Density Matrix Renormalization Group with Tensor Processing Units

TL;DR

This paper demonstrates how Google's TPUs can significantly accelerate the density matrix renormalization group method, enabling the computation of larger quantum many-body systems with unprecedented bond dimensions.

Contribution

It introduces the use of TPUs for DMRG, achieving large bond dimensions and faster optimization times for 2D lattice models, which was previously computationally challenging.

Findings

Achieved bond dimension D=65536 using TPUs.

Optimized a single MPS tensor in about 2 minutes.

Enabled large-scale DMRG calculations on 2D lattice models.

Abstract

Google's Tensor Processing Units (TPUs) are integrated circuits specifically built to accelerate and scale up machine learning workloads. They can perform fast distributed matrix multiplications and therefore be repurposed for other computationally intensive tasks. In this work we demonstrate the use of TPUs for accelerating and scaling up the density matrix renormalization group (DMRG), a powerful numerical approach to compute the ground state of a local quantum many-body Hamiltonian. The cost of DMRG scales with system size $N$ as $O(ND^3)$, where the so-called bond dimension $D$ regulates how expressive the underlying matrix product state (MPS) variational ansatz is. We consider lattice models in two spatial dimensions, with square lattices of size $10\times 10$ (free fermions) and $20\times 20$ (transverse field Ising model), for which the required MPS bond dimension is known to scale at least as $\exp(\sqrt{N})$. Using half of a TPU v3 pod (namely $1,\!024$ TPU v3 cores) we reached an unprecedentedly large bond dimension $D = 2^{16} = 65,\!536$, for which optimizing a single MPS tensor took about 2 minutes.