6,493 research outputs found
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 , we prove that if the environment of a
single tensor from the network can be evaluated with computational cost
, then the environment of any other tensor from can be
evaluated with identical cost . Moreover, we describe how the set of
all single tensor environments from can be simultaneously
evaluated with fixed cost . 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
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
qTorch: The Quantum Tensor Contraction Handler
Classical simulation of quantum computation is necessary for studying the
numerical behavior of quantum algorithms, as there does not yet exist a large
viable quantum computer on which to perform numerical tests. Tensor network
(TN) contraction is an algorithmic method that can efficiently simulate some
quantum circuits, often greatly reducing the computational cost over methods
that simulate the full Hilbert space. In this study we implement a tensor
network contraction program for simulating quantum circuits using multi-core
compute nodes. We show simulation results for the Max-Cut problem on 3- through
7-regular graphs using the quantum approximate optimization algorithm (QAOA),
successfully simulating up to 100 qubits. We test two different methods for
generating the ordering of tensor index contractions: one is based on the tree
decomposition of the line graph, while the other generates ordering using a
straight-forward stochastic scheme. Through studying instances of QAOA
circuits, we show the expected result that as the treewidth of the quantum
circuit's line graph decreases, TN contraction becomes significantly more
efficient than simulating the whole Hilbert space. The results in this work
suggest that tensor contraction methods are superior only when simulating
Max-Cut/QAOA with graphs of regularities approximately five and below. Insight
into this point of equal computational cost helps one determine which
simulation method will be more efficient for a given quantum circuit. The
stochastic contraction method outperforms the line graph based method only when
the time to calculate a reasonable tree decomposition is prohibitively
expensive. Finally, we release our software package, qTorch (Quantum TensOR
Contraction Handler), intended for general quantum circuit simulation.Comment: 21 pages, 8 figure
Towards an Efficient Use of the BLAS Library for Multilinear Tensor Contractions
Mathematical operators whose transformation rules constitute the building
blocks of a multi-linear algebra are widely used in physics and engineering
applications where they are very often represented as tensors. In the last
century, thanks to the advances in tensor calculus, it was possible to uncover
new research fields and make remarkable progress in the existing ones, from
electromagnetism to the dynamics of fluids and from the mechanics of rigid
bodies to quantum mechanics of many atoms. By now, the formal mathematical and
geometrical properties of tensors are well defined and understood; conversely,
in the context of scientific and high-performance computing, many tensor-
related problems are still open. In this paper, we address the problem of
efficiently computing contractions among two tensors of arbitrary dimension by
using kernels from the highly optimized BLAS library. In particular, we
establish precise conditions to determine if and when GEMM, the kernel for
matrix products, can be used. Such conditions take into consideration both the
nature of the operation and the storage scheme of the tensors, and induce a
classification of the contractions into three groups. For each group, we
provide a recipe to guide the users towards the most effective use of BLAS.Comment: 27 Pages, 7 figures and additional tikz generated diagrams. Submitted
to Applied Mathematics and Computatio
The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems
We present a compendium of numerical simulation techniques, based on tensor
network methods, aiming to address problems of many-body quantum mechanics on a
classical computer. The core setting of this anthology are lattice problems in
low spatial dimension at finite size, a physical scenario where tensor network
methods, both Density Matrix Renormalization Group and beyond, have long proven
to be winning strategies. Here we explore in detail the numerical frameworks
and methods employed to deal with low-dimension physical setups, from a
computational physics perspective. We focus on symmetries and closed-system
simulations in arbitrary boundary conditions, while discussing the numerical
data structures and linear algebra manipulation routines involved, which form
the core libraries of any tensor network code. At a higher level, we put the
spotlight on loop-free network geometries, discussing their advantages, and
presenting in detail algorithms to simulate low-energy equilibrium states.
Accompanied by discussions of data structures, numerical techniques and
performance, this anthology serves as a programmer's companion, as well as a
self-contained introduction and review of the basic and selected advanced
concepts in tensor networks, including examples of their applications.Comment: 115 pages, 56 figure
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