24,309 research outputs found
Tensor network states and geometry
Tensor network states are used to approximate ground states of local
Hamiltonians on a lattice in D spatial dimensions. Different types of tensor
network states can be seen to generate different geometries. Matrix product
states (MPS) in D=1 dimensions, as well as projected entangled pair states
(PEPS) in D>1 dimensions, reproduce the D-dimensional physical geometry of the
lattice model; in contrast, the multi-scale entanglement renormalization ansatz
(MERA) generates a (D+1)-dimensional holographic geometry. Here we focus on
homogeneous tensor networks, where all the tensors in the network are copies of
the same tensor, and argue that certain structural properties of the resulting
many-body states are preconditioned by the geometry of the tensor network and
are therefore largely independent of the choice of variational parameters.
Indeed, the asymptotic decay of correlations in homogeneous MPS and MERA for
D=1 systems is seen to be determined by the structure of geodesics in the
physical and holographic geometries, respectively; whereas the asymptotic
scaling of entanglement entropy is seen to always obey a simple boundary law --
that is, again in the relevant geometry. This geometrical interpretation offers
a simple and unifying framework to understand the structural properties of, and
helps clarify the relation between, different tensor network states. In
addition, it has recently motivated the branching MERA, a generalization of the
MERA capable of reproducing violations of the entropic boundary law in D>1
dimensions.Comment: 18 pages, 18 figure
On the geometry of Tensor Network States of Grids
We discuss the geometry of a class of tensor network states, called projected
entangled pair states in the Physics literature. We provide initial results
towards a question of Verstraete and Rizzi regarding the tensor network state
of an grid; we partially answer the question for a
grid. We also study the grid sitting on a torus and provide
initial results towards understanding the Zariski closure of the set of tensor
network states associated to this graph. Finally, we give explicit tensors that
provide optimal (using currently available methods) bounds on the border rank
of a generic tensor in the tensor network state of the grid.Comment: 12 pages, 5 figure
Holographic Geometries of one-dimensional gapped quantum systems from Tensor Network States
We investigate a recent conjecture connecting the AdS/CFT correspondence and
entanglement renormalization tensor network states (MERA). The proposal
interprets the tensor connectivity of the MERA states associated to quantum
many body systems at criticality, in terms of a dual holographic geometry which
accounts for the qualitative aspects of the entanglement and the correlations
in these systems. In this work, some generic features of the entanglement
entropy and the two point functions in the ground state of one dimensional
gapped systems are considered through a tensor network state. The tensor
network is builded up as an hybrid composed by a finite number of MERA layers
and a matrix product state (MPS) acting as a cap layer. Using the holographic
formula for the entanglement entropy, here it is shown that an asymptotically
AdS metric can be associated to the hybrid MERA-MPS state. The metric is
defined by a function that manages the growth of the minimal surfaces near the
capped region of the geometry. Namely, it is shown how the behaviour of the
entanglement entropy and the two point correlators in the tensor network,
remains consistent with a geometric computation which only depends on this
function. From these observations, an explicit connection between the
entanglement structure of the tensor network and the function which defines the
geometry is provided.Comment: 25 pages, 6 Postscript figures. The v2 is a major revisioned version
with a new title and results to be published in JHE
Groundstate fidelity phase diagram of the fully anisotropic two-leg spin-1/2 XXZ ladder
The fully anisotropic two-leg spin-1/2 ladder model is studied in terms
of an algorithm based on the tensor network representation of quantum many-body
states as an adaptation of projected entangled pair states to the geometry of
translationally invariant infinite-size quantum spin ladders. The tensor
network algorithm provides an effective method to generate the groundstate wave
function, which allows computation of the groundstate fidelity per lattice
site, a universal marker to detect phase transitions in quantum many-body
systems. The groundstate fidelity is used in conjunction with local order and
string order parameters to systematically map out the groundstate phase diagram
of the ladder model. The phase diagram exhibits a rich diversity of quantum
phases. These are the ferromagnetic, stripe ferromagnetic, rung singlet, rung
triplet, Nel, stripe Nel and Haldane phases,
along with the two phases and .Comment: 23 pages, 15 figures. Revised version with additional text and
figures. New Forma
Tensor network decompositions in the presence of a global symmetry
Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. We discuss how to incorporate a global symmetry, given by a compact, completely reducible group G, in tensor network decompositions and algorithms. This is achieved by considering tensors that are invariant under the action of the group G. Each symmetric tensor decomposes into two types of tensors: degeneracy tensors, containing all the degrees of freedom, and structural tensors, which only depend on the symmetry group. In numerical calculations, the use of symmetric tensors ensures the preservation of the symmetry, allows selection of a specific symmetry sector, and significantly reduces computational costs. On the other hand, the resulting tensor network can be interpreted as a superposition of exponentially many spin networks. Spin networks are used extensively in loop quantum gravity, where they represent states of quantum geometry. Our work highlights their importance in the context of tensor network algorithms as well, thus setting the stage for cross-fertilization between these two areas of research
Consistency Conditions for an AdS/MERA Correspondence
The Multi-scale Entanglement Renormalization Ansatz (MERA) is a tensor
network that provides an efficient way of variationally estimating the ground
state of a critical quantum system. The network geometry resembles a
discretization of spatial slices of an AdS spacetime and "geodesics" in the
MERA reproduce the Ryu-Takayanagi formula for the entanglement entropy of a
boundary region in terms of bulk properties. It has therefore been suggested
that there could be an AdS/MERA correspondence, relating states in the Hilbert
space of the boundary quantum system to ones defined on the bulk lattice. Here
we investigate this proposal and derive necessary conditions for it to apply,
using geometric features and entropy inequalities that we expect to hold in the
bulk. We show that, perhaps unsurprisingly, the MERA lattice can only describe
physics on length scales larger than the AdS radius. Further, using the
covariant entropy bound in the bulk, we show that there are no conventional
MERA parameters that completely reproduce bulk physics even on super-AdS
scales. We suggest modifications or generalizations of this kind of tensor
network that may be able to provide a more robust correspondence.Comment: 38 pages, 9 figure
Comb tensor networks
In this paper we propose a special type of a tree tensor network that has the
geometry of a comb---a 1D backbone with finite 1D teeth projecting out from it.
This tensor network is designed to provide an effective description of higher
dimensional objects with special limited interactions, or, alternatively,
one-dimensional systems composed of complicated zero-dimensional objects. We
provide details on the best numerical procedures for the proposed network,
including an algorithm for variational optimization of the wave-function as a
comb tensor network, and the transformation of the comb into a matrix product
state. We compare the complexity of using a comb versus alternative matrix
product state representations using density matrix renormalization group (DMRG)
algorithms. As an application, we study a spin-1 Heisenberg model system which
has a comb geometry. In the case where the ends of the teeth are terminated by
spin-1/2 spins, we find that Haldane edge states of the teeth along the
backbone form a critical spin-1/2 chain, whose properties can be tuned by the
coupling constant along the backbone. By adding next-nearest-neighbor
interactions along the backbone, the comb can be brought into a gapped phase
with a long-range dimerization along the backbone. The critical and dimerized
phases are separated by a Kosterlitz-Thouless phase transition, the presence of
which we confirm numerically. Finally, we show that when the teeth contain an
odd number of spins and are not terminated by spin-1/2's, a special type of
comb edge states emerge.Comment: 14 pages, 17 figure
Verifying Random Quantum Circuits with Arbitrary Geometry Using Tensor Network States Algorithm
The ability to efficiently simulate random quantum circuits using a classical
computer is increasingly important for developing Noisy Intermediate-Scale
Quantum devices. Here we present a tensor network states based algorithm
specifically designed to compute amplitudes for random quantum circuits with
arbitrary geometry. Singular value decomposition based compression together
with a two-sided circuit evolution algorithm are used to further compress the
resulting tensor network. To further accelerate the simulation, we also propose
a heuristic algorithm to compute the optimal tensor contraction path. We
demonstrate that our algorithm is up to orders of magnitudes faster than
the Schdinger-Feynman algorithm for verifying random quantum
circuits on the -qubit Sycamore processor, with circuit depths below .
We also simulate larger random quantum circuits up to qubits, showing
that this algorithm is an ideal tool to verify relatively shallow quantum
circuits on near-term quantum computers.Comment: 5 pages, 3 figures, 1 tabl
Fractional quantum Hall effect in the interacting Hofstadter model via tensor networks
We show via tensor network methods that the Harper-Hofstadter Hamiltonian for
hard-core bosons on a square geometry supports a topological phase realizing
the fractional quantum Hall effect on the lattice. We address the
robustness of the ground state degeneracy and of the energy gap, measure the
many-body Chern number, and characterize the system using Green functions,
showing that they decay algebraically at the edges of open geometries,
indicating the presence of gapless edge modes. Moreover, we estimate the
topological entanglement entropy by taking a combination of lattice
bipartitions that reproduces the topological structure of the original
proposals by Kitaev and Preskill, and Levin and Wen. The numerical results show
that the topological contribution is compatible with the expected value . Our results provide extensive evidence that FQH states are within reach
of state-of-the-art cold atom experiments.Comment: 9 pages, 11 figures; close to published versio
Space-time random tensor networks and holographic duality
In this paper we propose a space-time random tensor network approach for
understanding holographic duality. Using tensor networks with random link
projections, we define boundary theories with interesting holographic
properties, such as the Renyi entropies satisfying the covariant
Hubeny-Rangamani-Takayanagi formula, and operator correspondence with local
reconstruction properties. We also investigate the unitarity of boundary theory
in spacetime geometries with Lorenzian signature. Compared with the spatial
random tensor networks, the space-time generalization does not require a
particular time slicing, and provides a more covariant family of microscopic
models that may help us to understand holographic duality.Comment: 31 pages, 9 figure
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