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

Dissipative stabilization of finite-energy square-lattice GKP states using stabilizers conjugated by a cooling ([5], Appx. B) or damping operator, i.e., a damped exponential of the total occupation number [6,7]. Preparation of approximate square-lattice GKP states has been studied both theoretically and experimentally [2,810]. Various damped versions of GKP states are equivalent [11,12], and there exists a Fock-state expansion [13; Appx. A].Two Josephson junctions coupled by a gyrator [14].Periodic driving (a.k.a. Floquet engineering) [15].Approximate GKP states can be prepared using Gaussian operations and photon detectors [16].An optimal-size circuit using ancillary qubits can be used to prepare an approximate GKP state [17]. The size of the circuit is linear in the logarithm of the approximation parameters of the GKP codes.

Gates

By applying square-lattice GKP error correction to Gaussian input states, universality can be achieved without non-Gaussian elements [18].

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 [19] based on Knill error correction (a.k.a. telecorrection [20]), which is based on teleportation [21,22].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 [23]. It has been extended to utilize previously measured syndrome information [24].

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 \(\sqrt{\pi}/6\) [25].

Realizations

Motional degree of freedom of a trapped ion: square-lattice GKP encoding realized with the help of post-selection by Home group [26,27], 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 [7]. State preparation also realized by Tan group [28]. Universal gate set, including a two-qubit entangling gate, realized by Tan group [29]. State initialization and application to measuring displacements [30].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 [31]. Subsequent paper by Devoret group [23] 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 [32]). See Ref. [33] for another experiment. A feed-forward-free, i.e., fully autonomous protocol has also been implemented by Nord Quantique [34]. Qudit encodings with \(q=3,4\) have been realized, with logical error rates also beyond break even [35].GKP states and homodyne measurements have been realized in propagating telecom light by the Furusawa group [36].Single-qubit \(Z\)-gate has been demonstrated [37] in the single-photon subspace of an infinite-mode space [38], in which time and frequency become bosonic conjugate variables of a single effective bosonic mode. In this context, GKP position-state wavefunctions are called Dirac combs or frequency combs.

Notes

Single-mode GKP states have been introduced in quantum foundations research defining modular conjugate variables [39] and in coherent-state theory associated with the Heisenberg-Weyl group [40,41][42; Sec. 1.5 and 3.2].The basis formed by the code and error states of a single-mode GKP code is known as the Zak basis, Weil-Brezin transform, or \(kq\) representation in condensed-matter physics [43] and signal processing [44; Ch. 1][45; Eq. (1.112)]. Expansion of a function on \(\mathbb{R}\) in terms of this basis is called the Zak transform [44].

Parents

Cousins

  • Approximate quantum error-correcting code (AQECC) — Square-lattice GKP codes approximately protect against photon loss [2,3,46].
  • 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 — GKP codewords, when written in terms of coherent states, form a square lattice in phase space.
  • Fusion-based quantum computing (FBQC) code — GKP states can be used to perform computation in a fusion-based encoding [47].
  • Kitaev current-mirror qubit code — Current-mirror code phase gates utilize ancillary osillators in square-lattice GKP states [48,49].
  • Zero-pi qubit code — Zero-pi code phase gates utilize ancillary osillators in square-lattice GKP states [48,49].
  • 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 [50].
  • Asymmetric quantum code — GKP code parameters against position and momentum displacements can be tuned by the choice of lattice (e.g., square vs rectangular).
  • Modular-qudit GKP code — The square-lattice GKP code can be obtained from the modular-qudit code by taking the physical qudit dimension to be infinite [1; Sec. II].
  • Spin GKP code — Spin-GKP code constructions utilize the Holstein-Primakoff mapping [51] (see also [52]) to convert various expressions for square-lattice 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) arXiv:quant-ph/0008040 DOI
[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]
N. C. Menicucci, “Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States”, Physical Review Letters 112, (2014) arXiv:1310.7596 DOI
[6]
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
[7]
B. de Neeve, T. L. Nguyen, T. Behrle, and J. Home, “Error correction of a logical grid state qubit by dissipative pumping”, (2020) arXiv:2010.09681
[8]
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
[9]
P. Campagne-Ibarcq et al., “Quantum error correction of a qubit encoded in grid states of an oscillator”, Nature 584, 368 (2020) DOI
[10]
I. Tzitrin, J. E. Bourassa, N. C. Menicucci, and K. K. Sabapathy, “Progress towards practical qubit computation using approximate Gottesman-Kitaev-Preskill codes”, Physical Review A 101, (2020) arXiv:1910.03673 DOI
[11]
T. Matsuura, H. Yamasaki, and M. Koashi, “Equivalence of approximate Gottesman-Kitaev-Preskill codes”, Physical Review A 102, (2020) arXiv:1910.08301 DOI
[12]
L. J. Mensen, B. Q. Baragiola, and N. C. Menicucci, “Phase-space methods for representing, manipulating, and correcting Gottesman-Kitaev-Preskill qubits”, Physical Review A 104, (2021) arXiv:2012.12488 DOI
[13]
B. M. Terhal, J. Conrad, and C. Vuillot, “Towards scalable bosonic quantum error correction”, Quantum Science and Technology 5, 043001 (2020) arXiv:2002.11008 DOI
[14]
M. Rymarz, S. Bosco, A. Ciani, and D. P. DiVincenzo, “Hardware-Encoding Grid States in a Nonreciprocal Superconducting Circuit”, Physical Review X 11, (2021) arXiv:2002.07718 DOI
[15]
X. C. Kolesnikow, R. W. Bomantara, A. C. Doherty, and A. L. Grimsmo, “Gottesman-Kitaev-Preskill State Preparation Using Periodic Driving”, Physical Review Letters 132, (2024) arXiv:2303.03541 DOI
[16]
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
[17]
L. Brenner, L. Caha, X. Coiteux-Roy, and R. Koenig, “The complexity of Gottesman-Kitaev-Preskill states”, (2024) arXiv:2410.19610
[18]
B. Q. Baragiola, G. Pantaleoni, R. N. Alexander, A. Karanjai, and N. C. Menicucci, “All-Gaussian Universality and Fault Tolerance with the Gottesman-Kitaev-Preskill Code”, Physical Review Letters 123, (2019) arXiv:1903.00012 DOI
[19]
B. W. Walshe, B. Q. Baragiola, R. N. Alexander, and N. C. Menicucci, “Continuous-variable gate teleportation and bosonic-code error correction”, Physical Review A 102, (2020) arXiv:2008.12791 DOI
[20]
C. M. Dawson, H. L. Haselgrove, and M. A. Nielsen, “Noise thresholds for optical cluster-state quantum computation”, Physical Review A 73, (2006) arXiv:quant-ph/0601066 DOI
[21]
E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005) arXiv:quant-ph/0410199 DOI
[22]
E. Knill, “Scalable Quantum Computation in the Presence of Large Detected-Error Rates”, (2004) arXiv:quant-ph/0312190
[23]
V. V. Sivak et al., “Real-time quantum error correction beyond break-even”, Nature 616, 50 (2023) arXiv:2211.09116 DOI
[24]
M. Puviani, S. Borah, R. Zen, J. Olle, and F. Marquardt, “Boosting the Gottesman-Kitaev-Preskill quantum error correction with non-Markovian feedback”, (2023) arXiv:2312.07391
[25]
S. Glancy and E. Knill, “Error analysis for encoding a qubit in an oscillator”, Physical Review A 73, (2006) arXiv:quant-ph/0510107 DOI
[26]
C. Flühmann, T. L. Nguyen, M. Marinelli, V. Negnevitsky, K. Mehta, and J. P. Home, “Encoding a qubit in a trapped-ion mechanical oscillator”, Nature 566, 513 (2019) arXiv:1807.01033 DOI
[27]
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
[28]
V. G. Matsos, C. H. Valahu, T. Navickas, A. D. Rao, M. J. Millican, X. C. Kolesnikow, M. J. Biercuk, and T. R. Tan, “Robust and Deterministic Preparation of Bosonic Logical States in a Trapped Ion”, Physical Review Letters 133, (2024) arXiv:2310.15546 DOI
[29]
V. G. Matsos, C. H. Valahu, M. J. Millican, T. Navickas, X. C. Kolesnikow, M. J. Biercuk, and T. R. Tan, “Universal Quantum Gate Set for Gottesman-Kitaev-Preskill Logical Qubits”, (2024) arXiv:2409.05455
[30]
C. H. Valahu, M. P. Stafford, Z. Huang, V. G. Matsos, M. J. Millican, T. Chalermpusitarak, N. C. Menicucci, J. Combes, B. Q. Baragiola, and T. R. Tan, “Quantum-Enhanced Multi-Parameter Sensing in a Single Mode”, (2024) arXiv:2412.04865
[31]
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
[32]
Z. Ni et al., “Beating the break-even point with a discrete-variable-encoded logical qubit”, Nature 616, 56 (2023) arXiv:2211.09319 DOI
[33]
M. Kudra et al., “Robust Preparation of Wigner-Negative States with Optimized SNAP-Displacement Sequences”, PRX Quantum 3, (2022) arXiv:2111.07965 DOI
[34]
D. Lachance-Quirion et al., “Autonomous quantum error correction of Gottesman-Kitaev-Preskill states”, (2023) arXiv:2310.11400
[35]
B. L. Brock, S. Singh, A. Eickbusch, V. V. Sivak, A. Z. Ding, L. Frunzio, S. M. Girvin, and M. H. Devoret, “Quantum Error Correction of Qudits Beyond Break-even”, (2024) arXiv:2409.15065
[36]
S. Konno et al., “Logical states for fault-tolerant quantum computation with propagating light”, Science 383, 289 (2024) arXiv:2309.02306 DOI
[37]
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
[38]
É. Descamps, A. Keller, and P. Milman, “Gottesman-Kitaev-Preskill encoding in continuous modal variables of single photons”, (2024) arXiv:2310.12618
[39]
Y. Aharonov, H. Pendleton, and A. Petersen, “Modular variables in quantum theory”, International Journal of Theoretical Physics 2, 213 (1969) DOI
[40]
Cartier, Pierre. "Quantum mechanical commutation relations and theta functions." Proc. Sympos. Pure Math. Vol. 9. 1966.
[41]
A. M. Perelomov, “Coherent states and theta functions”, Functional Analysis and Its Applications 6, 292 (1973) DOI
[42]
A. Perelomov, Generalized Coherent States and Their Applications (Springer Berlin Heidelberg, 1986) DOI
[43]
J. Zak, “Finite Translations in Solid-State Physics”, Physical Review Letters 19, 1385 (1967) DOI
[44]
H. G. Feichtinger and T. Strohmer, editors , Gabor Analysis and Algorithms (Birkhäuser Boston, 1998) DOI
[45]
G. B. Folland, “Harmonic Analysis in Phase Space. (AM-122)”, (1989) DOI
[46]
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) arXiv:1801.07271 DOI
[47]
Doherty, A., Gimeno-segovia, M., Litinski, D., Nickerson, N., Pant, M., Rudolph, T. and Sparrow, C., Psiquantum, Corp., 2024. GENERATION AND MEASUREMENT OF ENTANGLED SYSTEMS OF PHOTONIC GKP QUBITS. U.S. Patent Application 18/273,753.
[48]
A. Kitaev, “Protected qubit based on a superconducting current mirror”, (2006) arXiv:cond-mat/0609441
[49]
P. Brooks, A. Kitaev, and J. Preskill, “Protected gates for superconducting qubits”, Physical Review A 87, (2013) arXiv:1302.4122 DOI
[50]
A. D. C. Tosta, T. O. Maciel, and L. Aolita, “Grand Unification of continuous-variable codes”, (2022) arXiv:2206.01751
[51]
T. Holstein and H. Primakoff, “Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet”, Physical Review 58, 1098 (1940) DOI
[52]
C. D. Cushen and R. L. Hudson, “A quantum-mechanical central limit theorem”, Journal of Applied Probability 8, 454 (1971) DOI
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

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} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/gkp

Cite as:

“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/edit/main/codes/quantum/oscillators/stabilizer/lattice/gkp.yml.