# GKP cluster-state code[1]

## Description

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 MBQC.

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.

## Gates

## Fault Tolerance

## Parents

- Gottesman-Kitaev-Preskill (GKP) code — The GKP cluster-state code is a concatenation of a cluster-state stabilizer code with a single-mode GKP code. A GKP-based cluster state is a multimode GKP codeword, although other codewords are not utilized in CV MBQC.
- Qudit-into-oscillator code
- Concatenated bosonic code

## Cousins

- Cluster-state code — The GKP cluster-state code is a concatenation of a cluster-state stabilizer code with a single-mode GKP code.
- Distance-balanced code — Weight reduction has been studied in the context of GKP cluster-state codes [5].

## References

- [1]
- N. C. Menicucci, “Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States”, Physical Review Letters 112, (2014) arXiv:1310.7596 DOI
- [2]
- N. C. Menicucci et al., “Universal Quantum Computation with Continuous-Variable Cluster States”, Physical Review Letters 97, (2006) arXiv:quant-ph/0605198 DOI
- [3]
- M. Gu et al., “Quantum computing with continuous-variable clusters”, Physical Review A 79, (2009) arXiv:0903.3233 DOI
- [4]
- J. E. Bourassa et al., “Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer”, Quantum 5, 392 (2021) arXiv:2010.02905 DOI
- [5]
- E. Sabo et al., “Weight Reduced Stabilizer Codes with Lower Overhead”, (2024) arXiv:2402.05228

## Page edit log

- Victor V. Albert (2022-06-28) — most recent

## Cite as:

“GKP cluster-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/gkp-cluster-state