Site-Order Optimization in the Density Matrix Renormalization Group via Multi-Site Rearrangement

TL;DR

This work addresses how site ordering impacts the accuracy of DMRG/MPS calculations by extending prior local two-site rearrangements to multi-site rearrangements. The authors implement a two-part iterative algorithm that alternates standard DMRG updates with a local rearrangement over a window of $N_s$ sites, evaluating all $N_s!$ candidate orders via entanglement entropies and sweeping across the MPS. Application to a spin-1/2 Heisenberg chain with random site permutation shows that increasing $N_s$ from 2 to 3 yields large reductions in the ground-state energy error (65%–94%), with further improvements from larger $\chi_{opt}$ and reduced average inter-site distance $\mathcal{D}$, even at modest optimization costs. The results establish multi-site rearrangement as a powerful preprocessing step to improve MPS-based simulations, with potential extensions to fermionic systems, TTNs, and accelerated implementations through lookups, truncation, and parallelization.

Abstract

In the approaches based on matrix-product states (MPSs), such as the density-matrix renormalization group (DMRG) method, the ordering of the sites crucially affects the computational accuracy. We investigate the performance of an algorithm that searches for the optimal site order by iterative local site rearrangement. We improve the algorithm by expanding the range of site rearrangement and apply it to a one-dimensional quantum Heisenberg model with random site permutation. The results indicate that increasing the range of the site rearrangement significantly improves the computational accuracy of the DMRG method. In particular, increasing the rearrangement range from two to three sites reduces the average relative error in the ground-state energy by 65% to 94% in the cases we tested. We also discuss the computational cost of the algorithm and its application as a preprocessing for MPS-based calculations.

Figures (8)

• Figure 1: Matrix-Product State (MPS) for a system with $N$ sites. Shaded triangles and red diamond represent the three-leg isometric tensors and the singular value diagonal matrix, respectively. Open circles are the physical sites arranged in the order $\{ r(i)\}$, which is an arbitrary permutation of the site index $i= 1, 2, ..., N$. The edge bonds are terminated by a trivial boundary vector $|0 \rangle$, which represents the dimension-one auxiliary index at the ends of the MPS.
• Figure 2: Schematic pictures of states with a single-stroke entanglement structure. The blue line connects the sites that are strongly entangled. Vertical dotted lines represent a cut of the system, and the numbers indicated for each dotted line denote the number of blue lines across the cut.
• Figure 3: Schematic pictures of site order optimization for a state with a single-stroke entanglement structure. (a) State for which a two-site swap can achieve the optimal site order. (b) State for which a three-site rearrangement is required to reach the optimal site order.
• Figure 4: (a) Part of MPS focused at a step in the site-order optimization, including $N_{\rm s}$ isometries and a singular value matrix. The indices of the physical sites in the region are $\{ r(i), ..., r(i+N_{\rm s}-1) \}$. (b) Contraction of the isometries and singular value matrix in the focused region. (c) Updated part of MPS in the focused region where the site order is modified to the optimized one, $\{ \tilde{r}(i), ..., \tilde{r}(i+N_{\rm s}-1)\}$, which is a permutation of $\{ r(i), ..., r(i+N_{\rm s}-1) \}$.
• Figure 5: (a) Spin model in a 1D lattice with random site permutation. (b) The same model as (a) but in a 1D lattice where the sites are arranged in ascending order. An example for the case of an eight-site system with $\{ p(1), p(2), ..., p(8)\} = \{ 4, 1, 2, 7, 5, 6, 8, 3 \}$ is shown.