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Encoding a qubit in a trapped-ion mechanical oscillator


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 oscillator1,2,3. In such a system, a powerful encoding has been devised based on periodically spaced superpositions of position eigenstates4,5,6. Various proposals have been made for realizing approximations to such states, but these have thus far remained out of reach7,8,9,10,11. Here we demonstrate such an encoded qubit using a superposition of displaced squeezed states of the harmonic motion of a single trapped 40Ca+ ion, controlling and measuring the mechanical oscillator through coupling to an ancillary internal-state qubit12. 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 variables13. The code states also have direct applications in quantum sensing, allowing simultaneous measurement of small displacements in both position and momentum14,15.

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Fig. 1: Grid state encoding and control.
Fig. 2: Logical readout.
Fig. 3: Arbitrary single-qubit operations.
Fig. 4: Process tomography of logical operations, characterized by the χ matrix.

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.


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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




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.

Corresponding authors

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

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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 sideband22. 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 , with δ the detuning from the angular motional frequency ωm. 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.

af, 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 Aet/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.

Extended Data Table 1 Creation of code states

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Flühmann, C., Nguyen, T.L., Marinelli, M. et al. Encoding a qubit in a trapped-ion mechanical oscillator. Nature 566, 513–517 (2019).

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