Accelerating two-dimensional tensor network optimization by preconditioning

TL;DR

This work tackles the high cost and slow convergence of gradient-based iPEPS optimization by introducing a preconditioner derived from the tangent-space metric. It evaluates full and local metric approximations, with the local metric (and $\chi=1$ environment) offering a practical, scalable boost to convergence via a near-linear solve of $P g' = g$ using Krylov methods. Across Heisenberg and Kitaev models and both single- and large-unit-cell iPEPS, the preconditioned approach reduces iteration counts and wall-time, enabling more efficient exploration of strongly correlated 2D quantum systems. The results underscore the viability of geometry-informed preconditioning for tensor-network variational methods, while highlighting open avenues for Hamiltonian-aware preconditioners that maintain low computational overhead.

Abstract

We revisit gradient-based optimization for infinite projected entangled pair states (iPEPS), a tensor network ansatz for simulating many-body quantum systems. This approach is hindered by two major challenges: the high computational cost of evaluating energies and gradients, and an ill-conditioned optimization landscape that slows convergence. To reduce the number of optimization steps, we introduce an efficient preconditioner derived from the leading term of the metric tensor. We benchmark our method against standard optimization techniques on the Heisenberg and Kitaev models, demonstrating substantial improvements in overall computational efficiency. Our approach is broadly applicable across various contraction schemes, unit cell sizes, and Hamiltonians, highlighting the potential of preconditioned optimization to advance tensor network algorithms for strongly correlated systems.

Figures (4)

• Figure 1: The optimization results for the square-lattice Heisenberg model with $D=3$ iPEPS using different optimization schemes. The inset shows the same data with energy difference to a reference energy on a logarithmic scale.
• Figure 2: The same optimization results as in \ref{['fig:different_methods_compare_iters']}, but plotted as a function of wall time. All runtimes are measured on an NVIDIA 4090 GPU.
• Figure 3: Optimization results for the square-lattice Heisenberg model with $D=5$ using the local-metric preconditioned L-BFGS algorithm and the standard L-BFGS algorithm. We use different shades of lines to indicate different random initializations.
• Figure 4: Optimization results for the square-lattice Heisenberg model using different preconditioners. From top to bottom, the bond dimensions of the iPEPS are $D=3, 5,$ and $7$, respectively. The inset shows the same data with energy difference to a reference energy on a logarithmic scale.