[Paper Review] DMRG Approach to Optimizing Two-Dimensional Tensor Networks

Katharine Hyatt, E. Miles Stoudenmire|arXiv (Cornell University)|Aug 23, 2019

Quantum many-body systems18 citations

TL;DR

This paper introduces a DMRG-like algorithm for optimizing two-dimensional PEPS tensor networks by incorporating key features of the original DMRG method—canonical form, regular eigenvalue solvers, and stable large updates. The approach achieves rapid, accurate convergence for 2D spin models, demonstrating exponential energy convergence and competitive accuracy with exact QMC results at bond dimension D=6.

ABSTRACT

Tensor network algorithms have been remarkably successful solving a variety of problems in quantum many-body physics. However, algorithms to optimize two-dimensional tensor networks known as PEPS lack many of the aspects that make the seminal density matrix renormalization group (DMRG) algorithm so powerful for optimizing one-dimensional tensor networks known as matrix product states. We implement a framework for optimizing two-dimensional PEPS tensor networks which includes all of steps that make DMRG so successful for optimizing one-dimension tensor networks. We present results for several 2D spin models and discuss possible extensions and applications.

Motivation & Objective

• To extend the powerful DMRG algorithm, successful for 1D MPS, to two-dimensional PEPS tensor networks.
• To address the lack of canonical forms, efficient eigensolvers, and stable large updates in existing PEPS optimization methods.
• To enable high-precision, scalable simulations of 2D quantum many-body systems using tensor networks.
• To establish a foundation for future improvements such as adaptive bond dimension and two-site updates.

Proposed method

• Adopt a one-site DMRG approach for PEPS, updating one tensor at a time while keeping others fixed.
• Enforce a canonical form on the PEPS network using recent advances in PEPS canonization, ensuring orthonormality and numerical stability.
• Project the Hamiltonian into the orthonormal basis defined by the fixed tensors, forming a regular eigenvalue problem.
• Solve the projected eigenvalue problem using iterative solvers like Lanczos or Davidson for efficiency and accuracy.
• Perform large, stable updates to individual tensors while maintaining overall network stability.
• Use a GPU-accelerated backend in the ITensor library to optimize performance, particularly for tensor contractions and MPO-MPO multiplications.

Experimental results

Research questions

• RQ1 Can the DMRG algorithm’s core advantages—canonical form, regular eigenvalue problems, and stable large updates—be successfully adapted to 2D PEPS?
• RQ2 How does the convergence rate of PEPS-DMRG compare to exact or QMC results for 2D spin models?
• RQ3 What is the impact of finite bond dimension and canonization accuracy on the energy and spin-spin correlators in PEPS-DMRG?
• RQ4 Can the method be extended to include adaptive bond dimension growth via a two-site algorithm?
• RQ5 How does the performance of finite-size PEPS-DMRG compare to infinite PEPS (iPEPS) in estimating bulk properties?

Key findings

• The PEPS-DMRG algorithm achieves exponential convergence of the energy with respect to DMRG sweeps, with only about ten sweeps needed to reach within 10⁻² of the final energy for L=10 systems.
• For the L=10 Heisenberg model, the energy per site converges to within 10⁻² of the exact QMC result using bond dimension D=6 and environment dimension χ=12.
• Spin-spin correlators computed with D=6 PEPS show a finite-bond-dimension discrepancy compared to exact MPS-DMRG results, comparable to state-of-the-art iPEPS methods.
• Occasional energy increases during optimization are attributed to imperfect canonization, indicating that improvements in canonization algorithms will enhance stability and accuracy.
• The total runtime is dominated by tensor contractions (46%) and MPO-MPO multiplications (14%), highlighting opportunities for algorithmic optimization.
• The GPU-accelerated ITensor backend significantly improves simulation speed, enabling efficient exploration of larger systems and higher bond dimensions.