Unitary Quantum Cellular Automata for Density Classification

Abstract

We investigate the density classification task (DCT) -- determining the majority bit in a one-dimensional binary lattice -- within a quantum cellular automaton (CA) framework. While there is no one-dimensional two-state, radius $r \geq 1$, deterministic CA with periodic boundary conditions that solves the DCT perfectly, we explore whether a unitary quantum model can succeed. We employ the Partitioned Unitary Quantum Cellular Automaton (PUQCA), a number-conserving model, and, via evolutionary search, find solutions to the DCT where the success condition is stipulated in terms of measurement probabilities rather than convergence to fixed-point configurations. Finally, we identify a classically simulable regime of the PUQCA in which we find rules that solve the DCT at fixed system sizes and analyze their performance.