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

Quadratic fermionic interactions yield effective Hamiltonians for adiabatic quantum computing