# Oscillator-into-oscillator GKP code[1]

Also known as GKP-stabilizer code.

## Description

Multimode GKP code with an infinite-dimensional logical space. Can be obtained by considering an \(n\)-mode GKP code with a finite-dimensional logical space, removing stabilizers such that the logical space becomes infinite dimensional, and applying a Gaussian circuit.

Simple oscillator-into-oscillator GKP codes include GKP-repetition codes and GKP two-mode-squeezing (TMS) codes [1]. Arbitrary oscillator-into-oscillator GKP codes can be reduced to generalized GKP TMS codes, and the optimal code design problem can be efficiently solved [2].

## Protection

Oscillator-into-oscillator GKP codes to protect one or more modes against displacement noise using GKP resource states.

## Encoding

Gaussian circuit applied to \(k\) modes storing logical information and \(n-k\) modes initialized in a fixed GKP state.

## Decoding

Syndromes can be read off using ancilla modes, yielding partial information about noise in the logical modes that can then be used in an efficient ML decoding procedure [3].

## Threshold

Thresholds against displacement noise cannot be obtained without ideal (i.e., non-normalizable) codewords [4].

## Notes

Introduction to and examples of oscillator-into-oscillator GKP codes [5].

## Parents

- Quantum lattice code — Oscillator-into-oscillator GKP codes are \(n\)-mode quantum lattice codes with less than \(2n\) stabilizers, i.e., constructed using a degenerate lattice (see Appx. A of Ref. [6]).
- Oscillator-into-oscillator code

## Child

- GKP CV-cluster-state 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 \(V_{jk}\).

## Cousins

- Concatenated GKP code — Oscillator-into-oscillator GKP codes concantenated with qubit-into-oscillator GKP codes can outperform more conventional concatenations of qubit-into-oscillator GKP codes with qubit stabilizer codes [3].
- \(D_4\) hyper-diamond GKP code — \(D_4\) hyper-diamond GKP codes may be optimal for GKP stabilizer codes utilizing two ancilla modes [2].
- Hexagonal GKP code — Hexagonal GKP codes may be optimal for GKP stabilizer codes utilizing one ancilla mode [2].
- Analog stabilizer code — Analog stabilizer codes protect logical modes against arbitrarily large displacements on a few modes, while oscillator-into-oscillator GKP codes protect a finite-dimensional logical space against sufficiently small displacements in any number of modes. Encoding in analog-stabilizer (oscillator-into-oscillator GKP) codes can be done by a Gaussian operation acting on a tensor product of an arbitrary state in the first mode and position states (GKP states) on the remaining modes. Protection of logical modes against small displacements cannot be done using only Gaussian resources [7–9], so oscillator-into-oscillator GKP codes can be thought of as analog stabilizer encodings utilizing non-Gaussian GKP resource states.
- Homological number-phase code — Rotor analogues of \(k\)-into-\(n\) oscillator-into-oscillator GKP codes can be constructed by initializing \(n-k\) physical rotors in superpositions of phase states and applying a Clifford semigroup encoding circuit [10].

## References

- [1]
- K. Noh, S. M. Girvin, and L. Jiang, “Encoding an Oscillator into Many Oscillators”, Physical Review Letters 125, (2020) arXiv:1903.12615 DOI
- [2]
- J. Wu, A. J. Brady, and Q. Zhuang, “Optimal encoding of oscillators into more oscillators”, Quantum 7, 1082 (2023) arXiv:2212.11970 DOI
- [3]
- Y. Xu et al., “Qubit-Oscillator Concatenated Codes: Decoding Formalism and Code Comparison”, PRX Quantum 4, (2023) arXiv:2209.04573 DOI
- [4]
- L. Hanggli and R. Konig, “Oscillator-to-Oscillator Codes Do Not Have a Threshold”, IEEE Transactions on Information Theory 68, 1068 (2022) arXiv:2102.05545 DOI
- [5]
- V. V. Albert, “Bosonic coding: introduction and use cases”, (2022) arXiv:2211.05714
- [6]
- J. Conrad, J. Eisert, and F. Arzani, “Gottesman-Kitaev-Preskill codes: A lattice perspective”, Quantum 6, 648 (2022) arXiv:2109.14645 DOI
- [7]
- C. Vuillot et al., “Quantum error correction with the toric Gottesman-Kitaev-Preskill code”, Physical Review A 99, (2019) arXiv:1810.00047 DOI
- [8]
- J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian States with Gaussian Operations is Impossible”, Physical Review Letters 89, (2002) arXiv:quant-ph/0204052 DOI
- [9]
- J. Niset, J. Fiurášek, and N. J. Cerf, “No-Go Theorem for Gaussian Quantum Error Correction”, Physical Review Letters 102, (2009) arXiv:0811.3128 DOI
- [10]
- Y. Xu, Y. Wang, and V. V. Albert, “Clifford operations and homological codes for rotors and oscillators”, (2024) arXiv:2311.07679

## Page edit log

- Victor V. Albert (2022-12-15) — most recent
- Armin Gerami (2022-12-15)
- Victor V. Albert (2022-09-15)
- Victor V. Albert (2022-03-24)

## Cite as:

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