GKP cluster-state concatenated code[1]

Multi-mode code encoding logical qubits into a cluster-state stabilizer code concatenated with a single-mode GKP code. Provides a way to perform a continuous-variable (CV) analogue of fault-tolerant measurement-based qubit computation.

A cluster state of GKP qubits on a graph is made by applying two-mode \(C_Z\)-type gates \(e^{\pm i \hat{x}\otimes\hat{x}}\) to a tensor product of \(|\overline{+}\rangle\) logical GKP states on each vertex. Logical Clifford gates are performed on the cluster state using CV measurement-based computation [2][3], i.e., via a combination of linear-optical gates and homodyne measurements on subsets of vertices. Magic-state distillation is required for universal computation. GKP error correction can be naturally combined with CV measurement-based protocols since the performance of both is quantified by a squeezing parameter.