Quadratic fermionic interactions yield effective Hamiltonians for adiabatic quantum computing

TL;DR

This paper demonstrates that quadratic fermionic Hamiltonians typically have polynomially-large energy gaps, enabling efficient adiabatic quantum computing for finding ground states of various fermionic and spin systems.

Contribution

It analytically proves polynomially-large gaps for quadratic fermionic Hamiltonians and shows how adiabatic quantum computing can efficiently find ground states of complex fermionic and spin systems.

Findings

Polynomially-large energy gaps in quadratic fermionic Hamiltonians.

Efficient polynomial-time adiabatic algorithms for fermionic and spin systems.

Polynomial-time preparation of the one-dimensional cluster state.

Abstract

Polynomially-large ground-state energy gaps are rare in many-body quantum systems, but useful for adiabatic quantum computing. We show analytically that the gap is generically polynomially-large for quadratic fermionic Hamiltonians. We then prove that adiabatic quantum computing can realize the ground states of Hamiltonians with certain random interactions, as well as the ground states of one, two, and three-dimensional fermionic interaction lattices, in polynomial time. Finally, we use the Jordan-Wigner transformation and a related transformation for spin-3/2 particles to show that our results can be restated using spin operators in a surprisingly simple manner. A direct consequence is that the one-dimensional cluster state can be found in polynomial time using adiabatic quantum computing.