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Error correction of a logical grid state qubit by dissipative pumping


Stabilization of encoded logical qubits using quantum error correction is crucial for the realization of reliable quantum computers. Although error-correcting codes implemented using individual physical qubits require many separate systems to be controlled, codes constructed using a quantum oscillator offer the possibility to perform error correction with a single physical entity. One powerful encoding approach for oscillators is the grid state or Gottesman–Kitaev–Preskill (GKP) encoding, which allows small displacement errors to be corrected. Here we introduce and implement a dissipative map designed for physically realistic finite GKP codes, which performs quantum error correction of a logical qubit encoded in the motion of a single trapped ion. The correction cycle involves two rounds, which correct small displacements in position and momentum. We demonstrate an extension in coherence time of logical states by a factor of three using both square and hexagonal GKP codes. The simple dissipative map used for this correction can be viewed as a type of reservoir engineering, which pumps into the manifold of highly non-classical GKP qubit states.

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Fig. 1: Comparison of infinite and finite state measurements.
Fig. 2: Finite state measurement of logical observables.
Fig. 3: Pumping to the logical subspace.
Fig. 4: Effect of error correction on logical states.

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Data availability

The datasets generated and analysed during this study will be made available on our group website ( and on the ETH Research Collection (

Code availability

The code used to perform experiments during this study is available from the corresponding author upon reasonable request.


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We thank P. Campagne-Ibarcq for stimulating discussions, and J. Hastrup for valuable feedback on the manuscript. As we were performing experiments and writing this paper, we became aware of parallel theoretical work on finite GKP state measurement28 and on finite GKP state stabilization29. J.P.H. thanks E. M. Home, P. D. Home and Y. Iida for support during the theoretical part of this work. We acknowledge support from the Swiss National Science Foundation through the National Centre of Competence in Research for Quantum Science and Technology (QSIT) grant 51NF40-160591 and from the Swiss National Science Foundation under grant no. 200020 165555/1.

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J.P.H. devised the pumping scheme and performed the initial simulations, which were then extended by T.-L.N. B.d.N. programmed the experimental sequences and experiments were carried out by T.-L.N., B.d.N. and T.B. T.-L.N. performed the data analysis. The paper was written by J.P.H. with input from all the authors.

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Correspondence to Jonathan P. Home.

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Extended data

Extended Data Fig. 1 Simulation of break-even (extended figure 1).

Simulation showing break-even where the lifetime of a square encoded YL state (red solid) can be extended longer than that of a \(\left(\left|0\right\rangle + \left|1\right\rangle\right)/\sqrt{2}\) Fock state superposition (dashed black). An exponential fit (dashed red) is also shown.

Extended Data Fig. 2 Displacements for projection to logical states (extended figure 2).

Illustration of displacements in phase space during logical state initialization into an eigenstate of a) XL, b) YL or c) ZL. Starting at the origin, we apply a global displacement (green arrows) followed by a measurement of the logical state (thick orange arrows, short bias displacements for finite measurements are not shown) and an appropriate feedback (blue arrows). The wave packet is split, following different feedback paths depending on whether it is a + eigenstate (red arrows) or − eigenstate (blue arrows) of the logical operator. We can initialize into either + or – eigenstates by tuning the feedback direction as shown by the solid or dashed arrows respectively.

Extended Data Fig. 3 Noise model and comparison with experiment (extended figure 3).

a) Trap frequency measured at a time delayed from a 50 Hz line trigger. The color shows the probability of finding the ion in \({\left|0\right\rangle }_{S}\) after the frequency calibration experiment described in the text. The solid line is a fit using the five lowest harmonics of 50 Hz. b) A measurement of motional coherence of the state \((\left|0\right\rangle +\left|1\right\rangle )/\sqrt{2}\), taken using a Ramsey sequence. An exponential fit yields a coherence time of 16.4(9) ms. The solid line is the result of the Monte-Carlo wavefunction simulation based on the noise parameters given in the text. c) Decay simulation of a GKP \(\left|{1}_{L}\right\rangle\) logical state under the independent action of each error channel: Markovian dephasing (orange), 50 Hz noise (green) and heating (blue) with (solid lines) and without (dashed lines) stabilization. For each error channel we optimise the frequency at which we apply the error correction. d-e) Comparison of data with simulation for the time evolution of logical readouts \(\left\langle {X}_{L}^{f}\right\rangle\) (orange), \(\left\langle {Y}_{L}^{f}\right\rangle\) (green) and \(\left\langle {Z}_{L}^{f}\right\rangle\) (blue) with stabilization (solid lines) and without stabilization (dashed lines) for both the d) square and e) hexagonal finite-GKP encoding. The simulation reproduces qualitatively the experimental data for both. The gray curves represent the coherence of the \((\left|0\right\rangle +\left|1\right\rangle )/\sqrt{2}\) Fock state superposition extracted from b).

Extended Data Fig. 4 Full data for logical readouts (extended figure 4).

Full datasets for logical readouts of \(\left\langle {X}_{L}^{f}\right\rangle\) (orange), \(\left\langle {Y}_{L}^{f}\right\rangle\) (green) and \(\left\langle {Z}_{L}^{f}\right\rangle\) (blue) with stabilization (circles) and without stabilization (crosses) for both a) square and b) hexagonal finite-GKP encoding. Exponential fits are also shown for the cases with stabilization (solid lines) and without stabilization (dashed lines). The short-time parts of both datasets are shown in figure 4 of the main text. The gray curves represent the coherence of the \((\left|0\right\rangle +\left|1\right\rangle )/\sqrt{2}\) Fock state superposition.

Extended Data Table 1 Parameters used to prepare an approximate GKP eigenstate \(\left|{\psi }_{L}\right\rangle\) in equation (16) using the measurement free method from19. In the notation of the main text \(\left|-{Z}_{L}\right\rangle =\left|{1}_{L}\right\rangle\)
Extended Data Table 2 Pulse sequence used for projective preparation of a GKP eigenstate \(\left|{\psi }_{L}\right\rangle\). It consists of three state-dependent displacements and one global displacement. Here \(\delta =\sqrt{\pi }/2\) and \(\epsilon \approx 0.06\sqrt{\pi }\). In the notation of the main text \(\left|+{Z}_{L}\right\rangle =\left|{0}_{L}\right\rangle\) and \(\left|-{Z}_{L}\right\rangle =\left|{1}_{L}\right\rangle\)

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de Neeve, B., Nguyen, TL., Behrle, T. et al. Error correction of a logical grid state qubit by dissipative pumping. Nat. Phys. 18, 296–300 (2022).

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