GKP cluster-state 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 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.


Single-mode logical Clifford gates can be performed using Gaussian operations and measurements on a 1D GKP cluster state, while two-mode logical Clifford gates require a 2D cluster state. Magic-state distillation using photon-counting can be used for a non-Clifford logical \(\pi/8\) gate.

Fault Tolerance

First encoding demonstrating the possibility of fault-tolerant measurement-based computation with CV cluster states. A fault-tolerance threshold can be achieved by concatenating existing fault-tolerant schemes for qubit-based cluster-state encodings with the GKP code [1].Hybrid cluster state consisting of GKP qubits at some modes and squeezed states at others has been proposed to work in a fault-tolerant scheme [4].


A lower bound on the squeezing required to obtain a particular error rate can be formulated in terms of the displacement noise strength. This in turn determines how much squeezing is required in order to be below threshold for a particular concatenated code. A threshold of \(10^{-6}\) yields a required squeezing of 20.5 dB [1].




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

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“GKP cluster-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/gkp-cluster-state
@incollection{eczoo_gkp-cluster-state, title={GKP cluster-state code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/gkp-cluster-state} }
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“GKP cluster-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/gkp-cluster-state

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/oscillators/stabilizer/lattice/gkp-cluster-state.yml.