Faster identification of optimal contraction sequences for tensor networks

TL;DR

This paper introduces a modified search algorithm with enhanced pruning that significantly speeds up the process of finding optimal contraction sequences for tensor networks, crucial for quantum physics and chemistry applications.

Contribution

The paper presents a novel, faster algorithm for identifying optimal tensor contraction sequences, improving efficiency over existing exhaustive search methods.

Findings

Performance increased by several orders of magnitude

Guarantees identification of optimal contraction sequences

Compatible MATLAB implementation provided

Abstract

The efficient evaluation of tensor expressions involving sums over multiple indices is of significant importance to many fields of research, including quantum many-body physics, loop quantum gravity, and quantum chemistry. The computational cost of evaluating an expression may depend strongly upon the order in which the index sums are evaluated, and determination of the operation-minimising contraction sequence for a single tensor network (single term, in quantum chemistry) is known to be NP-hard. The current preferred solution is an exhaustive search, using either an iterative depth-first approach with pruning or dynamic programming and memoisation, but these approaches are impractical for many of the larger tensor network Ansaetze encountered in quantum many-body physics. We present a modified search algorithm with enhanced pruning which exhibits a performance increase of several orders of magnitude while still guaranteeing identification of an optimal operation-minimising contraction sequence for a single tensor network. A reference implementation for MATLAB, compatible with the ncon() and multienv() network contractors of arXiv:1402.0939 and arXiv:1310.8023 respectively, is supplied.