  We investigate the spatial search problem on the two-dimensional square
lattice, using the Dirac evolution operator discretised according to the
staggered lattice fermion formalism. $d=2$ is the critical dimension for the
spatial search problem, where infrared divergence of the evolution operator
leads to logarithmic factors in the scaling behaviour. As a result, the
construction used in our accompanying article \cite{dgt2search} provides an
$O(\sqrt{N}\log N)$ algorithm, which is not optimal. The scaling behaviour can
be improved to $O(\sqrt{N\log N})$ by cleverly controlling the massless Dirac
evolution operator by an ancilla qubit, as proposed by Tulsi \cite{tulsi}. We
reinterpret the ancilla control as introduction of an effective mass at the
marked vertex, and optimise the proportionality constants of the scaling
behaviour of the algorithm by numerically tuning the parameters.

Quantum Physics