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arXiv:0804.0520 · Physical Review Letters

Quantum Multiscale Entanglement Renormalization Ansatz Channels

Vittorio Giovannetti, Simone Montangero, Rosario Fazio

Published Oct 30, 2008• 93 citations
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Abstract

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.

DOI
10.1103/physrevlett.101.180503

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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.
We show how to compute the critical exponents of one-dimensional quantum critical systems in the thermodynamic limit. The method is based on an iterative scheme applied to the multiscale entanglement renormalization Ansatz for the ground-state wave function. We test this scheme to compute the critical exponents of the Ising and XXZXXZXXZ model for which we can compare the method with the exact values. The agreement is at worst within few percent of the exact results already for moderate dimensions of the tensor indices.
In this paper, we review the properties of homogeneous multiscale entanglement renormalization ansatz (MERA) to describe quantum critical systems. We discuss in more detail our results for one-dimensional (1D) systems (the Ising and Heisenberg models) and present new data for the 2D Ising model. Together with the results for the critical exponents, we provide a detailed description of the numerical algorithm and a discussion of new optimization strategies. The relation between the critical properties of the system and the tensor structure of the MERA is expressed using the formalism of quantum channels, which we review and extend.
Homogeneous multiscale entanglement renormalization ansatz states have been recently introduced to describe quantum critical systems. Here we present an extensive analysis of the properties of such states by clarifying the definition of their transfer superoperator whose structure is studied within an informational theoretical approach. Explicit expressions for computing the expectation values of symmetric observables are given both in the case of finite size systems and in the thermodynamic limit of infinitely many particles.
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.
We show how to build a multiscale entanglement renormalization ansatz (MERA) representation of the ground state of a many-body Hamiltonian H by applying the recently proposed tensor network renormalization [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)] to the Euclidean time evolution operator e(-βH) for infinite β. This approach bypasses the costly energy minimization of previous MERA algorithms and, when applied to finite inverse temperature β, produces a MERA representation of a thermal Gibbs state. Our construction endows tensor network renormalization with a renormalization group flow in the space of wave functions and Hamiltonians (and not merely in the more abstract space of tensors) and extends the MERA formalism to classical statistical systems.