Tensor network states have proven to be powerful tools for investigating quantum many-body systems. Here, the authors propose a tensor network method, based on an adaptation of entanglement renormalization, for investigating strongly disordered systems. The method makes use of the strong disorder renormalization group to determine the order in which lattice sites are coarse grained, which sets the overall structure of the corresponding tensor network ansatz, before optimization using variational energy minimization. This approach leads to a new class of efficiently contractible tensor network ansatz for 1D systems, which may be understood as a generalization of the multiscale entanglement renormalization ansatz for disordered systems.
Tensor network states have proven to be powerful tools for investigating quantum many-body systems. Here, the authors propose a tensor network method, based on an adaptation of entanglement renormalization, for investigating strongly disordered systems. The method makes use of the strong disorder renormalization group to determine the order in which lattice sites are coarse grained, which sets the overall structure of the corresponding tensor network ansatz, before optimization using variational energy minimization. This approach leads to a new class of efficiently contractible tensor network ansatz for 1D systems, which may be understood as a generalization of the multiscale entanglement renormalization ansatz for disordered systems.
Tensor network (TN) states, including entanglement renormalization (ER), can encompass a wider variety of entangled states. When the entanglement structure of the quantum state of interest is nonuniform in real space, accurately representing the state with a limited number of degrees of freedom hinges on appropriately configuring the TN to align with the entanglement pattern. However, a proposal has yet to show a structural search of ER due to its high computational cost and the lack of flexibility in its algorithm. In this study, we conducted an optimal structural search of TN, including ER, based on the reconstruction of their local structures with respect to variational energy. First, we demonstrated that our algorithm for the spin- 1 / 2 tetramer singlets model could calculate exact ground energy using the multiscale entanglement renormalization ansatz (MERA) structure as an initial TN structure. Subsequently, we applied our algorithm to the random XY models with the two initial structures: MERA and the suitable structure underlying the strong disordered renormalization group. We found that, in both cases, our algorithm achieves improvements in variational energy, fidelity, and entanglement entropy. The degree of improvement in these quantities is superior in the latter case compared to the former, suggesting that utilizing an existing TN design method as a preprocessing step is important for maximizing our algorithm's performance. Published by the American Physical Society 2024
In this thesis, we use the new numerical method multiscale entanglement
renormalization ansatz (MERA) to study the one dimensional spin system. This
method, MERA, utilizes a network of tensors to represent quantum many-body
systems on a lattice. Other examples of tensor network state are matrix product
states (MPSs) for 1D systems, tree tensor networks (TTNs) for systems with a
tree shape, and projected entangled-pair states (PEPSs) for 2D systems and be-
yond. The three structures dier in the graph that denes how the tensors are
interconnected into a network: The graphs for MPSs, TTNs, and 2D PEPSs are,
respectively, a chain, a tree, and a 2D lattice. Importantly, from these tensor
networks the expectation value of local observables can be computed eciently.
Developing algorithms to simulate quantum many-body systems is important
for understanding and studying the physical principles. And these may become
powerful tools for the analysis of quantum computation and quantum information
theory. But in general ecient simulation is dicult, due to the fundamental
quantum physical laws.
The main method we have studied and worked on is multiscale entangle-
ment renormalization ansatz (MERA) which is a variational ansatz for many-
body states. MERA is a tree tensor network for 1D, 2D systems and beyond.
We have developed code of ternary-MERA to compute the ground state energy,
expected values of local observables, and correlators for 1D lattice systems. In
a nutshell, MERA is an algorithm for entanglement renormalization which is a
numerical technique based on locally reorganizing the Hilbert space of a quantum
many-body system with the aim to reduce the amount of entanglement in its wave
function. We use this method to study 1D Ising model, XXZ model, and J1 - J2
model, and then comparing the numerical results with exact diagonalization (ED)
and density matrix renormalization group (DMRG).
We propose an entanglement-based algorithm of the tensor-network strong-disorder renormalization group (tSDRG) method for quantum spin systems with quenched randomness. In contrast to the previous tSDRG algorithm based on the energy spectrum of renormalized block Hamiltonians, we directly utilize the entanglement structure associated with the blocks to be renormalized. We examine accuracy of the algorithm for the random antiferromagnetic Heisenberg models on one-dimensional, triangular, and square lattices. We then find that the entanglement-based tSDRG achieves better accuracy than the previous one for the square-lattice model with weak randomness, while it is less efficient for the one-dimensional and triangular-lattice models particularly in the strong-randomness region. The theoretical background and possible improvements of the algorithm are also discussed.
In a recent contribution [P. Corboz, G. Evenbly, F. Verstraete, and G. Vidal, arXiv:0904.4151 (unpublished)] entanglement renormalization was generalized to fermionic lattice systems in two spatial dimensions. Entanglement renormalization is a real-space coarse-graining transformation for lattice systems that produces a variational ansatz, the multiscale entanglement renormalization ansatz (MERA), for the ground states of local Hamiltonians. In this paper we describe in detail the fermionic version of the MERA formalism and algorithm. Starting from the bosonic MERA, which can be regarded both as a quantum circuit or in relation to a coarse-graining transformation, we indicate how the scheme needs to be modified to simulate fermions. To confirm the validity of the approach, we present benchmark results for free and interacting fermions on a square lattice with sizes between $6\ifmmode\times\else\texttimes\fi{}6$ and $162\ifmmode\times\else\texttimes\fi{}162$ and with periodic boundary conditions. The present formulation of the approach applies to generic tensor network algorithms.
Tensor network representations of many-body quantum systems can be described in terms of quantum channels. We focus on channels associated with the multiscale entanglement renormalization ansatz tensor network that has been recently introduced to efficiently describe critical systems. Our approach allows us to compute the multiscale entanglement renormalization ansatz correspondent to the thermodynamical limit of a critical system introducing a transfer matrix formalism, and to relate the system critical exponents to the convergence rates of the associated channels.
A method to study strongly interacting quantum many-body systems at and away from criticality is proposed. The method is based on a MERA-like tensor network that can be efficiently and reliably contracted on a noisy quantum computer using a number of qubits that is much smaller than the system size. We prove that the outcome of the contraction is stable to noise and that the estimated energy upper bounds the ground state energy. The stability, which we numerically substantiate, follows from the positivity of operator scaling dimensions under renormalization group flow. The variational upper bound follows from a particular assignment of physical qubits to different locations of the tensor network plus the assumption that the noise model is local. We postulate a scaling law for how well the tensor network can approximate ground states of lattice regulated conformal field theories in d spatial dimensions and provide evidence for the postulate. Under this postulate, a O(logd(1/δ))-qubit quantum computer can prepare a valid quantum-mechanical state with energy density δ above the ground state. In the presence of noise, δ=O(εlogd+1(1/ε)) can be achieved, where ε is the noise strength.