Quadratic fermionic interactions yield Hamiltonians with large ground-state energy gaps

Polynomially large ground-state energy gaps are rare in many-body quantum systems, but useful in quantum information and an interesting feature of the one-dimensional quantum Ising model. We show analytically that the gap is generically polynomially large not just for the quantum Ising model, but for one-, two-, and three-dimensional interaction lattices and Hamiltonians with certain random interactions. We extend the analysis to Hamiltonian evolutions and 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. These results also yield a new perspective on the one-dimensional cluster state.

Figure 1

(Color online) The ground-state energy-gap distribution is compared to the distribution for the other energy gaps, in reduced units. All the energy levels are computed for 1000 random $n=10$ (ten-qubit) Hamiltonians. Each Hamiltonian is chosen randomly as described in Theorem III.3. As predicted by Theorem III.3, the ground-state energy gaps are much larger than the other gaps.

Figure 2

(Color online) Energy spectrum as a function of $s$ for a two-dimensional lattice of interacting fermions, using the $A$ and $B$ matrices from Eq. (24), in reduced units. The minimum ground-state energy gap is larger than most of the other minimum-energy gaps.