GKP cluster-state code

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

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].

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

Cousins

Concatenated quantum code
Cluster-state code — The GKP cluster-state code is a concatenation of a cluster-state stabilizer code with a single-mode GKP code.

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

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

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