GKP cluster-state concatenated 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 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.

Gates

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.

Parents

Cousin

Zoo code information

Internal code ID: gkp-cluster-state

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: gkp-cluster-state

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

References

[1]
N. C. Menicucci, “Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States”, Physical Review Letters 112, (2014). DOI; 1310.7596
[2]
N. C. Menicucci et al., “Universal Quantum Computation with Continuous-Variable Cluster States”, Physical Review Letters 97, (2006). DOI; quant-ph/0605198
[3]
M. Gu et al., “Quantum computing with continuous-variable clusters”, Physical Review A 79, (2009). DOI; 0903.3233

Cite as:

“GKP cluster-state concatenated 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/tree/main/codes/quantum/oscillators/stabilizer/gkp-cluster-state.yml.