Improved contraction of finite projected entangled pair states

TL;DR

This paper improves the contraction and optimization of finite PEPS (fPEPS) supplemented by PEPOs by detailing memory-efficient contraction patterns and introducing a controlled bond expansion (CBE) powered by randomized SVD (RSVD). The dual innovations—memory-sliced contractions and RSVD-accelerated CBE—enable more scalable environment approximations and energy minimizations, with benchmarks on the Hubbard model up to $8\times8$ and environment bonds up to $χ=500$ and PEPS bonds up to $D=6$ (SU(2) symmetry). While energy convergence improves and memory demands decrease versus previous work, the results remain short of the DMRG bounds for large 2D systems, indicating ongoing gaps to practical competitiveness. The authors discuss future directions, including Monte Carlo contraction and belief-propagation methods, to further enhance the reach of the fPEPS-PEPO framework in describing large, heterogeneous two-dimensional quantum systems.

Abstract

We present an improved version of the algorithm contracting and optimizing finite projected entangled pair states (fPEPS) in conjunction with projected entangled pair operators (PEPOs). Our work has two components to it. First, we explain in detail the characteristic contraction patterns that occur in fPEPS calculations and how to slice them such that peak memory occupation remains minimal while ensuring efficient parallel computation. Second, we combine controlled bond expansion [A. Gleis, J.-W. Li, and J. von Delft, Phys. Rev. Lett. 130, 246402 (2023)] with randomized singular value decomposition [V. Rokhlin, A. Szlam, and M. Tygert, SIAM J. Matrix Anal. Appl. (2009)] and apply it throughout the fPEPS algorithm. We present benchmark results for the Hubbard model for system sizes up to 8x8 and SU(2) symmetric bond dimension of up to D = 6 for PEPS bonds and $χ$ = 500 for the environment bonds. Finally, we comment on the state and future of the fPEPS-PEPO framework.

Figures (12)

• Figure 1: PEPS-PEPO network for a 4$\times$4 lattice. Black lines connect PEPS-tensors, blue lines connect PEPO tensors and green lines connect PEPS and PEPO tensors.
• Figure 2: Characteristic contraction patterns appearing in the fPEPS algorithm, with the environment contraction depicted in (a) and the contraction of an effective single-site Hamiltonian and a PEPS tensor depicted in (b). $T$, $L$, $B$ and $R$ are environment tensors, while the tensor in the center $C$ is a sandwich of a PEPS tensor, PEPO tensor and adjoint PEPS tensor for (a) and a sandwich of a PEPS tensor and PEPO tensor for (b).
• Figure 3: Controlled bond expansion for the environment approximation.
• Figure 4: Completeness relation for environment tensors.
• Figure 5: RSVD for the CBE within environment approximation (a) Initial setup of $A \, \Omega$. (b) Separation of right orthogonal projector into identity and tangential projector via the completeness relation in Fig. \ref{['fig:env_completeness']}. (c) Contraction of tangential projector and $\Omega$. (d) Contraction of both bracketed clusters. (e) Subtraction of both bracketed contraction results. (f) Contraction of upper four tensors from (e) and separation of left orthogonal projector into identity and tangential projector. (g) Final contractions and subtraction.