The complexity of translationally invariant lowdimensional spin lattices in 3D
Abstract
In this theoretical paper, we consider spin systems in three spatial dimensions and consider the computational complexity of estimating the ground state energy, known as the local Hamiltonian problem, for translationally invariant Hamiltonians. We prove that the local Hamiltonian problem for 3D lattices with facecentered cubic unit cells and 4local translationally invariant interactions between spin3/2 particles and open boundary conditions is QMA_{EXP}complete, where QMA_{EXP} is the class of problems which can be verified in exponential time on a quantum computer. We go beyond a mere embedding of past hard 1D history state constructions, for which the local spin dimension is enormous: even stateoftheart constructions have local dimension 42. We avoid such a large local dimension by combining some different techniques in a novel way. For the verifier circuit which we embed into the ground space of the local Hamiltonian, we utilize a recently developed computational model, called a quantum ring machine, which is especially well suited for translationally invariant history state constructions. This is encoded with a new and particularly simple universal gate set, which consists of a single 2qubit gate applied only to nearestneighbour qubits. The Hamiltonian construction involves a classical Wang tiling problem as a binary counter which translates one cube side length into a binary description for the encoded verifier input and a carefully engineered history state construction that implements the ring machine on the cubic lattice faces. 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 on par with the best nontranslationally invariant construction.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 November 2017
 DOI:
 10.1063/1.5011338
 arXiv:
 arXiv:1702.08830
 Bibcode:
 2017JMP....58k1901B
 Keywords:

 Quantum Physics;
 Computer Science  Computational Complexity;
 68Q17;
 81V70;
 68Q10;
 82D25
 EPrint:
 20 pages. 3 figures in main text