# The Complexity of Translationally-Invariant Low-Dimensional Spin Lattices in 3D

## TL;DR

This paper proves that the local Hamiltonian problem for 3D face-centered cubic spin lattices with translational invariance and spin-3/2 particles is QMAEXP-complete, introducing novel techniques to reduce spin dimension.

## Contribution

It introduces new methods combining classical tiling and quantum circuit encoding to lower the local spin dimension in 3D translationally-invariant spin systems.

## Key findings

- Proves QMAEXP-completeness for 3D face-centered cubic lattices.
- Reduces the local spin dimension by two orders of magnitude.
- Achieves parity with non-translationally-invariant models.

## Abstract

In this paper, we consider spin systems in three spatial dimensions, and prove that the local Hamiltonian problem for 3D lattices with face-centered cubic unit cells, 4-local translationally-invariant interactions between spin-3/2 particles and open boundary conditions is QMAEXP-complete. We go beyond a mere embedding of past hard 1D history state constructions, and utilize a classical Wang tiling problem as binary counter in order to translate one cube side length into a binary description for the verifier input. We further make use of a recently-developed computational model especially well-suited for history state constructions, and combine it with a specific circuit encoding shown to be universal for quantum computation. These novel techniques allow us to significantly lower the local spin dimension, surpassing the best translationally-invariant result to date by two orders of magnitude (in the number of degrees of freedom per coupling). This brings our models en par with the best non-translationally-invariant construction.

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Source: https://tomesphere.com/paper/1702.08830