Deterministic Universal Logical Gates for Finite-Energy GKP Qubits in Trapped Neutral Atoms

Abstract

We present a universal logical gate set for finite-energy Gottesman-Kitaev-Preskill (GKP) qubits encoded in the harmonic motional states of trapped neutral atoms. Internal electronic states serve as ancilla for implementing state-dependent conditional displacements in phase space, enabling arbitrary single-qubit phase gates. A Transient Rydberg excitation-mediated atomic dipole-dipole interaction enables a controlled-Z gate without invoking the blockade mechanism. The gates operate under magic-trapping conditions, allowing continuous trapping throughout the gate sequence, thereby preserving the motional encoding without intermediate measurement or feedforward. We assess the experimental feasibility of the proposed protocols using neutral $^{88}\mathrm{Sr}$ atoms and obtain average gate fidelities of $0.986$ for single-qubit phase gates and $0.985$ for the controlled-Z gate at a finite squeezing parameter of $Δ=0.25$ (12 dB of squeezing), increasing to $0.997$ and $0.995$ respectively as $Δ$ reaches $0.1$. These results provide a route toward universal, measurement-free logical control of motional GKP qubits in neutral-atom architectures.