81 research outputs found
Thermofield-based chain mapping approach for open quantum systems
We consider a thermofield approach to analyze the evolution of an open
quantum system coupled to an environment at finite temperature. In this
approach, the finite temperature environment is exactly mapped onto two virtual
environments at zero temperature. These two environments are then unitarily
transformed into two different chains of oscillators, leading to a one
dimensional structure that can be numerically studied using tensor network
techniques
Tensor Network Algorithms: a Route Map
Tensor networks provide extremely powerful tools for the study of complex
classical and quantum many-body problems. Over the last two decades, the
increment in the number of techniques and applications has been relentless, and
especially the last ten years have seen an explosion of new ideas and results
that may be overwhelming for the newcomer. This short review introduces the
basic ideas, the best established methods and some of the most significant
algorithmic developments that are expanding the boundaries of the tensor
network potential. The goal is to help the reader not only appreciate the many
possibilities offered by tensor networks, but also find their way through
state-of-the-art codes, their applicability and some avenues of ongoing
progress.Comment: Review paper. To appear (after edition) in Annu. Rev. Condens. Matter
Phys. v2 fixes a problem with the placement of figure
Wigner function for a particle in an infinite lattice
We study the Wigner function for a quantum system with a discrete, infinite
dimensional Hilbert space, such as a spinless particle moving on a one
dimensional infinite lattice. We discuss the peculiarities of this scenario and
of the associated phase space construction, propose a meaningful definition of
the Wigner function in this case, and characterize the set of pure states for
which it is non-negative. We propose a measure of non-classicality for states
in this system which is consistent with the continuum limit. The prescriptions
introduced here are illustrated by applying them to localized and Gaussian
states, and to their superpositions.Comment: 19 pages (single column), 7 figure
Variational Matrix Product Operators for the Steady State of Dissipative Quantum Systems
We present a new variational method, based on the matrix product operator
(MPO) ansatz, for finding the steady state of dissipative quantum chains
governed by master equations of the Lindblad form. Instead of requiring an
accurate representation of the system evolution until the stationary state is
attained, the algorithm directly targets the final state, thus allowing for a
faster convergence when the steady state is a MPO with small bond dimension.
Our numerical simulations for several dissipative spin models over a wide range
of parameters illustrate the performance of the method and show that indeed the
stationary state is often well described by a MPO of very moderate dimensions.Comment: Accepted versio
Unifying Projected Entangled Pair States contractions
The approximate contraction of a Projected Entangled Pair States (PEPS)
tensor network is a fundamental ingredient of any PEPS algorithm, required for
the optimization of the tensors in ground state search or time evolution, as
well as for the evaluation of expectation values. An exact contraction is in
general impossible, and the choice of the approximating procedure determines
the efficiency and accuracy of the algorithm. We analyze different previous
proposals for this approximation, and show that they can be understood via the
form of their environment, i.e. the operator that results from contracting part
of the network. This provides physical insight into the limitation of various
approaches, and allows us to introduce a new strategy, based on the idea of
clusters, that unifies previous methods. The resulting contraction algorithm
interpolates naturally between the cheapest and most imprecise and the most
costly and most precise method. We benchmark the different algorithms with
finite PEPS, and show how the cluster strategy can be used for both the tensor
optimization and the calculation of expectation values. Additionally, we
discuss its applicability to the parallelization of PEPS and to infinite
systems (iPEPS).Comment: 28 pages, 15 figures, accepted versio
Algorithms for finite Projected Entangled Pair States
Projected Entangled Pair States (PEPS) are a promising ansatz for the study
of strongly correlated quantum many-body systems in two dimensions. But due to
their high computational cost, developing and improving PEPS algorithms is
necessary to make the ansatz widely usable in practice. Here we analyze several
algorithmic aspects of the method. On the one hand, we quantify the connection
between the correlation length of the PEPS and the accuracy of its approximate
contraction, and discuss how purifications can be used in the latter. On the
other, we present algorithmic improvements for the update of the tensor that
introduce drastic gains in the numerical conditioning and the efficiency of the
algorithms. Finally, the state-of-the-art general PEPS code is benchmarked with
the Heisenberg and quantum Ising models on lattices of up to
sites.Comment: 18 pages, 20 figures, accepted versio
Quantum simulation of the Schwinger model: A study of feasibility
We analyze some crucial questions regarding the practical feasibility of
quantum simulation for lattice gauge models. Our analysis focuses on two models
suitable for the quantum simulation of the Schwinger Hamiltonian, or QED in 1+1
dimensions, which we investigate numerically using tensor networks. In
particular, we explore the effect of representing the gauge degrees of freedom
with finite-dimensional systems and show that the results converge rapidly;
thus even with small dimensions it is possible to obtain a reasonable accuracy.
We also discuss the time scales required for the adiabatic preparation of the
interacting vacuum state and observe that for a suitable ramping of the
interaction the required time is almost insensitive to the system size and the
dimension of the physical systems. Finally, we address the possible presence of
noninvariant terms in the Hamiltonian that is realized in the experiment and
show that for low levels of noise it is still possible to achieve a good
precision for some ground-state observables, even if the gauge symmetry is not
exact in the implemented model.Comment: 10 pages, 10 figures, published versio
How much entanglement is needed to reduce the energy variance?
We explore the relation between the entanglement of a pure state and its
energy variance for a local one dimensional Hamiltonian, as the system size
increases. In particular, we introduce a construction which creates a matrix
product state of arbitrarily small energy variance for spins,
with bond dimension scaling as , where is a
constant. This implies that a polynomially increasing bond dimension is enough
to construct states with energy variance that vanishes with the inverse of the
logarithm of the system size. We run numerical simulations to probe the
construction on two different models, and compare the local reduced density
matrices of the resulting states to the corresponding thermal equilibrium. Our
results suggest that the spatially homogeneous states with logarithmically
decreasing variance, which can be constructed efficiently, do converge to the
thermal equilibrium in the thermodynamic limit, while the same is not true if
the variance remains constant.Comment: small changes to fix typos and bibliographic reference
Using matrix product states to study the dynamical large deviations of kinetically constrained models
Here we demonstrate that tensor network techniques | originally devised for the analysis of quantum many-body problems | are well suited for the detailed study of rare event statistics in kinetically constrained models (KCMs). As concrete examples we consider the Fredrickson- Andersen and East models, two paradigmatic KCMs relevant to the modelling of glasses. We show how variational matrix product states allow to numerically approximate | systematically and with high accuracy | the leading eigenstates of the tilted dynamical generators which encode the large deviation statistics of the dynamics. Via this approach we can study system sizes beyond what is possible with other methods, allowing us to characterise in detail the _nite size scaling of the trajectory-space phase transition of these models, the behaviour of spectral gaps, and the spatial structure and \entanglement" properties of dynamical phases. We discuss the broader implications of our results
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