data Recorded During Generalized Tonic-Clonic Seizures

A prerequisite for applying some signal analysis methods to electroencephalographic (EEG) data is that the data be statistically stationary. We have investigated the stationarity of 21-electrode multivariate EEG data recorded from ten patients during generalized tonic-clonic (GTC) seizures elicited by electroconvulsive therapy (ECT). Stationarity was examined by calculating probability density functions (pdfs) and power spectra over small equal-length non-overlapping time windows and then by studying visually and quantitatively the evolution of these quantities over the duration of the seizures. Our analysis shows that most of the seizures had time intervals of at least a few seconds that were statistically stationary by several criteria and simultaneously for different electrodes, and that some leads were delayed in manifesting the statistical changes associated with seizure onset evident in other leads. The stationarity across electrodes was further examined by studying redundancy of the EEG leads and how that redundancy evolved over the course of the GTC seizures. Using several different measures, we found a substantial redundancy which suggests that fewer than 21 electrodes will likely suffice for extracting dynamical and clinical insights. The redundancy analysis also demonstrates for the first time posterior-to-anterior time delays in the mid-ictal region of GTC seizures, which suggests the existence of propagating waves. The implications of these results are discussed for understanding GTC seizures and ECT treatment.Comment: 44 pages, 10 pages of figure

Universal Dynamics of Heavy Operators in CFT2_2

We obtain an asymptotic formula for the average value of the operator product expansion coefficients of any unitary, compact two dimensional CFT with $c>1$. This formula is valid when one or more of the operators has large dimension or -- in the presence of a twist gap -- has large spin. Our formula is universal in the sense that it depends only on the central charge and not on any other details of the theory. This result unifies all previous asymptotic formulas for CFT$_2$ structure constants, including those derived from crossing symmetry of four point functions, modular covariance of torus correlation functions, and higher genus modular invariance. We determine this formula at finite central charge by deriving crossing kernels for higher genus crossing equations, which give analytic control over the structure constants even in the absence of exact knowledge of the conformal blocks. The higher genus modular kernels are obtained by sewing together the elementary kernels for four-point crossing and modular transforms of torus one-point functions. Our asymptotic formula is related to the DOZZ formula for the structure constants of Liouville theory, and makes precise the sense in which Liouville theory governs the universal dynamics of heavy operators in any CFT. The large central charge limit provides a link with 3D gravity, where the averaging over heavy states corresponds to a coarse-graining over black hole microstates in holographic theories. Our formula also provides an improved understanding of the Eigenstate Thermalization Hypothesis (ETH) in CFT$_2$, and suggests that ETH can be generalized to other kinematic regimes in two dimensional CFTs.Comment: 60 pages, 8 figures. v2: fixed typos, added sections 1.6 and 5 elaborating on the connection between the universal formula and Liouville theor

Deep Learning Beyond Lefschetz Thimbles

The generalized thimble method to treat field theories with sign problems requires repeatedly solving the computationally-expensive holomorphic flow equations. We present a machine learning technique to bypass this problem. The central idea is to obtain a few field configurations via the flow equations to train a feed-forward neural network. The trained network defines a new manifold of integration which reduces the sign problem and can be rapidly sampled. We present results for the $1+1$ dimensional Thirring model with Wilson fermions on sizable lattices. In addition to the gain in speed, the parameterization of the integration manifold we use avoids the "trapping" of Monte Carlo chains which plagues large-flow calculations, a considerable shortcoming of the previous attempts.Comment: 12 pages, 4 figure

Sigma models on quantum computers

We formulate a discretization of sigma models suitable for simulation by quantum computers. Space is substituted by a lattice, as usually done in lattice field theory, while the target space (a sphere) is replaced by the "fuzzy sphere", a construction well known from non-commutative geometry. Contrary to more naive discretizations of the sphere, in this construction the exact $O(3)$ symmetry is maintained, which suggests that the discretized model is in the same universality class as the continuum model. That would allow for continuum results to be obtained for very rough discretizations of the target space as long as the space discretization is made fine enough. The cost of performing time-evolution, measured as the number of CNOT operations necessary, is $12 L T/\Delta t$, where $L$ is the number of spatial sites, $T$ the maximum time extent and $\Delta t$ the time spacing.Comment: 5 pages, 2 figures, v2 includes additional references and refined discussio