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Entanglement and iSWAP gate between molecular qubits

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Abstract

Quantum computation and simulation rely on long-lived qubits with controllable interactions. Trapped polar molecules have been proposed as a promising quantum computing platform, offering scalability and single-particle addressability while still leveraging inherent complexity and strong couplings of molecules1,2,3,4,5. Recent progress in the single quantum state preparation and coherence of the hyperfine-rotational states of individually trapped molecules allows them to serve as promising qubits6,7,8,9,10,11, with intermolecular dipolar interactions creating entanglement12,13. However, universal two-qubit gates have not been demonstrated with molecules. Here we harness intrinsic molecular resources to implement a two-qubit iSWAP gate using individually trapped X1Σ+ NaCs molecules. By allowing the molecules to interact for 664 μs at a distance of 1.9 μm, we create a maximally entangled Bell state with a fidelity of 94(3)% in trials in which both molecules are present. Using motion–rotation coupling, we measure residual excitation of the lowest few motional states along the axial trapping direction and find them to be the primary source of decoherence. Finally, we identify two non-interacting hyperfine states within the ground rotational level in which we encode a qubit. The interaction is toggled by transferring between interacting and non-interacting states to realize an iSWAP gate. We verify the gate performance by measuring its logical truth table.

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Fig. 1: NaCs molecules in optical tweezers for dipolar interactions and quantum logic gates.
Fig. 2: Characterization of dipole–dipole-mediated rotational exchange interaction and entanglement, using data obtained by post-selection to include only cases in which both molecules are detected.
Fig. 3: Coherent motion–rotation coupling to characterize molecule temperature and trap displacement ζ.
Fig. 4: Engineering dipolar interaction and the iSWAP gate between hyperfine-state-encoded qubits.

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Data availability

All data presented in this paper are available in the Harvard Dataverse71. Other supporting data are available from the corresponding author upon reasonable request.

Code availability

The code used for modelling in this study is available in the Harvard Dataverse71.

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Acknowledgements

We thank J. Zhang for early experimental contributions; T. Rosenband, G. Pupillo, S. Jandura, M. Bergonzoni and B. Zhu for their discussions; and A. Carroll and A. Carter for careful reading of the paper. This work is supported by AFOSR (FA9550-23-1-0538), NSF (PHY-2110225 & PFC-PHY-2317149) and AFOSR-MURI (FA9550-20-1-0323 and FA9550-21-1-0069). Note that ref. 72 is a related molecule work leveraging hyperfine states as qubits.

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  1. These authors contributed equally: Lewis R. B. Picard, Annie J. Park

Authors and Affiliations

  1. Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA, USA

    Lewis R. B. Picard, Annie J. Park, Gabriel E. Patenotte, Samuel Gebretsadkan & Kang-Kuen Ni

  2. Department of Physics, Harvard University, Cambridge, MA, USA

    Lewis R. B. Picard, Annie J. Park, Gabriel E. Patenotte, Samuel Gebretsadkan & Kang-Kuen Ni

  3. Harvard-MIT Center for Ultracold Atoms, Cambridge, MA, USA

    Lewis R. B. Picard, Annie J. Park, Gabriel E. Patenotte, Samuel Gebretsadkan & Kang-Kuen Ni

  4. JILA, National Institute of Standards and Technology and Department of Physics, University of Colorado, Boulder, CO, USA

    David Wellnitz & Ana Maria Rey

  5. Center for Theory of Quantum Matter, University of Colorado, Boulder, CO, USA

    David Wellnitz & Ana Maria Rey

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Contributions

L.R.B.P. and A.J.P. collected and analysed the data. G.E.P. assisted with the data collection and developed molecular structure and aberration models. S.G. provided experimental support. D.W. performed the theoretical work and numerical modelling. A.M.R. supervised the theoretical work. K.-K.N. proposed and supervised the experiment. All authors contributed to the preparation of the paper.

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Correspondence to Annie J. Park or Kang-Kuen Ni.

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Extended data figures and tables

Extended Data Fig. 1 Dipole-dipole interactions using the same sequence as illustrated in Fig. 2 for intermolecular distances of 2.2 and 2.5 μm.

Solid lines are fits to a master equation model of interaction with phenomenological dephasing parameters.

Extended Data Fig. 2 Bell state infidelity at the optimal time predicted as a function of inverse temperature.

The solid purple curve illustrates the prediction of the full model including astigmatism of 0.13λ. The green curve illustrates the corresponding result without any astigmatism. The two dashed curves show simulations without single-molecule decoherence γdeph = 0, with (orange) and without (blue) astigmatism. Numbers quoted in the main text at 80% ground state fraction correspond to ħωax/(kBT) = 1.6. Computed with motional level cutoffs n = 25 [ħωax/(kBT) = 0.4]; n = 20 [0.5 ≤ ħωax/(kBT) ≤ 1.2]; n = 15 [1.3 ≤ ħωax/(kBT) ≤ 2]; n = 10 [2 ≤ ħωax/(kBT)].

Extended Data Fig. 3 Hyperfine-changing microwave pulse from \(| {\boldsymbol{e}}\rangle \) to \(| {\bf{1}}\rangle \).

Fit is a \({\cos }^{2}\) function with symmetric exponential decay, giving a π-time of 0.307(9) ms and 1/e decay time of 1.2(4) ms.

Extended Data Fig. 4 Full outcomes of truth table measurements for hyperfine gate.

(a) Populations measured in each two-qubit state following state-preparation in that state, relative to population in \(| 00\rangle \), representing the relative SPAM fidelity of each state. (b) Populations measured in each two-qubit state as a fraction of the total detected molecule population in the \((| 0\rangle ,| 1\rangle )\) manifold for each of the input states to the iSWAP gate. This corrects for leakage to the \(| e\rangle \) state during state-preparation and measurement, as well as during the hyperfine gate itself. (c) Populations measured in each two-qubit state following application of the iSWAP gate sequence, relative to the initial population in \(| 00\rangle \) in (a), illustrating the amount lost to leakage.

Extended Data Fig. 5 Rotational-hyperfine structure and relevant coupling strengths of NaCs in the experimental regime.

(a) Hyperfine sublevels of N = 0 and N = 1 rotational manifolds. Color-shaded regions denote states of the same rotational sub-level. The x-axis represents the combined nuclear-spin contribution to the Zeeman shift, and is chosen to make the rotational structure apparent. (b) Estimated impurity of the states \(| 1\rangle \) and \(| e\rangle \) when dressed by the microwave field used for the \(| 1\rangle \leftrightarrow | e\rangle \) transition, which depicts all available transitions from these states. The hyperfine-changing transition, labeled by 3, is red-detuned from \(| 0\rangle \leftrightarrow | e\rangle \), labelled 4, by 1.9 MHz and comparably narrow. Off-resonant coupling is primarily due to the transitions 1, 2, and 4.

Extended Data Table 1 Fit values and bootstrapped uncertainty estimates for master equation model of interactions, for the three distances shown in Fig. 2 and Extended Data Fig. 1
Full size table

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Picard, L.R.B., Park, A.J., Patenotte, G.E. et al. Entanglement and iSWAP gate between molecular qubits. Nature (2024). https://doi.org/10.1038/s41586-024-08177-3

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