Tensor network method for reversible classical computation

TL;DR

This paper introduces a tensor network approach with an iterative compression-decimation scheme to efficiently solve and count solutions for reversible classical computational problems modeled as vertex models, surpassing naive enumeration.

Contribution

The authors develop a novel tensor network method with an ICD algorithm to exactly solve large reversible classical computation problems efficiently.

Findings

Successfully computes the number of solutions for large systems.

Efficiently compresses and decimates tensor networks to avoid exponential complexity.

Provides a scalable method for solving vertex model-based computational problems.

Abstract

We develop a tensor network technique that can solve universal reversible classical computational problems, formulated as vertex models on a square lattice [Nat. Commun. 8, 15303 (2017)]. By encoding the truth table of each vertex constraint in a tensor, the total number of solutions compatible with partial inputs/outputs at the boundary can be represented as the full contraction of a tensor network. We introduce an iterative compression-decimation (ICD) scheme that performs this contraction efficiently. The ICD algorithm first propagates local constraints to longer ranges via repeated contraction-decomposition sweeps over all lattice bonds, thus achieving compression on a given length scale. It then decimates the lattice via coarse-graining tensor contractions. Repeated iterations of these two steps gradually collapse the tensor network and ultimately yield the exact tensor trace for large systems, without the need for manual control of tensor dimensions. Our protocol allows us to obtain the exact number of solutions for computations where a naive enumeration would take astronomically long times.