Density Matrix Renormalization Group with Tensor Processing Units

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 10×10 (free fermions) and 20×20 (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 1024 TPU v3 cores), we reach an unprecedentedly large bond dimension D=2^{16}=65536, for which optimizing a single MPS tensor takes about 2 min.