We describe a numerical many-body technique that is based on both tensor networks and quantum Monte Carlo. The variational ansatz is a tensor network that can harvest volume-law entanglement. It is constructed from a tensor train to which one applies a set of non-local operators that force several indices of the tensor train to represent the same physical index, hence its name -- replica tensor train (RTT). From the tensor network toolbox, it inherits the possibility to make linear combinations of these states and apply a certain class of operators. We can therefore find the ground-state of a local Hamiltonian in a purely algebraic way as in standard tensor network algorithms -- i.e. without using gradient descent methods. On the other hand, the volume-law structure forbids calculating physical observables directly. In much the same way as on a quantum computer where one can prepare a state but can only sample it at the end, here we have to use Markov Chain Monte Carlo to compute the observables. We further show that the approach can be extended to build Krylov-subspace ground-state methods within the variational manifold. We illustrate the different algorithms on a two-dimensional spin model with a transverse magnetic field, which can be solved by this approach at low computational cost.
We describe a numerical many-body technique that is based on both tensor networks and quantum Monte Carlo. The variational ansatz is a tensor network that can harvest volume-law entanglement. It is constructed from a tensor train to which one applies a set of non-local operators that force several indices of the tensor train to represent the same physical index, hence its name -- replica tensor train (RTT). From the tensor network toolbox, it inherits the possibility to make linear combinations of these states and apply a certain class of operators. We can therefore find the ground-state of a local Hamiltonian in a purely algebraic way as in standard tensor network algorithms -- i.e. without using gradient descent methods. On the other hand, the volume-law structure forbids calculating physical observables directly. In much the same way as on a quantum computer where one can prepare a state but can only sample it at the end, here we have to use Markov Chain Monte Carlo to compute the observables. We further show that the approach can be extended to build Krylov-subspace ground-state methods within the variational manifold. We illustrate the different algorithms on a two-dimensional spin model with a transverse magnetic field, which can be solved by this approach at low computational cost.
Monte Carlo sampling techniques have been proposed as a strategy to reduce the computational cost of contractions in tensor network approaches to solving many-body systems. Here, we put forward a variational Monte Carlo approach for the multiscale entanglement renormalization ansatz (MERA), which is a unitary tensor network. Two major adjustments are required compared to previous proposals with nonunitary tensor networks. First, instead of sampling over configurations of the original lattice, made of L sites, we sample over configurations of an effective lattice, which is made of just ln(L) sites. Second, the optimization of unitary tensors must account for their unitary character while being robust to statistical noise, which we accomplish with a modified steepest descent method within the set of unitary tensors. We demonstrate the performance of the variational Monte Carlo MERA approach in the relatively simple context of a finite quantum spin chain at criticality, and discuss future, more challenging applications, including two-dimensional systems.
Variational minimization of tensor network states enables the exploration of low energy states of lattice gauge theories. However, the exact numerical evaluation of high-dimensional tensor network states remains challenging in general. In [E. Zohar and J. I. Cirac, Phys. Rev. D 97, 034510 (2018)] it was shown how, by combining gauged Gaussian projected entangled pair states with a variational Monte Carlo procedure, it is possible to efficiently compute physical observables. In this paper we demonstrate how this approach can be used to investigate numerically the ground state of a lattice gauge theory. More concretely, we explicitly carry out the variational Monte Carlo procedure based on such contraction methods for a pure gauge Kogut-Susskind Hamiltonian with a Z3 gauge field in two spatial dimensions. This is a first proof of principle to the method, which provides an inherent way to increase the number of variational parameters and can be readily extended to systems with physical fermions.
It appears that the tensor network ansatz, while not quite complete, is an efficient coordinate system for the tiny subset of a many-body Hilbert space which can be realized as a low energy state of a local Hamiltonian.However, we don't fully understand precisely which phases are captured by the tensor network ansatz, how to compute their physical observables (even numerically), or how to compute a tensor network representation for a ground state given a microscopic Hamiltonian. These questions are algorithmic in nature, but their resolution is intimately related to understanding the nature of quantum entanglement in many-body systems.For this reason it is useful to compute the tensor network representation of various `model' wavefunctions representative of different phases of matter; this allows us to understand how the entanglement properties of each phase are expressed in the tensor network ansatz, and can serve as test cases for algorithm development.Condensed matter physics has many illuminating model wavefunctions, such as Laughlin's celebrated wave function for the fractional quantum Hall effect, the Bardeen-Cooper-Schrieffer wave function for superconductivity, and Anderson's resonating valence bond ansatz for spin liquids.This thesis presents some results on exact tensor network representations of these model wavefunctions. In addition, a tensor network representation is given for the time evolution operator of a long-range one-dimensional Hamiltonian, which allows one to numerically simulate the time evolution of power-law interacting spin chains as well as two-dimensional strips and cylinders.
Strongly correlated systems exhibit phenomena -- such as high-T_c superconductivity or the fractional quantum Hall effect -- that are not explicable by classical and semi-classical methods. Moreover, due to the exponential scaling of the associated Hilbert space, solving the proposed model Hamiltonians by brute-force numerical methods is bound to fail. Thus, it is important to develop novel numerical and analytical methods that can explain the physics in this regime. Tensor Network states are quantum many-body states that help to overcome some of these difficulties by defining a family of states that depend only on a small number of parameters. Their use is twofold: they are used as variational ansatzes in numerical algorithms as well as providing a framework to represent a large class of exactly solvable models that are believed to represent all possible phases of matter. The present thesis investigates mathematical properties of these states thus deepening the understanding of how and why Tensor Networks are suitable for the description of quantum many-body systems. It is believed that tensor networks can represent ground states of local Hamiltonians, but how good is this representation? This question is of fundamental importance as variational algorithms based on tensor networks can only perform well if any ground state can be approximated efficiently in such a way. While any state can be written as a tensor network state, the number of parameters needed for the description might be too large. This is not the case for one-dimensional systems: only a few parameters are required to have a good approximation of their ground states; that, in turn, allows for numerical algorithms based on tensor networks performing well. The situation in two dimensions is somewhat more complicated, but it is known that ground states of local Hamiltonians can be expressed as tensor networks with sub-exponentially many parameters. In the present thesis, we improve on these existing bounds strengthening the claim that the language of tensor networks is suitable to describe many-body systems. Another central question is how symmetries of the system such as translational invariance, time-reversal symmetry or local unitary symmetry can be reflected in tensor networks. This question is important as systems appearing in nature might intrinsically possess certain symmetries; on one hand, understanding these symmetries simplifies the description of these systems. On the other hand, the presence of symmetries leads to the appearance of novel phases -- symmetry-protected topological (SPT) order, -- and tensor networks provide the right language to classify these phases. In one dimension and for certain classes of two-dimensional tensor networks (states generated by so-called injective tensors) it is well understood how symmetries of the state can be described. A general framework, however, has yet to be developed. In the present thesis, we contribute to the development of the theory in two ways. We first investigate the question for injective tensors, and generalize the existing proof for any geometry including the hyperbolic geometry used in the AdS/CFT correspondence. Second, we introduce a class of tensor network states that include previously known examples of states exhibiting SPT order. We show how symmetries are reflected in these states thus deepening the understanding of SPT order in two dimensions.
We introduce a novel method of efficiently simulating the non-equilibrium steady state of large many-body open quantum systems with highly non-local interactions, based on a variational Monte Carlo optimization of a matrix product operator ansatz. Our approach outperforms and offers several advantages over comparable algorithms, such as an improved scaling of the computational cost with respect to the bond dimension for periodic systems. We showcase the versatility of our approach by studying the phase diagrams and correlation functions of the dissipative quantum Ising model with collective dephasing and long-ranged power law interactions for spin chains of up to N=100 spins.
Tensor network states form a variational ansatz class widely used, both analytically and numerically, in the study of quantum many-body systems. It is known that if the underlying graph contains a cycle, e.g., as in projected entangled pair states, then the set of tensor network states of given bond dimension is not closed. Its closure is the tensor network variety. Recent work has shown that states on the boundary of this variety can yield more efficient representations for states of physical interest, but it remained unclear how to systematically find and optimize over such representations. We address this issue by defining an ansatz class of states that includes states at the boundary of the tensor network variety of given bond dimension. We show how to optimize over this class in order to find ground states of local Hamiltonians by only slightly modifying standard algorithms and code for tensor networks. We apply this method to different models and observe favorable energies and runtimes when compared with standard tensor network methods.