## Abstract

The stable operation of quantum computers will rely on error correction, in which single quantum bits of information are stored redundantly in the Hilbert space of a larger system. Such encoded qubits are commonly based on arrays of many physical qubits, but can also be realized using a single higher-dimensional quantum system, such as a harmonic oscillator^{1,2,3}. In such a system, a powerful encoding has been devised based on periodically spaced superpositions of position eigenstates^{4,5,6}. Various proposals have been made for realizing approximations to such states, but these have thus far remained out of reach^{7,8,9,10,11}. Here we demonstrate such an encoded qubit using a superposition of displaced squeezed states of the harmonic motion of a single trapped ^{40}Ca^{+} ion, controlling and measuring the mechanical oscillator through coupling to an ancillary internal-state qubit^{12}. We prepare and reconstruct logical states with an average squared fidelity of 87.3 ± 0.7 per cent. Also, we demonstrate a universal logical single-qubit gate set, which we analyse using process tomography. For Pauli gates we reach process fidelities of about 97 per cent, whereas for continuous rotations we use gate teleportation and achieve fidelities of approximately 89 per cent. This control method opens a route for exploring continuous variable error correction as well as hybrid quantum information schemes using both discrete and continuous variables^{13}. The code states also have direct applications in quantum sensing, allowing simultaneous measurement of small displacements in both position and momentum^{14,15}.

## Access options

Subscribe to Journal

Get full journal access for 1 year

$199.00

only $3.90 per issue

All prices are NET prices.

VAT will be added later in the checkout.

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

## Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon request.

## Additional information

**Publisher’s note:** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## References

- 1.
Chuang, I. L., Leung, D. W. & Yamamoto, Y. Bosonic quantum codes for amplitude damping.

*Phys. Rev. A***56**, 1114–1125 (1997). - 2.
Michael, M. H. et al. New class of quantum error-correcting codes for a bosonic mode.

*Phys. Rev. X***6**, 031006 (2016). - 3.
Lund, A. P., Ralph, T. C. & Haselgrove, H. L. Fault-tolerant linear optical quantum computing with small-amplitude coherent states.

*Phys. Rev. Lett*.**100**, 030503 (2008) - 4.
Gottesman, D., Kitaev, A. & Preskill, J. Encoding a qubit in an oscillator.

*Phys. Rev. A***64**, 012310 (2001). - 5.
Albert, V. V. et al. Performance and structure of single-mode bosonic codes.

*Phys. Rev. A***97**, 032346 (2018). - 6.
Noh, K., Albert, V. V. & Jiang, L. Improved quantum capacity bounds of Gaussian loss channels and achievable rates with Gottesman–Kitaev–Preskill codes. Preprint at https://arxiv.org/abs/1801.07271 (2018).

- 7.
Travaglione, B. C. & Milburn, G. J. Preparing encoded states in an oscillator.

*Phys. Rev. A***66**, 052322 (2002). - 8.
Pirandola, S., Mancini, S., Vitali, D. & Tombesi, P. Continuous variable encoding by ponderomotive interaction.

*Eur. Phys. J. D***37**, 283–290 (2006). - 9.
Vasconcelos, H. M., Sanz, L. & Glancy, S. All-optical generation of states for “encoding a qubit in an oscillator”.

*Opt. Lett*.**35**, 3261–3263 (2010). - 10.
Terhal, B. M. & Weigand, D. Encoding a qubit into a cavity mode in circuit QED using phase estimation.

*Phys. Rev. A***93**, 012315 (2016). - 11.
Motes, K. R., Baragiola, B. Q., Gilchrist, A. & Menicucci, N. C. Encoding qubits into oscillators with atomic ensembles and squeezed light.

*Phys. Rev. A***95**, 053819 (2017). - 12.
Flühmann, C., Negnevitsky, V., Marinelli, M. & Home, J. P. Sequential modular position and momentum measurements of a trapped ion mechanical oscillator.

*Phys. Rev. X***8**, 021001 (2018). - 13.
Andersen, U. L., Neergaard-Nielsen, J. S., van Loock, P. & Furusawa, A. Hybrid discrete- and continuous-variable quantum information.

*Nat. Phys*.**11**, 713 (2015). - 14.
Duivenvoorden, K., Terhal, B. M. & Weigand, D. Single-mode displacement sensor.

*Phys. Rev. A***95**, 012305 (2017). - 15.
Neumann, J. V.

*Mathematical Foundations of Quantum Mechanics*(Princeton Univ. Press, Princeton, 1996). - 16.
Devitt, S. J., Munro, W. J. & Nemoto, K. Quantum error correction for beginners.

*Rep. Prog. Phys*.**76**, 076001 (2013). - 17.
Heeres, R. W. et al. Implementing a universal gate set on a logical qubit encoded in an oscillator.

*Nat. Commun*.**8**, 94 (2017). - 18.
Ofek, N. et al. Extending the lifetime of a quantum bit with error correction in superconducting circuits.

*Nature***536**, 441–445 (2016). - 19.
Haljan, P. C., Brickman, K.-A., Deslauriers, L., Lee, P. J. & Monroe, C. Spin-dependent forces on trapped ions for phase-stable quantum gates and entangled states of spin and motion.

*Phys. Rev. Lett*.**94**, 153602 (2005). - 20.
Schleich, W. P.

*WKB and Berry Phase*171–188 (Wiley-VCH, Berlin, 2005). - 21.
Glancy, S. & Knill, E. Error analysis for encoding a qubit in an oscillator.

*Phys. Rev. A***73**, 012325 (2006). - 22.
Kienzler, D. et al. Quantum harmonic oscillator state synthesis by reservoir engineering.

*Science***347**, 53–56 (2015). - 23.
Kienzler, D. et al. Observation of quantum interference between separated mechanical oscillator wave packets.

*Phys. Rev. Lett*.**116**, 140402 (2016). - 24.
Lo, H.-Y. et al. Spin-motion entanglement and state diagnosis with squeezed oscillator wavepackets.

*Nature***521**, 336–339 (2015). - 25.
Wallentowitz, S. & Vogel, W. Reconstruction of the quantum mechanical state of a trapped ion.

*Phys. Rev. Lett*.**75**, 2932–2935 (1995). - 26.
Leibfried, D. et al. Experimental determination of the motional quantum state of a trapped atom.

*Phys. Rev. Lett*.**77**, 4281–4285 (1996). - 27.
Knill, E., Laflamme, R. & Zurek, W. H. Resilient quantum computation: error models and thresholds.

*Proc. R. Soc. Lond. A***454**, 365–384 (1998). - 28.
Nielsen, M. A. & Chuang, I. L.

*Quantum Computation and Quantum Information: 10th Anniversary Edition*(Cambridge Univ. Press, Cambridge, 2010). - 29.
Brown, K. R. et al. Coupled quantized mechanical oscillators.

*Nature***471**, 196–199 (2011). - 30.
Harlander, M., Lechner, R., Brownnutt, M., Blatt, R. & Hänsel, W. Trapped-ion antennae for the transmission of quantum information.

*Nature***471**, 200–203 (2011). - 31.
Toyoda, K., Hiji, R., Noguchi, A. & Urabe, S. Hong–Ou–Mandel interference of two phonons in trapped ions.

*Nature***527**, 74 (2015). - 32.
Fukui, K., Tomita, A., Okamoto, A. & Fujii, K. High-threshold fault-tolerant quantum computation with analog quantum error correction.

*Phys. Rev. X***8**, 021054 (2018). - 33.
Vuillot, C., Asasi, H., Wang, Y., Pryadko, L. P. & Terhal, B. M. Quantum error correction with the Toric-GKP code. Preprint at http://arxiv.org/abs/1810.00047 (2018).

- 34.
Ketterer, A.

*Modular Variables in Quantum Information*. PhD thesis, Univ. Sorbonne Paris Cité and Univ. Paris Diderot (2016). - 35.
Kienzler, D.

*Quantum Harmonic Oscillator State Synthesis by Reservoir Engineering*. PhD thesis, ETH Zürich (2015). - 36.
Wineland, D. J. et al. Experimental issues in coherent quantum-state manipulation of trapped atomic ions.

*J. Res. Natl Inst. Stand. Technol*.**103**, 259–328 (1998). - 37.
Kienzler, D. et al. Quantum harmonic oscillator state control in a squeezed Fock basis.

*Phys. Rev. Lett*.**119**, 033602 (2017). - 38.
Wolf, F. et al. Motional Fock states for quantum-enhanced amplitude and phase measurements with trapped ions. Preprint at http://arxiv.org/abs/1807.01875 (2018).

- 39.
Gerritsma, R. et al. Quantum simulation of the Klein paradox with trapped ions.

*Phys. Rev. Lett*.**106**, 060503 (2011). - 40.
Zähringer, F. et al. Realization of a quantum random walk with one and two trapped ions.

*Phys. Rev. Lett*.**104**, 100503 (2010). - 41.
Gerritsma, R. et al. Quantum simulation of the Dirac equation.

*Nature***463**, 68–71 (2010). - 42.
Efron, B. & Tibshirani, R. J.

*An Introduction to the Bootstrap*(Chapman & Hall/CRC, Boca Raton, 1993). - 43.
James, D. F. V., Kwiat, P. G., Munro, W. J. & White, A. G. Measurement of qubits.

*Phys. Rev. A***64**, 052312 (2001). - 44.
Bhandari, R. & Peters, N. A. On the general constraints in single qubit quantum process tomography.

*Sci. Rep*.**6**, 26004 (2016).

## Acknowledgements

We thank D. Kienzler, L. de Clercq and H.-Y. Lo for important contributions to the apparatus. 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. We acknowledge support from the Swiss National Science Foundation under grant no. 200020_165555/1. K.M. was supported by an ETH Zürich Postdoctoral Fellowship 17-1-FEL050. The research is partly based upon work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the US Army Research Office grant W911NF-16-1-0070. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA or the US Government. The US Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the view of the US Army Research Office.

### Reviewer information

*Nature* thanks Alessandro Ferraro and the other anonymous reviewer(s) for their contribution to the peer review of this work.

## Author information

### Affiliations

#### Institute for Quantum Electronics, ETH Zürich, Zürich, Switzerland

- C. Flühmann
- , T. L. Nguyen
- , M. Marinelli
- , V. Negnevitsky
- , K. Mehta
- & J. P. Home

### Authors

### Search for C. Flühmann in:

### Search for T. L. Nguyen in:

### Search for M. Marinelli in:

### Search for V. Negnevitsky in:

### Search for K. Mehta in:

### Search for J. P. Home in:

### Contributions

Experimental data were taken and analysed by C.F., using an apparatus with significant contributions from V.N., M.M., C.F., T.L.N. and K.M. The paper was written by C.F. and J.P.H., with input from all authors. Experiments were conceived by C.F. and J.P.H.

### Competing interests

The authors declare no competing interests.

### Corresponding authors

Correspondence to C. Flühmann or J. P. Home.

## Extended data figures and tables

### Extended Data Fig. 1 Calibration techniques in phase space.

Here we show how the required properties of the tickling pulse are determined (‘calibrated’) and the use of the tickling pulse for motional frequency calibrations.

**a**, Matching of the tickling pulse to the SDF pulse (see Methods for nomenclature). The squeezed ion motional state (dashed state labelled 0) is displaced using an SDF together with two π/2 internal state rotations (SDFz). This realizes the displaced squeezed state 1. A subsequent tickling pulse is calibrated in order to revert the displacement implemented by the laser. After this shift the oscillator is in state 2. Whether or not the squeezed state returns to the squeezed vacuum can be probed using the squeezed basis analogue of the red sideband^{22}. Shown is the case of a laser displacement along the squeezed axis, which enhances sensitivity for the tickling coupling strength.**b**, Similarly, a laser displacement perpendicular to the squeezed axis is used to calibrate the direction of the tickling pulse.**c**, Motional frequency calibration. The ion is ground state cooled (0), then a coherent state (1) is created by a first tickling pulse. The state evolves freely during the wait time*T*and rotates by an angle*Tδ*, with*δ*the detuning from the angular motional frequency*ω*_{m}. A second tickling pulse inverts the first displacement. Because of the detuning, the final state (3) does not return to the ground state, which can be detected applying a red sideband probe pulse.### Extended Data Fig. 2 Example of a pulse sequence.

This pulse sequence is used during process tomography of the T-gate. The blue line shows laser pulses based on the 397 nm laser used for cooling and fluorescence detection of the internal states. The red line shows manipulations using the 729 nm laser used for SDF pulses as well as carrier rotations, while the black line denotes tickling pulses implemented using an RF voltage. The upper row shows the sequence used for preparing \({\left|0\right\rangle }_{L}\), including initial cooling, squeezed state preparation (‘squeezed pumping’) and modular variable measurements (‘Mod(l)’). The lower row shows first the implementation of \({\hat{U}}_{L}^{X}({\rm{\pi }}/2,{\rm{\pi }}/2)\) by gate teleportation (creating \({|{\Phi }_{+}\rangle }_{L}\)), followed by application of a teleported T-gate (‘\(\hat{T}\)-gate’) and subsequently the readout of the states.

### Extended Data Fig. 3 Grid qubit lifetime measurements.

**a**–**f**, States are prepared (left column, \({\left|0\right\rangle }_{L}\); right column, \({\left|+\right\rangle }_{L}\)), and after a variable wait time the state is read out. The resulting measurement data (blue points with s.e.m. error bars) are fitted with an exponential decay*A*e^{−t/T}(solid line). For prepared state \({\left|0\right\rangle }_{L}\):**a**, readout of \({\hat{Z}}_{L}\), from which we find*T*= 3.7 ± 0.2 ms;**c**, readout of \({\hat{S}}_{X}\) with*T*= 0.8 ± 0.1 ms; and**e**, readout of \({\hat{S}}_{Z}\) with*T*= 1.1 ± 0.1 ms. For prepared state \({\left|+\right\rangle }_{L}\):**b**, readout of \({\hat{X}}_{L}\) with*T*= 3.6 ± 0.3 ms;**d**, readout of \({\hat{S}}_{X}\) with*T*= 1.0 ± 0.1 ms;**f**, readout of \({\hat{S}}_{Z}\) with*T*= 0.7 ± 0.1 ms.### Extended Data Fig. 4 Two qubit gate implemented in two modes of a single trapped ion.

This circuit implements \({\hat{\sigma }}_{L}^{i}\)-controlled \(\pm {\hat{\sigma }}_{L}^{j}\) operations between two grid state qubits \(\left|{Q}_{1}\right\rangle \) and \(\left|{Q}_{2}\right\rangle \) mediated by one internal ancillary qubit \(\left|0\right\rangle \leftrightarrow \left|1\right\rangle \). The required operations are shown in the left circuit, while on the right the equivalent logical operation is shown. The sign of the operation is determined by the ancillary qubit readout. See Methods for details.

## Rights and permissions

To obtain permission to re-use content from this article visit RightsLink.

## About this article

## Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.