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Entanglement of bosonic modes through an engineered exchange interaction


Quantum computation presents a powerful new paradigm for information processing. A robust universal quantum computer can be realized with any well controlled quantum system, but a successful platform will ultimately require the combination of highly coherent, error-correctable quantum elements with at least one entangling operation between them1,2. Quantum information stored in a continuous-variable system—for example, a harmonic oscillator—can take advantage of hardware-efficient quantum error correction protocols that encode information in the large available Hilbert space of each element3,4,5. However, such encoded states typically have no controllable direct couplings, making deterministic entangling operations between them particularly challenging. Here we develop an efficient implementation of the exponential-SWAP operation6 and present its experimental realization between bosonic qubits stored in two superconducting microwave cavities. This engineered operation is analogous to the exchange interaction between discrete spin systems, but acts within any encoded subspace of the continuous-variable modes. Based on a control rotation, the operation produces a coherent superposition of identity and SWAP operations between arbitrary states of two harmonic oscillator modes and can be used to enact a deterministic entangling gate within quantum error correction codes. These results provide a valuable building block for universal quantum computation using bosonic modes.

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Fig. 1: Sketch (not to scale) of device architecture and experimental protocol.
Fig. 2: Deterministic entanglement in Fock encoding.
Fig. 3: Characterization of eSWAP unitary in coherent state encoding.
Fig. 4: Quantum process tomography.

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Data availability

The data that support the findings of this study are available from the corresponding authors on reasonable request.


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We thank R. Filip for helpful discussions; N. Frattini, K. Sliwa, M. J. Hatridge and A. Narla for providing the Josephson parametric converters; and N. Ofek and P. Reinhold for providing the logic and control interface for the field programmable gate array used in this experiment. This research was supported by the US Army Research Office (W911NF-14-1-0011 and W911NF-16-10349). Y.Y.G. was supported by an ASTAR NSS Fellowship; B.J.L. was supported by a Yale QIMP Fellowship; S.M.G. was supported by the National Science Foundation (DMR-1609326); and L.J. was supported by the Alfred P. Sloan Foundation (BR 2013-049) and the Packard Foundation (2013-39273). Facilities use was supported by the Yale Institute for Nanoscience and Quantum Engineering (YINQE), the Yale SEAS cleanroom, and the National Science Foundation (MRSECDMR-1119826).

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Nature thanks Kero Lau and Gheorghe Paraoanu for their contribution to the peer review of this work.

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Authors and Affiliations



Y.Y.G. and B.J.L. fabricated the transmon qubits, assembled the experimental apparatus, and performed the experiments under the supervision of L.F., M.H.D. and R.J.S. Y.Y.G., B.J.L. and K.S.C. analysed the data. L.J. and S.M.G. provided theoretical support. Y.Y.G., B.J.L. and R.J.S. wrote the manuscript with feedback from all authors.

Corresponding authors

Correspondence to Yvonne Y. Gao, Brian J. Lester or Robert J. Schoelkopf.

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Competing interests

R.J.S., M.H.D. and L.F. are co-founders of, and R.J.S. and L.F. are equity shareholders in, Quantum Circuits, Inc. Y.Y.G., B.J.L., L.J., S.M.G. and R.J.S. are inventors on patent application no. 62/613,866 submitted by Yale University, which covers the design and methods for Robust Quantum Logical Gates.

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Extended data figures and tables

Extended Data Fig. 1 Top view of the 3D double-cavity cQED system.

The centre transmon ancilla (qC) provides nonlinear coupling between the modes of Alice (orange) and Bob (blue). The package accommodates two additional transmon ancillae, qA and qB, which are each coupled to one of the cavities Alice and Bob, respectively. Each transmon ancilla is measured via a neighbouring readout resonator (RA, RB, RC). The RF drives (ω1, ω2) are coupled to the system through the drive port of qC. See Methods for details.

Extended Data Fig. 2 A quantum Fredkin gate.

a, The resonance condition for a parametrically driven SWAP operation measured as a function of one of the drive frequencies and the drive duration with the ancillary transmon initialized in \((|g\rangle +|e\rangle )/\sqrt{2}\). The colour scale shows the probability for the ancilla to be excited (Pe) after a rotation that follows the parametric drive. b, The reconstructed three-mode density matrix after the Fredkin gate for the initial states \(\left|g\right\rangle \otimes \left|0\right\rangle \otimes \left|1\right\rangle \)(upper left), \(\left|e\right\rangle \otimes \left|0\right\rangle \otimes \left|1\right\rangle \)(upper right), \(\frac{1}{\sqrt{2}}\left(\left|g\right\rangle +\left|e\right\rangle \right)\otimes \left|0\right\rangle \otimes \left|1\right\rangle \)(bottom left) and \(\frac{1}{\sqrt{2}}\left(\left|g\right\rangle -\left|e\right\rangle \right)\otimes \left|0\right\rangle \otimes \left|1\right\rangle \)(bottom right). The colour scale shows the real value of each element in the reconstructed density matrices.

Extended Data Fig. 3 The quantum process matrix for Fock encoding.

The real (left) and imaginary (right) components of the complex process matrix χ are shown for the input (a), the identity operation (b), \(\sqrt{\mathrm{SWAP}}\) (c) and SWAP (d). For each operation, we show the measured (top) and ideal (bottom) matrices for comparison. Each of the measured process matrices are calculated without correction for SPAM errors. The colour scale shows the value for each (real or imaginary) component of the respective process matrices.

Extended Data Fig. 4 QPT for binomial code.

Left to right: the process matrix in the Pauli transfer representation for \({U}_{{\rm{E}}}\left(0\right)\), \({U}_{{\rm{E}}}({\rm{\pi }}/4)\) and \({U}_{{\rm{E}}}({\rm{\pi }}/2)\) acting on Alice and Bob, encoded in the binomial basis. From these results, we obtain a process fidelity of 0.70, 0.58 and 0.65 for the three operations without correction for SPAM errors. The colour scale shows the value for each element of the respective process matrices.

Extended Data Table 1 Hamiltonian parameters of all cQED components
Extended Data Table 2 Coherence properties of the system

Supplementary information

Supplementary Information

This file contains Supplementary Text with additional details about the operation and an error budget, as well as additional references, Supplementary Figures 1-2 and Supplementary Tables 1-2

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Gao, Y.Y., Lester, B.J., Chou, K.S. et al. Entanglement of bosonic modes through an engineered exchange interaction. Nature 566, 509–512 (2019).

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