**School of Physics Thesis Dissertation Defense**

**Presenter**: Wangwei Lan

**Title**: More efficient contraction algorithms for 2D tensor networks

**Date/Time**: Monday, November 28, 2022 at 1:00 p.m.

**Location**: Howey N201/202

**Virtual Link**: https://gatech.zoom.us/j/2500120410?pwd=RXFXMWFBeGl2RHBqWlRsL25uVitZZz09

Meeting ID: 250 012 0410 / Passcode: 765708

**Committee**: Dr. Glen Evenbly, School of Physics, Georgia Institute of Technology (Advisor)

Dr. Zhigang Jiang, School of Physics, Georgia Institute of Technology

Dr. Martin Mourigal, School of Physics, Georgia Institute of Technology

Dr. Dragomir Davidovic, School of Physics, Georgia Institute of Technology

Dr. Spencer Bryngelson, School of Computational Science and Engineering, Georgia Institute of Technology

**Abstract**:

Tensor network algorithms are important numerical tools for studying quantum many-body

problems. However, the high computational costs have prevented its applications in two-

dimensional (2D) systems. In this thesis, we discussed our work on more efficient con-

tractions of 2D tensor networks. In particular, for 2D statistical mechanics, we propose

a modified form of a tensor renormalization group algorithm for evaluating partition func-

tions of classical statistical mechanical models on 2D lattices. This algorithm coarse-grains

only the rows and columns of the lattice adjacent to a single core tensor at each step, such

that the lattice size shrinks linearly with the number of coarse-graining steps as opposed

to shrinking exponentially as in the usual tensor renormalization group (TRG). However,

the cost of this new approach only scales as O(χ4) in terms of the bond dimension χ, sig-

nificantly cheaper than the O(χ6) cost scaling of TRG, whereas numerical benchmarking

indicates that both approaches have comparable accuracy for the same bond dimension χ.

In 2D quantum mechanics, we propose a pair of approximations that allows the leading

order computational cost of contracting an infinite projected entangled-pair state (iPEPS)

to be reduced from O(χ3D6) to O(χ3D3) when using a corner-transfer approach. The first

approximation involves (i) reducing the environment needed for truncation of the bound-

ary tensors (ii) relies on the sequential contraction and truncation of bra and ket indices,

rather than doing both together as with the established algorithm. Our benchmark results

are comparable to the standard iPEPS algorithm. The improvement in computational cost

enables us to perform large bond dimension calculations, extending its potential to solve

challenging problems.