Hello,

In all papers on adiabatic quantum computation I am aware of, an
eigenvalue gap condition is assumed between the ground state and first
excited state to prevent state transitions, i.e.,

There is a time-variant hamiltonian H(t), 0<=t<=1, with continuous
ground state, |e0(t)>, and first excited state, |e1(t)>, where e0(t)
and e1(t) denote eigenvalues. The gap condition: e1(t)-e0(t) > 0
insures |e0(0)> evolves to |e0(1)> if the evolution is slow enough.

However, forms of the adiabatic theorem hold for hamiltonians which
violate the gap condition. I believe that (under fairly general
conditions) continuity of |e0(t)> is sufficient, even if there is no
gap, i.e., e0(t) crosses e1(t) and other eigenvalues. See links below.

My question is whether there are any implementable time-varying
hamiltonians which exploit this generalized adiabatic condition to
solve real problems.

Thanks for any answers or opinions,
Lou Pagnucco

"A note on the adiabatic theorem without gap condition" - Stefan Teufel
http://www.maphy.uni-tuebingen.de/Me...AWOGmp.ps/view

"Adiabatic Theorem without a Gap Condition" - J. Avron, A. Elgart
http://www.maphy.uni-tuebingen.de/Me...AWOGmp.ps/view