1202.3646 (C. R. Laumann et al.)

The quantum adiabatic algorithm and scaling of gaps at first order
quantum phase transitions    [PDF]

C. R. Laumann, R. Moessner, A. Scardicchio, S. L. Sondhi

Motivated by the quantum adiabatic algorithm (QAA), we consider the scaling
of the Hamiltonian gap at quantum first order transitions, generally expected
to be exponentially small in the size of the system. However, we show that a
quantum antiferromagnetic Ising chain in a staggered field can exhibit a first
order transition with only an algebraically small gap. In addition, we
construct a simple classical translationally invariant one-dimensional
Hamiltonian containing nearest-neighbour interactions only, which exhibits an
exponential gap at a thermodynamic quantum first-order transition of
essentially topological origin. This establishes that (i) the QAA can be
successful even across first order transitions but also that (ii) it can fail
on exceedingly simple problems readily solved by inspection, or by classical
annealing.

View original: http://arxiv.org/abs/1202.3646