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arXiv:1602.01166 · Physical Review Letters

Entanglement Renormalization and Wavelets

Glen Evenbly, Steven R. White

Published Apr 6, 2016• 79 citations
arXivHTMLPDF

Abstract

We establish a precise connection between discrete wavelet transforms and entanglement renormalization, a real-space renormalization group transformation for quantum systems on the lattice, in the context of free particle systems. Specifically, we employ Daubechies wavelets to build approximations to the ground state of the critical Ising model, then demonstrate that these states correspond to instances of the multiscale entanglement renormalization ansatz (MERA), producing the first known analytic MERA for critical systems.

DOI
10.1103/physrevlett.116.140403

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We establish a precise connection between discrete wavelet transforms and entanglement renormalization, a real-space renormalization group transformation for quantum systems on the lattice, in the context of free particle systems. Specifically, we employ Daubechies wavelets to build approximations to the ground state of the critical Ising model, then demonstrate that these states correspond to instances of the multiscale entanglement renormalization ansatz (MERA), producing the first known analytic MERA for critical systems.
A bstract We propose an adaptation of Entanglement Renormalization for quantum field theories that, through the use of discrete wavelet transforms, strongly parallels the tensor network architecture of the Multiscale Entanglement Renormalization Ansatz (a.k.a. MERA). Our approach, called wMERA, has several advantages of over previous attempts to adapt MERA to continuum systems. In particular, (i) wMERA is formulated directly in position space, hence preserving the quasi-locality and sparsity of entanglers; and (ii) it enables a built-in RG flow in the implementation of real-time evolution and in computations of correlation functions, which is key for efficient numerical implementations. As examples, we describe in detail two concrete implementations of our wMERA algorithm for free scalar and fermionic theories in (1+1) spacetime dimensions. Possible avenues for constructing wMERAs for interacting field theories are also discussed.
We propose an adaptation of Entanglement Renormalization for quantum field theories that, through the use of discrete wavelet transforms, strongly parallels the tensor network architecture of the \emph{Multiscale Entanglement Renormalization Ansatz} (a.k.a. MERA). Our approach, called wMERA, has several advantages of over previous attempts to adapt MERA to continuum systems. In particular, (i) wMERA is formulated directly in position space, hence preserving the quasi-locality and sparsity of entanglers; and (ii) it enables a built-in RG flow in the implementation of real-time evolution and in computations of correlation functions, which is key for efficient numerical implementations. As examples, we describe in detail two concrete implementations of our wMERA algorithm for free scalar and fermionic theories in (1+1) spacetime dimensions. Possible avenues for constructing wMERAs for interacting field theories are also discussed.
We propose the use of entanglement renormalization techniques to study boundary critical phenomena on a lattice system. The multiscale entanglement renormalization ansatz (MERA), in its scale invariant version, offers a very compact approximation to quantum critical ground states. Here we show that, by adding a boundary to the MERA, an accurate approximation to the ground state of a semi-infinite critical chain with an open boundary is obtained, from which one can extract boundary scaling operators and their scaling dimensions. As in Wilson's renormalization-group formulation of the Kondo problem, our construction produces, as a side result, an effective chain displaying explicit separation of energy scales. We present benchmark results for the quantum Ising and quantum XX models with free and fixed boundary conditions.
Entanglement renormalization is a unitary real-space renormalization scheme. The corresponding tensor network states are known as MERA, and are particularly well-suited to describe quantum systems at criticality. In this work we explain how one can construct Gaussian bosonic quantum circuits that implement entanglement renormalization for ground states of arbitrary free bosonic chains. The construction is based on biorthogonal wavelet filters, and the dispersion relation of the Hamiltonian is translated to a filter design problem. We prove an approximation result that relates properties of the filters to the accumulation of errors in the MERA state. Finally, we explain how the continuum limit (a free bosonic quantum field) emerges naturally from the wavelet construction.
Entanglement renormalization is a unitary real-space renormalization scheme. The corresponding quantum circuits or tensor networks are known as MERA, and they are particularly well-suited to describing quantum systems at criticality. In this work we show how to construct Gaussian bosonic quantum circuits that implement entanglement renormalization for ground states of arbitrary free bosonic chains. The construction is based on wavelet theory, and the dispersion relation of the Hamiltonian is translated into a filter design problem. We give a general algorithm that approximately solves this design problem and provide an approximation theory that relates the properties of the filters to the accuracy of the corresponding quantum circuits. Finally, we explain how the continuum limit (a free bosonic quantum field) emerges naturally from the wavelet construction.
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.