Continuous-variable analogue of measurement-based state preparation for GKP encoded states*

Quantum error correction (QEC) is essential for achieving fault-tolerant quantum computation by protecting quantum information against noise and decoherence. The Gottesman-Kitaev-Preskill (GKP) encoding provides a leading solution in continuous-variable (CV) systems, offering robustness against small displacement errors. This makes GKP codes particularly well-suited for photonic and bosonic platforms. However, a critical challenge remains: generating non-Pauli eigenstates, which are necessary to promote the Clifford operations of GKP encoding to universal quantum computation. Conventional schemes require nonlinear resources to access such states, posing major experimental obstacles.

In this work, we explore a surprising and advantageous consequence of working with non-ideal GKP states—specifically, those affected by Fock damping. We show that, contrary to common intuition, the imperfections in realistic GKP states play a crucial role in enabling access to non-Pauli logical states. In fact, our analysis reveals that using perfect GKP states prevents any possibility of projecting onto non-Pauli eigenstates within standard Gaussian measurement frameworks. It is precisely the structured deviation from ideality—introduced by Fock damping—that creates interference patterns and phase-space support necessary for post-measurement collapse into non-Clifford states, including potential magic states.

These findings highlight that certain imperfections, rather than being detrimental, can serve as a resource for universal quantum computation. This perspective opens new directions for leveraging experimentally realistic states in measurement-based quantum protocols, and underscores the utility of noise-tailored state preparation in GKP-encoded photonic systems.