Improving the efficiency of variational tensor network algorithms

We present several results relating to the contraction of generic tensor networks and discuss their application to the simulation of quantum many-body systems using variational approaches based upon tensor network states. Given a closed tensor network $\mathcal{T}$, we prove that if the environment of a single tensor from the network can be evaluated with computational cost $\kappa$, then the environment of any other tensor from $\mathcal{T}$ can be evaluated with identical cost $\kappa$. Moreover, we describe how the set of all single tensor environments from $\mathcal{T}$ can be simultaneously evaluated with fixed cost $3\kappa$. The usefulness of these results, which are applicable to a variety of tensor network methods, is demonstrated for the optimization of a Multi-scale Entanglement Renormalization Ansatz (MERA) for the ground state of a 1D quantum system, where they are shown to substantially reduce the computation time.Comment: 12 pages, 8 figures, RevTex 4.1, includes reference implementation. Software updated to v1.02: Resolved two scenarios in which multienv would generate errors for valid input

Symmetry-Protected Local Minima in Infinite DMRG

The infinite Density Matrix Renormalisation Group (iDMRG) algorithm is a highly successful numerical algorithm for the study of low-dimensional quantum systems, and is also frequently used to initialise the more popular finite DMRG algorithm. Implementations of both finite and infinite DMRG frequently incorporate support for the protection and exploitation of symmetries of the Hamiltonian. In common with other variational tensor network algorithms, convergence of iDMRG to the ground state is not guaranteed, with the risk that the algorithm may become stuck in a local minimum. In this paper I demonstrate the existence of a particularly harmful class of physically irrelevant local minima affecting both iDMRG and to a lesser extent also infinite Time-Evolving Block Decimation (iTEBD), for which the ground state is compatible with the protected symmetries of the Hamiltonian but cannot be reached using the conventional iDMRG or iTEBD algorithms. I describe a modified iDMRG algorithm which evades these local minima, and which also admits a natural interpretation on topologically ordered systems with a boundary.Comment: 13 pages, 9 figures, 1 table, RevTeX 4.1. New title, greatly expanded explanations, fixed some typos (incl. reference to equation in caption of Fig.3). Reversed orientation convention for arrow on accessory site to match arrows on physical sites: all site arrows are now inboun

Theory and Practice of Transactional Method Caching

Nowadays, tiered architectures are widely accepted for constructing large scale information systems. In this context application servers often form the bottleneck for a system's efficiency. An application server exposes an object oriented interface consisting of set of methods which are accessed by potentially remote clients. The idea of method caching is to store results of read-only method invocations with respect to the application server's interface on the client side. If the client invokes the same method with the same arguments again, the corresponding result can be taken from the cache without contacting the server. It has been shown that this approach can considerably improve a real world system's efficiency. This paper extends the concept of method caching by addressing the case where clients wrap related method invocations in ACID transactions. Demarcating sequences of method calls in this way is supported by many important application server standards. In this context the paper presents an architecture, a theory and an efficient protocol for maintaining full transactional consistency and in particular serializability when using a method cache on the client side. In order to create a protocol for scheduling cached method results, the paper extends a classical transaction formalism. Based on this extension, a recovery protocol and an optimistic serializability protocol are derived. The latter one differs from traditional transactional cache protocols in many essential ways. An efficiency experiment validates the approach: Using the cache a system's performance and scalability are considerably improved

Faster identification of optimal contraction sequences for tensor networks

The efficient evaluation of tensor expressions involving sums over multiple indices is of significant importance to many fields of research, including quantum many-body physics, loop quantum gravity, and quantum chemistry. The computational cost of evaluating an expression may depend strongly upon the order in which the index sums are evaluated, and determination of the operation-minimising contraction sequence for a single tensor network (single term, in quantum chemistry) is known to be NP-hard. The current preferred solution is an exhaustive search, using either an iterative depth-first approach with pruning or dynamic programming and memoisation, but these approaches are impractical for many of the larger tensor network Ansaetze encountered in quantum many-body physics. We present a modified search algorithm with enhanced pruning which exhibits a performance increase of several orders of magnitude while still guaranteeing identification of an optimal operation-minimising contraction sequence for a single tensor network. A reference implementation for MATLAB, compatible with the ncon() and multienv() network contractors of arXiv:1402.0939 and arXiv:1310.8023 respectively, is supplied.Comment: 25 pages, 12 figs, 2 tables, includes reference implementation of algorithm, v2.01. Update corrects the display of contraction sequences involving single-tensor traces (i.e. where an index in the input appears twice on the same tensor

Traffic noise exposure, education and annoyance: longitudinal experience from crosssectional surveys over time (1989-2004)

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Order-disorder transitions in a sheared many body system

Motivated by experiments on sheared suspensions that show a transition between ordered and disordered phases, we here study the long-time behavior of a sheared and overdamped 2-d system of particles interacting by repulsive forces. As a function of interaction strength and shear rate we find transitions between phases with vanishing and large single-particle diffusion. In the phases with vanishing single-particle diffusion, the system evolves towards regular lattices, usually on very slow time scales. Different lattices can be approached, depending on interaction strength and forcing amplitude. The disordered state appears in parameter regions where the regular lattices are unstable. Correlation functions between the particles reveal the formation of shear bands. In contrast to single particle densities, the spatially resolved two-particle correlation functions vary with time and allow to determine the phase within a period. As in the case of the suspensions, motion in the state with low diffusivity is essentially reversible, whereas in the state with strong diffusion it is not.Comment: 12 pages, 13 figures; Supplemental Movies: https://youtu.be/oFcrWo9Vs6E, https://youtu.be/tcowb7o05JQ, https://youtu.be/GkEUwycn7V4, https://youtu.be/k-XCo8CWFU

Matrix product states for anyonic systems and efficient simulation of dynamics

Matrix product states (MPS) have proven to be a very successful tool to study lattice systems with local degrees of freedom such as spins or bosons. Topologically ordered systems can support anyonic particles which are labeled by conserved topological charges and collectively carry non-local degrees of freedom. In this paper we extend the formalism of MPS to lattice systems of anyons. The anyonic MPS is constructed from tensors that explicitly conserve topological charge. We describe how to adapt the time-evolving block decimation (TEBD) algorithm to the anyonic MPS in order to simulate dynamics under a local and charge-conserving Hamiltonian. To demonstrate the effectiveness of anyonic TEBD algorithm, we used it to simulate (i) the ground state (using imaginary time evolution) of an infinite 1D critical system of (a) Ising anyons and (b) Fibonacci anyons both of which are well studied, and (ii) the real time dynamics of an anyonic Hubbard-like model of a single Ising anyon hopping on a ladder geometry with an anyonic flux threading each island of the ladder. Our results pertaining to (ii) give insight into the transport properties of anyons. The anyonic MPS formalism can be readily adapted to study systems with conserved symmetry charges, as this is equivalent to a specialization of the more general anyonic case.Comment: 18 pages, 15 figue