  We propose a method to speed up the quantum adiabatic algorithm using
catalysis by many-body delocalization. This is applied to random-field
antiferromagnetic Ising spin models. The algorithm is catalyzed in such a way
that the evolution approximates a Heisenberg model in the middle of its course,
and the model is in a delocalized phase. We show numerically that we can speed
up the standard algorithm for finding the ground state of the random-field
Ising model using this idea. We also demonstrate that the speedup is due to gap
amplification, even though the underlying model is not frustration-free. The
crossover to speedup occurs at roughly the value of the interaction which is
known to be the critical one for the delocalization transition. We also
calculate the participation ratio and entanglement entropy as a function of
time: their time dependencies indicate that the system is exploring more states
and that they are more entangled than when there is no catalyst. Together, all
these pieces of evidence demonstrate that the speedup is related to
delocalization. Even though only relatively small systems can be investigated,
the evidence suggests that the scaling of the method with system size is
favorable. Our method is illustrated by experimental results from a small
online IBM quantum computer, showing how to verify the method in future as such
machines improve. The cost of the catalytic method compared to the standard
algorithm is only a constant factor.
