Search on a Hypercubic Lattice through a Quantum Random Walk: II. d=2

TL;DR

This paper improves quantum spatial search algorithms on a 2D lattice by controlling the Dirac operator with an ancilla qubit, achieving a more optimal $O(\sqrt{N\log N})$ scaling through numerical tuning.

Contribution

It introduces a method to optimize the quantum search algorithm on a 2D lattice by controlling the Dirac evolution with an ancilla, reducing the complexity from $O(\sqrt{N}\log N)$ to $O(\sqrt{N\log N})$.

Findings

Achieved improved $O(\sqrt{N\log N})$ scaling for 2D lattice search.

Reinterpreted ancilla control as effective mass at the marked vertex.

Numerically optimized parameters for better algorithm performance.

Abstract

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.