Holographic local quench and effective complexity

We study the evolution of holographic complexity of pure and mixed states in $1+1$-dimensional conformal field theory following a local quench using both the "complexity equals volume" (CV) and the "complexity equals action" (CA) conjectures. We compare the complexity evolution to the evolution of entanglement entropy and entanglement density, discuss the Lloyd computational bound and demonstrate its saturation in certain regimes. We argue that the conjectured holographic complexities exhibit some non-trivial features indicating that they capture important properties of what is expected to be effective (or physical) complexity.Comment: 33 pages, 19 figures; v2: typos corrected; 35 pages, references added, new appendix. Version to match published in JHE

Managing urban socio-technical change? Comparing energy technology controversies in three European contexts

A {\em local graph partitioning algorithm} finds a set of vertices with small conductance (i.e. a sparse cut) by adaptively exploring part of a large graph $G$, starting from a specified vertex. For the algorithm to be local, its complexity must be bounded in terms of the size of the set that it outputs, with at most a weak dependence on the number $n$ of vertices in $G$. Previous local partitioning algorithms find sparse cuts using random walks and personalized PageRank. In this paper, we introduce a randomized local partitioning algorithm that finds a sparse cut by simulating the {\em volume-biased evolving set process}, which is a Markov chain on sets of vertices. We prove that for any set of vertices $A$ that has conductance at most $\phi$, for at least half of the starting vertices in $A$ our algorithm will output (with probability at least half), a set of conductance $O(\phi^{1/2} \log^{1/2} n)$. We prove that for a given run of the algorithm, the expected ratio between its computational complexity and the volume of the set that it outputs is $O(\phi^{-1/2} polylog(n))$. In comparison, the best previous local partitioning algorithm, due to Andersen, Chung, and Lang, has the same approximation guarantee, but a larger ratio of $O(\phi^{-1} polylog(n))$ between the complexity and output volume. Using our local partitioning algorithm as a subroutine, we construct a fast algorithm for finding balanced cuts. Given a fixed value of $\phi$, the resulting algorithm has complexity $O((m+n\phi^{-1/2}) polylog(n))$ and returns a cut with conductance $O(\phi^{1/2} \log^{1/2} n)$ and volume at least $v_{\phi}/2$, where $v_{\phi}$ is the largest volume of any set with conductance at most $\phi$.Comment: 20 pages, no figure

Reconstructing black hole exteriors and interiors using entanglement and complexity

Based on the AdS/CFT correspondence, we study how to reconstruct bulk spacetime metrics by various quantum information measures on the boundary field theories, which include entanglement entropy, mutual information, entanglement of purification, and computational complexity according to the proposals of complexity=volume 2.0 and complexity=generalized volume. We present several reconstruction methods, all of which are free of UV divergence and most of which are driven by the derivatives of the measures with respect to the boundary scales. We illustrate that the exterior and interior of a black hole can be reconstructed using the measures of spatial entanglement and time-evolved complexity, respectively. We find that these measures always probe the spacetime in a local way: reconstructing the bulk metric in different radial positions requires the information at different boundary scales. We also show that the reconstruction method using complexity=volume 2.0 proposal is the simplest and has the strongest locality.Comment: 25 pages, 18 figures, 1 tabl

Efficient wideband electromagnetic scattering computation for frequency dependent lossy dielectrics using WCAWE

This paper presents a model order reduction algorithm for the volume electric field integral equation (EFIE) formulation, that achieves fast and accurate frequency sweep calculations of electromagnetic wave scattering. An inhomogeneous, two-dimensional, lossy dielectric object whose material is characterized by a complex permittivity which varies with frequency is considered. The variation in the dielectric properties of the ceramic BaxLa4Ti 2+xO 12+3x in the <1 GHz frequency range is investigated for various values of x in a frequency sweep analysis. We apply the well-conditioned asymptotic waveform evaluation (WCAWE) method to circumvent the computational complexity associated with the numerical solution of such formulations. A multipoint automatic WCAWE method is also demonstrated which can produce an accurate solution over a much broader bandwidth. Several numerical examples are given on order to illustrate the accuracy and robustness of the proposed methods

Spatio-angular Minimum-variance Tomographic Controller for Multi-Object Adaptive Optics systems

Multi-object astronomical adaptive-optics (MOAO) is now a mature wide-field observation mode to enlarge the adaptive-optics-corrected field in a few specific locations over tens of arc-minutes. The work-scope provided by open-loop tomography and pupil conjugation is amenable to a spatio-angular Linear-Quadratic Gaussian (SA-LQG) formulation aiming to provide enhanced correction across the field with improved performance over static reconstruction methods and less stringent computational complexity scaling laws. Starting from our previous work [1], we use stochastic time-progression models coupled to approximate sparse measurement operators to outline a suitable SA-LQG formulation capable of delivering near optimal correction. Under the spatio-angular framework the wave-fronts are never explicitly estimated in the volume,providing considerable computational savings on 10m-class telescopes and beyond. We find that for Raven, a 10m-class MOAO system with two science channels, the SA-LQG improves the limiting magnitude by two stellar magnitudes when both Strehl-ratio and Ensquared-energy are used as figures of merit. The sky-coverage is therefore improved by a factor of 5.Comment: 30 pages, 7 figures, submitted to Applied Optic

Quantum Entanglement Phase Transitions and Computational Complexity: Insights from Ising Models

In this paper, we construct 2-dimensional bipartite cluster states and perform single-qubit measurements on the bulk qubits. We explore the entanglement scaling of the unmeasured 1-dimensional boundary state and show that under certain conditions, the boundary state can undergo a volume-law to an area-law entanglement transition driven by variations in the measurement angle. We bridge this boundary state entanglement transition and the measurement-induced phase transition in the non-unitary 1+1-dimensional circuit via the transfer matrix method. We also explore the application of this entanglement transition on the computational complexity problems. Specifically, we establish a relation between the boundary state entanglement transition and the sampling complexity of the bipartite $2$d cluster state, which is directly related to the computational complexity of the corresponding Ising partition function with complex parameters. By examining the boundary state entanglement scaling, we numerically identify the parameter regime for which the $2$d quantum state can be efficiently sampled, which indicates that the Ising partition function can be evaluated efficiently in such a region