Coherent manipulation of graph states composed of finite-energy Gottesman-Kitaev-Preskill-encoded qubits

Graph states are a central resource in measurement-based quantum information processing. In the photonic qubit architecture based on Gottesman-Kitaev-Preskill (GKP) encoding, the generation of high-fidelity graph states composed of realistic, finite-energy approximate GKP-encoded qubits thus constitutes a key task. We consider the finite-energy approximation of GKP qubit states given by a coherent superposition of shifted finite-squeezed vacuum states, where the displacements are Gaussian distributed. We present an exact description of graph states composed of such approximate GKP qubits as a coherent superposition of a Gaussian ensemble of randomly displaced ideal GKP-qubit graph states. We determine the transformation rules for the covariance matrix and the mean displacement vector of the Gaussian distribution of the ensemble under tools such as GKP-Steane error correction and fusion operations that can be used to grow large, high-fidelity GKP-qubit graph states. The former captures the noise in the graph state due to the finite-energy approximation of GKP qubits, while the latter relates to the possible absolute displacement errors on the individual qubits due to the homodyne measurements that are a part of these tools. The rules thus help in pinning down an exact coherent error model for graph states generated from truly finite-energy GKP qubits, which can shed light on their error correction properties.