Square-lattice GKP code[1] 

Description

Single-mode GKP qudit-into-oscillator code based on the rectangular lattice. Its stabilizer generators 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 a constraint that \(\alpha\beta=2q\pi\) where \(q\) is an integer denoting the logical dimension.

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.

The \(q=1\) trivial encoding is spanned by the canonical GKP state or grid state, \begin{align} |GKP\rangle=\sum_{n\in\mathbb{Z}}|x=n\sqrt{2\pi}\rangle~, \tag*{(1)}\end{align} where \(|x\rangle\) are single-mode position states.

Protection

For stabilizer \(\hat{S}_q(2\alpha),\hat{S}_p(2\beta)\), code can correct displacement errors up to \(\alpha/2\) in the \(q\)-direction and \(\beta/2\) 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 square-lattice GKP states is studied both theoretically and experimentally by putting the GKP lattice inside a Gaussian envelope [2,5–7].Dissipative stabilization of finite-energy square-lattice 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].Periodic driving (a.k.a. Floquet engineering) [12].Approximate GKP states can be prepared using Gaussian operations and photon detectors [13].

Gates

By applying square-lattice GKP error correction to Gaussian input states, universality can be achieved without non-Gaussian elements [14].

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].Decoder [15] based on Knill error correction (a.k.a. telecorrection [16]), which is based on teleportation [17,18].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 [19].

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 \(\sqrt{\pi}/6\) [20].

Realizations

Motional degree of freedom of a trapped ion: square-lattice GKP encoding realized with the help of post-selection [21,22], 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 square-lattice GKP error correction, where displacement error syndromes are measured to one bit of precision using an ancillary transmon [23]. Subsequent paper [19] 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 [24]).Single-qubit \(Z\)-gate has been demonstrated in the single-photon subspace of an infinite-mode space [25], 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 [26].

Notes

Reviews on GKP codes presented in Refs. [4,27].

Parents

Cousins

  • Approximate quantum error-correcting code (AQECC) — Square-lattice GKP codes approximately protect against photon loss [2,3,28].
  • Rotor code — Because square-lattice GKP error states are parameterized by two modular (i.e., periodic) variables of position and momentum, measuring one of the GKP stabilizers constrains the oscillator Hilbert space into that of a rotor.
  • \(\mathbb{Z}^n\) hypercubic lattice code — GKP codewords, when written in terms of coherent states, form a square lattice in phase space.
  • Kitaev current-mirror qubit code — Current-mirror code phase gates utilize ancillary osillators in square-lattice GKP states [29,30].
  • Zero-pi qubit code — Zero-pi code phase gates utilize ancillary osillators in square-lattice GKP states [29,30].
  • Rotor GKP code — GKP (rotor GKP) codes protect against shifts in linear (angular) degrees of freedom.
  • Number-phase code — Square-lattice GKP codes utilize translational symmetry in phase space, while number-phase codes utilize rotational symmetry. The two are related via a mapping [31].
  • Spin GKP code — Spin-GKP code constructions utilize the Holstein-Primakoff mapping [32] (see also [33]) to convert various expressions for square-lattice GKP states into codes for spin systems.

References

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[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) arXiv:1506.05033 DOI
[3]
V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018) arXiv:1708.05010 DOI
[4]
A. L. Grimsmo and S. Puri, “Quantum Error Correction with the Gottesman-Kitaev-Preskill Code”, PRX Quantum 2, (2021) arXiv:2106.12989 DOI
[5]
D. J. Weigand and B. M. Terhal, “Generating grid states from Schrödinger-cat states without postselection”, Physical Review A 97, (2018) arXiv:1709.08580 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
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I. Tzitrin et al., “Progress towards practical qubit computation using approximate Gottesman-Kitaev-Preskill codes”, Physical Review A 101, (2020) arXiv:1910.03673 DOI
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N. C. Menicucci, “Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States”, Physical Review Letters 112, (2014) arXiv:1310.7596 DOI
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B. Royer, S. Singh, and S. M. Girvin, “Stabilization of Finite-Energy Gottesman-Kitaev-Preskill States”, Physical Review Letters 125, (2020) arXiv:2009.07941 DOI
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B. de Neeve et al., “Error correction of a logical grid state qubit by dissipative pumping”, (2020) arXiv:2010.09681
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M. Rymarz et al., “Hardware-Encoding Grid States in a Nonreciprocal Superconducting Circuit”, Physical Review X 11, (2021) arXiv:2002.07718 DOI
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X. C. Kolesnikow et al., “Gottesman-Kitaev-Preskill state preparation using periodic driving”, (2023) arXiv:2303.03541
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D. Su, C. R. Myers, and K. K. Sabapathy, “Conversion of Gaussian states to non-Gaussian states using photon-number-resolving detectors”, Physical Review A 100, (2019) arXiv:1902.02323 DOI
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B. Q. Baragiola et al., “All-Gaussian Universality and Fault Tolerance with the Gottesman-Kitaev-Preskill Code”, Physical Review Letters 123, (2019) arXiv:1903.00012 DOI
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B. W. Walshe et al., “Continuous-variable gate teleportation and bosonic-code error correction”, Physical Review A 102, (2020) arXiv:2008.12791 DOI
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E. Knill, “Scalable Quantum Computation in the Presence of Large Detected-Error Rates”, (2004) arXiv:quant-ph/0312190
[19]
V. V. Sivak et al., “Real-time quantum error correction beyond break-even”, Nature 616, 50 (2023) arXiv:2211.09116 DOI
[20]
S. Glancy and E. Knill, “Error analysis for encoding a qubit in an oscillator”, Physical Review A 73, (2006) arXiv:quant-ph/0510107 DOI
[21]
C. Flühmann et al., “Encoding a qubit in a trapped-ion mechanical oscillator”, Nature 566, 513 (2019) arXiv:1807.01033 DOI
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C. Flühmann and J. P. Home, “Direct Characteristic-Function Tomography of Quantum States of the Trapped-Ion Motional Oscillator”, Physical Review Letters 125, (2020) arXiv:1907.06478 DOI
[23]
P. Campagne-Ibarcq et al., “Quantum error correction of a qubit encoded in grid states of an oscillator”, Nature 584, 368 (2020) arXiv:1907.12487 DOI
[24]
Z. Ni et al., “Beating the break-even point with a discrete-variable-encoded logical qubit”, Nature 616, 56 (2023) arXiv:2211.09319 DOI
[25]
N. Fabre et al., “Generation of a time-frequency grid state with integrated biphoton frequency combs”, Physical Review A 102, (2020) arXiv:1904.01351 DOI
[26]
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[28]
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[30]
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[33]
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Zoo Code ID: gkp

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

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