Theorem on the existence of a nonzero energy gap in adiabatic quantum computation

Adiabatic quantum computation, based on the adiabatic theorem, is a promising alternative to conventional quantum computation. The validity of an adiabatic algorithm depends on the existence of a nonzero energy gap between the ground and excited states. However, it is difficult to ascertain the exact value of the energy gap. In this paper, we put forward a theorem on the existence of nonzero energy gap for the Hamiltonians used in adiabatic quantum computation. It can help to effectively identify a large class of the Hamiltonians without energy-level crossing between the ground and excited states.

Figure 1

A sketch of a spectrum with a nonzero energy gap.

Figure 2

A sketch of a spectrum with an energy-level crossing.

Figure 3

Numerical simulation of energy levels for the illustrative example.