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Deterministic generation of multidimensional photonic cluster states with a single quantum emitter


Entanglement is a key resource in quantum computing and other prospective technologies. Multidimensional photonic graph states, such as cluster states, have a special entanglement structure that makes them a valuable resource for quantum metrology, secure quantum communication and measurement-based quantum computation. However, to date, the generation of multidimensional photonic cluster states has relied on probabilistic methods that limit the scalability of typical optical generation methods. Here we present an experimental implementation in the microwave domain of a resource-efficient scheme for the deterministic generation of two-dimensional photonic cluster states. Using a coupled resonator array as a slow-light waveguide, a single flux-tunable transmon qubit as a quantum emitter and a second auxiliary transmon as a switchable mirror, we achieve rapid, shaped emission of entangled photon wavepackets, and selective time-delayed feedback of photon wavepackets to the emitter qubit. We use these capabilities to generate a two-dimensional cluster state of four photons with 70% fidelity, as verified by the tomographic reconstruction of the quantum state.

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Fig. 1: Deterministic generation of 2D cluster states with a single emitter qubit.
Fig. 2: Emission of shaped photons pulses via flux modulation.
Fig. 3: CZ gate between emitter qubit and previously emitted photons via time-delayed feedback.
Fig. 4: Deterministic generation of a four-photon 2D cluster state.

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Data availability

The data that support the findings of this study are available from the corresponding author (O.P.) upon reasonable request.

Code availability

The codes used to perform the experiments and to analyse the data in this work are available from the corresponding author (O.P.) upon reasonable request.


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We thank E. Kim for helpful discussions regarding experimental setup, and we thank M. Chen for his collaboration in fridge-related work. This work was supported by the AFOSR MURI Quantum Photonic Matter (grant 16RT0696), through a grant from the Department of Energy (grant DE-SC0020152) and through a sponsored research agreement with Amazon Web Services. V.F. gratefully acknowledges support from NSF GFRP Fellowship.

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Authors and Affiliations



V.F., G.K., A.B., H.P. and O.P. contributed to the concept and planning of the experiment and the writing of the manuscript. V.F., G.K. and A.B. contributed to the device design and fabrication and the measurements and analysis of data.

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Correspondence to Oskar Painter.

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O.P. is currently employed by Amazon Web Services (AWS) as Director of their quantum hardware programme. AWS provided partial funding support for this work through a sponsored research grant.

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Nature Physics thanks Christopher Eichler and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary text and data, Supplementary Figs. 1–13 and additional references.

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Ferreira, V.S., Kim, G., Butler, A. et al. Deterministic generation of multidimensional photonic cluster states with a single quantum emitter. Nat. Phys. 20, 865–870 (2024).

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