Algorithms for entanglement renormalization

TL;DR

The paper develops and analyzes an iterative MERA-based algorithm to approximate the low-energy subspace of local Hamiltonians on D-dimensional lattices, achieving efficient computation of local observables, gaps, and correlators even at criticality. By exploiting the MERA circuit structure with disentanglers and isometries, it derives scalable procedures for computing expectations, two-point correlators, and optimization steps, with costs scaling as $O( abla^8 \, ext{log}\,N)$ for translation-invariant systems and $O( abla^8 N)$ in general. It introduces and benchmarks several variants, including TI-MERA, scale-invariant MERA, and finite-range MERA, demonstrating exponential energy convergence in $ abla$, accurate critical exponents, and long-range correlator fidelity in 1D Ising, Potts, XX, and Heisenberg models. The methods enable studying large 1D and 2D systems, including infinite and critical ones, with controlled accuracy and without resorting to Gaussian-state assumptions. The work provides a practical, self-contained framework for implementing MERA-based ground-state and low-energy subspace calculations across a range of lattice models.

Abstract

We describe an iterative method to optimize the multi-scale entanglement renormalization ansatz (MERA) for the low-energy subspace of local Hamiltonians on a D-dimensional lattice. For translation invariant systems the cost of this optimization is logarithmic in the linear system size. Specialized algorithms for the treatment of infinite systems are also described. Benchmark simulation results are presented for a variety of 1D systems, namely Ising, Potts, XX and Heisenberg models. The potential to compute expected values of local observables, energy gaps and correlators is investigated.

Figures (28)

• Figure 1: (Colour online) Quantum circuit $\mathcal{C}$ corresponding to a specific realization of the MERA, namely the binary 1D MERA of Fig. \ref{['fig:2MERA']}. In this particular example, circuit $\mathcal{C}$ is made of gates involving two incoming wires and two outgoing wires, $p=p_{in}=p_{out}=2$. Some of the unitary gates in this circuit have one incoming wire in the fixed state $\hbox{$| 0 \rangle$}$ and can be replaced with an isometry $w$ of type (1,2). By making this replacement, we obtain the isometric circuit of Fig. \ref{['fig:2MERA']}.
• Figure 2: (Colour online) Top: Example of a binary 1D MERA for a lattice $\mathcal{L}$ with $N=16$ sites. It contains two types of isometric tensors, organized in $T=4$ layers. The input (output) wires of a tensor are those that enter it from the top (leave it from the bottom). The top tensor is of type $(1,2)$ and the rank $\chi_T$ of its upper index determines the dimension of the subspace $\mathbb{V}_{U}\subseteq \mathbb{V}_{\mathcal{L}}$ represented by the MERA. The isometries$w$ are of type $(1,2)$ and are used to replace each block of two sites with a single effective site. Finally, the disentanglers$u$ are of type (2,2) and are used to disentangle the blocks of sites before coarse-graining. Bottom: Under the renormalization group transformation induced by the binary 1D MERA, three-site operators are mapped into three-site operators.
• Figure 3: (Colour online) Top: Example of ternary 1D MERA (rank $\chi_T$, $T=3$) for a lattice of 18 sites. It differs from the binary 1D MERA of Fig. \ref{['fig:2MERA']} in that the isometries are of type $(1,3)$, so that blocks of three sites are replaced with one effective site. Bottom: As a result, two-site operators are mapped into two-site operators during the coarse-graining.
• Figure 4: (Colour online) The tensors which comprise a MERA are constrained to be isometric, cf. Eq. \ref{['eq:isometry']}. The constraints for the isometries $w$ and disentanglers $u$ of the ternary MERA can be equivalently expressed (i) diagramatically or (ii) with equations. In this paper we will mostly use the diagramatic notation, which remains simple for complicated tensor networks.
• Figure 5: (Colour online) The past causal cone of a group of sites in $\mathcal{L}_0 \equiv \mathcal{L}$ is the subset of wires and gates that can affect the state of those sites. The example shows the causal cone of a pair of nearest neighbor sites of $\mathcal{L}_0$ for the ternary 1D MERA. Notice that for each lattice $\mathcal{L}_{\tau}$, $\tau=0,1,2,3,4$, the causal cone involves at most 2 sites. This can be seen to be the case for any pair of contiguous sites of $\mathcal{L}_0$. We refer to this property by saying that the causal cones of the MERA have bounded width.