Abstract
Experiments on disordered alloys1,2,3 suggest that spin glasses can be brought into low-energy states faster by annealing quantum fluctuations than by conventional thermal annealing. Owing to the importance of spin glasses as a paradigmatic computational testbed, reproducing this phenomenon in a programmable system has remained a central challenge in quantum optimization4,5,6,7,8,9,10,11,12,13. Here we achieve this goal by realizing quantum-critical spin-glass dynamics on thousands of qubits with a superconducting quantum annealer. We first demonstrate quantitative agreement between quantum annealing and time evolution of the Schrödinger equation in small spin glasses. We then measure dynamics in three-dimensional spin glasses on thousands of qubits, for which classical simulation of many-body quantum dynamics is intractable. We extract critical exponents that clearly distinguish quantum annealing from the slower stochastic dynamics of analogous Monte Carlo algorithms, providing both theoretical and experimental support for large-scale quantum simulation and a scaling advantage in energy optimization.
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Data availability
Data supporting the findings are available in the Zenodo online repository https://doi.org/10.5281/zenodo.7640779.
Code availability
An open-source version of the SQA code used in this work is available at https://github.com/dwavesystems/dwave-pimc.
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Acknowledgements
We are grateful to P. Young, H. Nishimori, S. Suzuki and J. Charfreitag for helpful discussions. A.W.S. was supported by the Simons Foundation under Simons Investigator Award no. 511064. Some of the numerical simulations were carried out using the Shared Computing Cluster managed by Boston Universityâs Research Computing Services. We thank both the technical and the non-technical staff at D-Wave, without whom this work would not be possible.
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A.D.K., J.R., T.L., R.H., A.Z., A.W.S. and M.H.A. conceived and designed the experiments and analysed the data. A.D.K., J.R., T.L. and A.W.S. performed the experiments and simulations. T.L., R.H., F.A., A.J.B., K.B., S.E., C.E., E.H., S.H., E.L., A.J.R.M., G.M., R.M., T.O., G.P.-L., M.R., C.R., Y.S., N.T., M.V., J.D.W., J.Y. and M.H.A. contributed to the design, fabrication, deployment and calibration of the QA system. A.D.K., J.R., R.H., A.W.S. and M.H.A. wrote the manuscript.
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A.W.S. declares no competing interests. All other authors have received stock options in D-Wave as current or former employees.
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Extended data figures and tables
Extended Data Fig. 1 State probabilities for 16-spin glasses.
a, We show observed probabilities for ground states and first excited states in Schrödinger evolution (taâ=â14âns), SA (taâ=â200 MCS), and SQA (taâ=â100 MCS) compared with experimental measurements from QA (taâ=â14âns). The three columns contain data for the three exemplary instances, with colors corresponding to those in Fig. 2. Annealing times for SA and SQA were chosen to have good agreement with Schrödinger evolution in average ground state probability. Each dynamics was run 19,200 times; dashed lines indicate the statistical floor, i.e., the probability if a state is seen exactly once. Unobserved states are represented by points below the statistical floor. b, Kullback-Leibler (KL) divergence in the probability distribution among first excited states, with QA used as a reference distribution, measures deviation between two dynamics for a given realization. Empirical CDF (proportion of 100 realizations below a given KL divergence) is shown. Data indicate that coherent quantum (Schrödinger) dynamics agrees more closely with experimental data better than does SA or SQA.
Extended Data Fig. 2 Data collapse for 3D spin glasses with JG = 1.
Best-fit exponent μ collapses U by rescaling data horizontally based on L, and r collapses â¨q2â© by scaling vertically based on L, given a horizontal rescaling by Lâμ.
Extended Data Fig. 3 Data collapse for 3D spin glasses with JGâ=â1/2.
Best-fit exponent μ collapses U by rescaling data horizontally based on L, and r collapses â¨q2â© by scaling vertically based on L, given a horizontal rescaling by Lâμ.
Extended Data Fig. 4 Data collapse for 3D spin glasses on simple cubic lattices.
Best-fit exponent μ collapses U by rescaling data horizontally based on L, and r collapses â¨q2â© by scaling vertically based on L, given a horizontal rescaling by Lâμ.
Extended Data Fig. 5 3D lattice structure in qubit connectivity graph.
\(L\times L\times (\max (L,12))\times 2\) lattices are found by heuristic search given a basic structure in which horizontal and vertical (long) couplings form two dimensions, and the interior of unit cells forms the third dimension. One dimer (thick gray line) and its six neighboring dimers are shown as thick gray lines. As in Fig. 1, purple and yellow lines represent glass couplings ofâ±âJG andâ±âJG/2 respectively.
Extended Data Fig. 6 Measurement of effective QA annealing time.
Two independent measurement methods are used to estimate ta for fast anneals (â<â20âns). First is a direct measurement using a witness qubit. Second is an extrapolative measurement that assumes a quantum KZ scaling in a 1D chain and assumes a kink density \(n\propto {t}_{a}^{-1/2}\) for taâ<â20âns. For the fastest anneals, ta deviates significantly from the values requested from the control electronics. However, the two independent measurement approaches give consistent results.
Extended Data Fig. 7 Extracting Ising model from flux-qubit model.
a, Eight-qubit gadget used to extract an effective annealing schedule in the transverse-field Ising model based on a many-body flux-qubit Hamiltonian. Dimers indicated by dashed ellipses are treated as six-level objects and combined to diagonalize a many-body Hamiltonian. b, General-purpose (nominal) annealing schedule based on single-qubit measurements, and extracted many-body effective schedule, used for Schrödinger evolution.
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King, A.D., Raymond, J., Lanting, T. et al. Quantum critical dynamics in a 5,000-qubit programmable spin glass. Nature 617, 61â66 (2023). https://doi.org/10.1038/s41586-023-05867-2
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DOI: https://doi.org/10.1038/s41586-023-05867-2
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