Energy Spectrum and Exact Cover in an Extended Quantum Ising Model

TL;DR

This paper studies an extended quantum Ising model with beyond-nearest-neighbour interactions and a site-dependent magnetic field, revealing how these factors influence the energy gap and potential applications to quantum computing problems.

Contribution

It introduces an exact treatment of the extended quantum Ising model with new insights into energy gaps and links to quantum satisfiability problems.

Findings

Beyond-nearest-neighbour interactions increase the minimal energy gap.

The longitudinal magnetic field provides a size-independent correction to the gap.

Application potential to quantum satisfiability and flux qubit chains.

Abstract

We investigate an extended version of the quantum Ising model which includes beyond-nearest neighbour interactions and an additional site-dependent longitudinal magnetic field. Treating the interaction exactly and using perturbation theory in the longitudinal field, we calculate the energy spectrum and find that the presence of beyond-nearest-neighbour interactions enhances the minimal gap between the ground state and the first excited state, irrespective of the nature of decay of these interactions along the chain. The longitudinal field adds a correction to this gap that is independent of the number of qubits. We discuss the application of our model to implementing specific instances of 3-satisfiability problems (Exact Cover) and make a connection to a chain of flux qubits.