# GKP CV-cluster-state code[1]

Also known as Hybrid cluster-state code.

## Description

Cluster-state code can consists of a generalized analog cluster state that is initialized in GKP (resource) states for some of its physical modes. Alternatively, it can be thought of as an oscillator-into-oscillator GKP code whose encoding consists of initializing \(k\) modes in momentum states (or, in the normalizable case, squeezed vacua), \(n-k\) modes in (normalizable) GKP states, and applying a Gaussian circuit consisting of two-body \(e^{i V_{jk} \hat{x}_j \hat{x}_k }\) for some angles \(V_{jk}\). Provides a way to perform fault-tolerant MBQC, with the required number \(n-k\) of GKP-encoded physical modes determined by the particular protocol [1–4].

## Encoding

Initializing \(k\) modes in momentum states (or, in the normalizable case, squeezed vacua), \(n-k\) modes in (normalizable) GKP states, and applying a Gaussian circuit consisting of two-body \(e^{i V_{jk} \hat{x}_j \hat{x}_k }\) for some angles \(V_{jk}\).

## Gates

Logical Clifford gates are performed on the cluster state via a combination of linear-optical gates and homodyne measurements on subsets of vertices [5,6]. Magic-state distillation is required for universal computation.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.Gate teleportation and error correction can be performed without active squeezing [7].

## Decoding

GKP error correction can be naturally combined with CV MBQC protocols since the performance of both is quantified by a squeezing parameter [1].

## Threshold

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]. Anti-squeezing does not affect the threshold [8]. The calculation has been generalized to various Markovian noise [9].

## Parents

- Oscillator-into-oscillator GKP code — A GKP CV-cluster-state code can be created by initializing \(k\) modes in momentum states (or, in the normalizable case, squeezed vacua), \(n-k\) modes in (normalizable) GKP states, and applying a Gaussian circuit consisting of two-body \(e^{i V_{jk} \hat{x}_j \hat{x}_k }\) for some angles \(V_{jk}\).
- Qudit-into-oscillator code

## Cousins

- Analog-cluster-state code — GKP CV-cluster-state codes reduce to analog-cluster-state codes when all physical modes are initialized in momentum states.
- Cluster-state code — GKP CV-cluster-state codes reduce to cluster-state codes concatenated with single-mode GKP codes [3] when all physical modes are initialized in GKP states.
- Concatenated GKP code — GKP CV-cluster-state codes reduce to cluster-state codes concatenated with single-mode GKP codes [3] when all physical modes are initialized in GKP states.
- Distance-balanced code — Weight reduction has been studied in the context of GKP CV-cluster-state codes [10].

## 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]
- J. E. Bourassa et al., “Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer”, Quantum 5, 392 (2021) arXiv:2010.02905 DOI
- [3]
- K. Fukui et al., “High-Threshold Fault-Tolerant Quantum Computation with Analog Quantum Error Correction”, Physical Review X 8, (2018) arXiv:1712.00294 DOI
- [4]
- I. Tzitrin et al., “Fault-Tolerant Quantum Computation with Static Linear Optics”, PRX Quantum 2, (2021) arXiv:2104.03241 DOI
- [5]
- N. C. Menicucci et al., “Universal Quantum Computation with Continuous-Variable Cluster States”, Physical Review Letters 97, (2006) arXiv:quant-ph/0605198 DOI
- [6]
- M. Gu et al., “Quantum computing with continuous-variable clusters”, Physical Review A 79, (2009) arXiv:0903.3233 DOI
- [7]
- B. W. Walshe et al., “Continuous-variable gate teleportation and bosonic-code error correction”, Physical Review A 102, (2020) arXiv:2008.12791 DOI
- [8]
- B. W. Walshe et al., “Robust fault tolerance for continuous-variable cluster states with excess antisqueezing”, Physical Review A 100, (2019) arXiv:1903.02162 DOI
- [9]
- T. Matsuura, N. C. Menicucci, and H. Yamasaki, “Continuous-Variable Fault-Tolerant Quantum Computation under General Noise”, (2024) arXiv:2410.12365
- [10]
- 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 CV-cluster-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/gkp-cluster-state