Spin lattices with two-body Hamiltonians for which the ground state encodes a cluster state

TL;DR

This paper introduces a method to construct qubit lattices with two-body Hamiltonians whose ground states encode cluster states, facilitating measurement-based quantum computation across various lattice geometries.

Contribution

It provides a general procedure applicable to any graph for creating such lattices, including detailed analysis of energy gaps and computational usefulness.

Findings

Energy gap is independent of lattice size

Gap depends on lattice type and coordination number

Different lattice types vary in suitability for quantum computation

Abstract

We present a general procedure for constructing lattices of qubits with a Hamiltonian composed of nearest-neighbour two-body interactions such that the ground state encodes a cluster state. We give specific details for lattices in one-, two-, and three-dimensions, investigating both periodic and fixed boundary conditions, as well as present a proof for the applicability of this procedure to any graph. We determine the energy gap of these systems, which is shown to be independent of the size of the lattice but dependent on the type of lattice (in particular, the coordination number), and investigate the scaling of this gap in terms of the coupling constants of the Hamiltonian. We provide a comparative analysis of the different lattice types with respect to their usefulness for measurement-based quantum computation.