# Gottesman-Kitaev-Preskill (GKP) code[1]

## Description

Single-mode GKP qudit-into-oscillator code based on the rectangular lattice. Its stabilizers are oscillator displacement operators \(\hat{S}_q(2\alpha)=e^{-2i\alpha \hat{p}}\) and \(\hat{S}_p(2\beta)=e^{2i\beta \hat{x}}\). To ensure \(\hat{S}_q(2\alpha)\) and \(\hat{S}_p(2\beta)\) generate a stabilizer group that is Abelian, there is another constraint that \(\alpha\beta=2k\pi\) where \(k\) is an integer. Codewords can be expressed as equal weight superpositions of coherent states on a rectangular lattice in phase space with spatial period \(2\sqrt{\pi}\). The exact GKP state is non-normalizable, so approximate constructs have to be considered.

## Protection

For stabilizer \(\hat{S}_q(2\alpha),\hat{S}_p(2\beta)\), code can correct displacement errors up to \(\frac{\alpha}{2}\) in the \(q\)-direction and \(\frac{\beta}{2}\) at \(p\)-direction. Approximately protects against photon loss errors [2][3], outperforming most other codes designed to explicitly protect against loss [3]. Very sensitive to dephasing errors [4]. A biased-noise GKP error correcting code can be prepared by choosing \(\alpha\neq \beta\).

## Encoding

Preparation of approximate GKP states is studied both theoretically and experimentally by putting the GKP lattice inside a Gaussian envelope [2][5][6][7].Dissipative stabilization of finite-energy GKP states using stabilizers conjugated by a cooling ([8], Appx. B) or damping operator, i.e., a damped exponential of the total occupation number [9][10].Two Josephson junctions coupled by a gyrator [11].

## Gates

By applying GKP error correction to Gaussian input states, universality can be achieved without non-Gaussian elements [12].

## Decoding

Syndrome measurement can be done by applying a controlled-displacement controlled by an ancilla qubit. The syndrome information can be obtained by measuring the ancilla qubit after controlled-displacement opearation. See Section. 2D in [4].Pauli \(X\),\(Y\) and \(Z\) measurements can be performed by measuring \(-\hat{p},\hat{x}-\hat{p}\) and \(\hat{x}\) repectively. If the measurement outcome is closed to an even multiple of \(\sqrt{\pi}\), then the outcome is +1. If the measurement outcome is closed to an odd multiple of \(\sqrt{\pi}\), then the outcome is -1. See Section. 2D in [4].Reinforcement learning decoder that uses only one ancilla qubit [13].

## Fault Tolerance

Clifford gates can be realized by performing linear-optical operations, sympletic transformations and displacements, all of which are Gaussian operations. Pauli gates can be performed using displacement operators. Clifford gates are fault tolerant in the sense that they map bounded-size errors to bounded-size errors [1].Error correction scheme is fault-tolerant to displacement noise as long as all input states have displacement errors less than \(\sqrt{\pi}/6\) [14].

## Realizations

Motional degree of freedom of a trapped ion: GKP encoding realized with the help of post-selection [15][16], followed by realization of reduced form of GKP error correction, where displacement error syndromes are measured to one bit of precision using an ion electronic state [10].Microwave cavity coupled to superconducting circuits: reduced form of GKP error correction, where displacement error syndromes are measured to one bit of precision using an ancillary transmon [17]. Subsequent paper [13] uses reinforcement learning for error-correction cycle design and is the first to go beyond break-even error-correction, with the lifetime of a logical qubit exceeding the cavity lifetime by about a factor of two (see also [18]).Single-qubit \(Z\)-gate has been demonstrated in the single-photon subspace of an infinite-mode space [19], in which time and frequency become bosonic conjugate variables of a single effective bosonic mode.In signal processing, GKP state position-state wavefunctions are related to Dirac combs [20].

## Notes

## Parents

## Cousins

- Approximate quantum error-correcting code (AQECC) — GKP codes approximately protect against photon loss [2][3][22].
- Number-phase code — GKP codes utilize translational symmetry in phase space, while number-phase codes utilize rotational symmetry. The two are related via a mapping [23].
- Quadrature-amplitude modulation (QAM) code — GKP codes are quantum analogues of lattice-based QAM codes.
- Rotor GKP code — GKP (rotor GKP) codes protect against shifts in linear (angular) degrees of freedom.
- Spin GKP code — Spin-GKP code constructions utilize the Holstein-Primakoff mapping [24] (see also [25]) to convert various expressions for GKP states into codes for spin systems.

## References

- [1]
- D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator”, Physical Review A 64, (2001). DOI; quant-ph/0008040
- [2]
- B. M. Terhal and D. Weigand, “Encoding a qubit into a cavity mode in circuit QED using phase estimation”, Physical Review A 93, (2016). DOI; 1506.05033
- [3]
- V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018). DOI; 1708.05010
- [4]
- A. L. Grimsmo and S. Puri, “Quantum Error Correction with the Gottesman-Kitaev-Preskill Code”, PRX Quantum 2, (2021). DOI; 2106.12989
- [5]
- D. J. Weigand and B. M. Terhal, “Generating grid states from Schrödinger-cat states without postselection”, Physical Review A 97, (2018). DOI
- [6]
- P. Campagne-Ibarcq et al., “Quantum error correction of a qubit encoded in grid states of an oscillator”, Nature 584, 368 (2020). DOI; 1907.12487
- [7]
- I. Tzitrin et al., “Progress towards practical qubit computation using approximate Gottesman-Kitaev-Preskill codes”, Physical Review A 101, (2020). DOI; 1910.03673
- [8]
- N. C. Menicucci, “Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States”, Physical Review Letters 112, (2014). DOI; 1310.7596
- [9]
- B. Royer, S. Singh, and S. M. Girvin, “Stabilization of Finite-Energy Gottesman-Kitaev-Preskill States”, Physical Review Letters 125, (2020). DOI; 2009.07941
- [10]
- Brennan de Neeve et al., “Error correction of a logical grid state qubit by dissipative pumping”. 2010.09681
- [11]
- M. Rymarz et al., “Hardware-Encoding Grid States in a Nonreciprocal Superconducting Circuit”, Physical Review X 11, (2021). DOI; 2002.07718
- [12]
- B. Q. Baragiola et al., “All-Gaussian Universality and Fault Tolerance with the Gottesman-Kitaev-Preskill Code”, Physical Review Letters 123, (2019). DOI; 1903.00012
- [13]
- V. V. Sivak et al., “Real-time quantum error correction beyond break-even”. 2211.09116
- [14]
- S. Glancy and E. Knill, “Error analysis for encoding a qubit in an oscillator”, Physical Review A 73, (2006). DOI; quant-ph/0510107
- [15]
- C. Flühmann et al., “Encoding a qubit in a trapped-ion mechanical oscillator”, Nature 566, 513 (2019). DOI; 1807.01033
- [16]
- C. Flühmann and J. P. Home, “Direct Characteristic-Function Tomography of Quantum States of the Trapped-Ion Motional Oscillator”, Physical Review Letters 125, (2020). DOI; 1907.06478
- [17]
- P. Campagne-Ibarcq et al., “Quantum error correction of a qubit encoded in grid states of an oscillator”, Nature 584, 368 (2020). DOI; 1907.12487
- [18]
- Zhongchu Ni et al., “Beating the break-even point with a discrete-variable-encoded logical qubit”. 2211.09319
- [19]
- N. Fabre et al., “Generation of a time-frequency grid state with integrated biphoton frequency combs”, Physical Review A 102, (2020). DOI; 1904.01351
- [20]
- H. G. Feichtinger and T. Strohmer, editors , Gabor Analysis and Algorithms (Birkhäuser Boston, 1998). DOI
- [21]
- B. M. Terhal, J. Conrad, and C. Vuillot, “Towards scalable bosonic quantum error correction”, Quantum Science and Technology 5, 043001 (2020). DOI; 2002.11008
- [22]
- K. Noh, V. V. Albert, and L. Jiang, “Quantum Capacity Bounds of Gaussian Thermal Loss Channels and Achievable Rates With Gottesman-Kitaev-Preskill Codes”, IEEE Transactions on Information Theory 65, 2563 (2019). DOI; 1801.07271
- [23]
- Allan D. C. Tosta, Thiago O. Maciel, and Leandro Aolita, “Grand Unification of continuous-variable codes”. 2206.01751
- [24]
- T. Holstein and H. Primakoff, “Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet”, Physical Review 58, 1098 (1940). DOI
- [25]
- C. D. Cushen and R. L. Hudson, “A quantum-mechanical central limit theorem”, Journal of Applied Probability 8, 454 (1971). DOI

## Page edit log

- Victor V. Albert (2022-08-02) — most recent
- Victor V. Albert (2022-03-22)
- Victor V. Albert (2021-12-15)
- Yijia Xu (2021-12-14)

## Zoo code information

## Cite as:

“Gottesman-Kitaev-Preskill (GKP) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/gkp