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Coherent quantum annealing in a programmable 2,000 qubit Ising chain


Quantum simulation has emerged as a valuable arena for demonstrating and understanding the capabilities of near-term quantum computers1,2,3. Quantum annealing4,5 has been successfully used in simulating a range of open quantum systems, both at equilibrium6,7,8 and out of equilibrium9,10,11. However, in all previous experiments, annealing has been too slow to coherently simulate a closed quantum system, due to the onset of thermal effects from the environment. Here we demonstrate coherent evolution through a quantum phase transition in the paradigmatic setting of a one-dimensional transverse-field Ising chain, using up to 2,000 superconducting flux qubits in a programmable quantum annealer. In large systems, we observe the quantum Kibble–Zurek mechanism with theoretically predicted kink statistics, as well as characteristic positive kink–kink correlations, independent of temperature. In small chains, excitation statistics validate the picture of a Landau–Zener transition at a minimum gap. In both cases, the results are in quantitative agreement with analytical solutions to the closed-system quantum model. For slower anneals, we observe anti-Kibble–Zurek scaling in a crossover to the open quantum regime. The coherent dynamics of large-scale quantum annealers demonstrated here can be exploited to perform approximate quantum optimization, machine learning and simulation tasks.

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Fig. 1: QPT in an annealed Ising chain.
Fig. 2: Kink density scaling and distribution.
Fig. 3: Normalized kink–kink correlations.
Fig. 4: Crossover to adiabaticity.

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Data availability

Data supporting the findings of this paper are available from the corresponding author upon request. Source data are provided with this paper.

Code availability

The TEBD code used in this paper is available from the corresponding author upon reasonable request. An open-source version of the PIMC code used in the Supplementary Information is available via GitHub at The version for this work is archived in Zenodo at


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We thank H. Oshiyama, N. Shibata, A. del Campo, L. Addario-Berry, T. Albash and A. W. Sandvik for fruitful discussions, and acknowledge the contributions of both technical and non-technical staff at D-Wave. We acknowledge the Center for Advanced Research Computing (CARC) at the University of Southern California for providing computing resources that have contributed to the research results reported within this publication. URL: DAL acknowledges support from the National Science Foundation ‘the Quantum Leap Big Idea’ under Grant No. OMA-1936388, and by DARPA under the RQMLS program, Agreement No. HR00112190071.

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Authors and Affiliations



A.D.K., S.S., J.R., A.Z., T.L. F.A., A.J.B., S.E., R.H., D.A.L., H.N. and M.H.A. conceived and designed the experiments and analysed the data. A.D.K., S.S., J.R., A.Z., T.L. and R.H. performed the experiments. T.L., F.A., A.J.B., S.E., E.H., S.H., E.L., A.J.R.M., G.M., T.O., G.P.-L., M.R., C.R., Y.S., J.D.W., J.Y., R.H. and M.H.A. contributed to the design, fabrication, deployment and calibration of the quantum annealing system. A.D.K., S.S., J.R., D.A.L., H.N. and M.H.A. wrote the manuscript.

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Correspondence to Andrew D. King.

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Competing interests

S.S., D.A.L. and H.N. declare no competing interests. A.D.K., J.R., A.Z., T.L., F.A., A.J.B., S.E., E.H., S.H., E.L., A.J.R.M., G.M., T.O., G.P.-L., M.R., C.R., Y.S., J.D.W., J.Y., R.H. and M.H.A. affiliated with D-Wave hold stock options in D-Wave and declare a competing financial interest on that basis.

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Nature Physics thanks Guglielmo Mazzola, David Bernal Neira and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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King, A.D., Suzuki, S., Raymond, J. et al. Coherent quantum annealing in a programmable 2,000 qubit Ising chain. Nat. Phys. 18, 1324–1328 (2022).

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