Abstract
Quantum computing involves the preparation of entangled states across many qubits. This requires efficient preparation protocols that are stable to noise and gate imperfections. Here we demonstrate the generation of the simplest long-range order—Ising order—using a measurement-based protocol on 54 system qubits in the presence of coherent and incoherent errors. We implement a constant-depth preparation protocol that uses classical decoding of measurements to identify long-range order that is otherwise hidden by the randomness of quantum measurements. By experimentally tuning the error rates, we demonstrate the stability of this decoded long-range order in two spatial dimensions, up to a critical phase transition belonging to the unusual Nishimori universality class. Although in classical systems Nishimori physics requires fine-tuning multiple parameters, here it arises as a direct result of the Born rule for measurement probabilities. Our study demonstrates the emergent phenomena that can be explored on quantum processors beyond a hundred qubits.
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Data availability
The data supporting the findings of this study can be found via figshare at https://doi.org/10.6084/m9.figshare.24293524 (ref. 55).
Code availability
Simulation and data analysis code may be made available upon reasonable request.
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Acknowledgements
We thank M. Ware, P. Jurcevic, Y. Kim, A. Eddins, H. Nayfeh, I. Lauer, D. McKay, G. Jones and J. Summerour for assistance with performing experiments and B. Mitchell, D. Zajac, J. Wootton, L. Govia, X. Wei, R. Gupta, T. Yoder, T. Soejima, K. Siva, M. Motta, Z. Minev, S. Pappalardi, S. Garratt, E. Altman, F. Valenti and H. Nishimori for thoughtful discussions. We thank H. Nishimori for careful reading of the paper. The Cologne group was partially funded by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy – Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1 – 390534769 and within the CRC network TR 183 (project grant no. 277101999: G.-Y.Z. and S.T.) as part of projects A04 and B01. The classical simulations were performed on the JUWELS cluster at the Forschungszentrum Juelich. R.V. is supported by the Harvard Quantum Initiative Postdoctoral Fellowship in Science and Engineering. A.V. is supported by a Simons Investigator grant and by NSF-DMR 2220703. A.V. and R.V. are supported by the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation (618615, A.V.). G.Z. is supported by the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract number DE-SC0012704. We acknowledge the use of IBM Quantum services for this work.
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E.H.C. led the execution and analysis of the experimental data. G.-Y.Z. and R.V. led the theoretical developments; G.-Y.Z. developed all numerical simulations and additional experimental analysis code. A.S., E.B. and D.L. contributed technical insights and code related to characterizing the states. N.T., G.Z., A.V. and S.T. contributed theoretical insights related to the critical transition; S.T. provided additional insights related to entanglement generation and verification. S.S. and A.K. provided experimental support and access, and contributed to the design of the experiments. E.H.C., G.-Y.Z. and R.V. drafted the paper and supplementary material; all authors contributed to revising both.
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Extended data
Extended Data Fig. 1 Typical device error rates.
Plotted in cumulative format, for Echoed Cross Resonance (ECR, blue), square-root of Pauli-X (SX, red), and measurement (Meas, black) gates. Dashed lines represent medians of distributions.
Extended Data Fig. 2 Absence of finite threshold in one-dimensional protocol.
As discussed in the main text, the 2D protocol exhibited robustness over the 1D protocol (seen here); the key signature being based on the scaling of the average of two-point correlations, f, as a function of system size. For comparison with Fig. 4 in the main text. (a) f grows with increasing system size but converges to finite value that depends on tA. (b) The peak of g converges to tA = π/4 indicative of absence of finite threshold for coherent error. For both (a) and (b), the three system sizes were measured with 20,000 experimental samples giving rise to the small standard deviations (bars).
Extended Data Fig. 3 Magnetization of 1D experiments with and without decoding at different tA values.
Plotting the same data set from Extended Data Fig. 2, we observe that the 1D behavior exhibited no growth in f with system size from 28 to 54 and had peak variances at the GHZ value of tA = π/4. (a) Two-point correlations in 1D experiments for sweeps of tA. The histograms at values of tA where variances were maximized for undecoded (b) and decoded (c). Although the bimodal distribution persisted up to a system size of 28, at 54 the distribution became uniform. And as expected, both the undecoded (d) and decoded (e) exhibited a binomial distribution in the trivial state (tA = 0). The error bars in (a) are standard deviations based on 20,000 experimental samples, while the histograms in the other subplots are based on the same data sets.
Extended Data Fig. 4 Magnetization of 2D experiments with and without decoding at different tA values.
(a) Two-point correlations in 2D experiments for sweeps of tA. The histograms at values of tA where variances were maximized for undecoded (b) and decoded (c). In contrast to the 1D cases (Extended Data Fig. 3), the bimodal distribution persisted up to a system size of 54. And similarly to the 1D case, both the undecoded (d) and decoded (e) exhibited a binomial distribution in the trivial state (tA = 0). The error bars in (a) are standard deviations based on 20,000 experimental samples, while the histograms in the other subplots are based on the same data sets.
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Supplemental Figs. 1–13, Discussion and Tables 1–4.
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Chen, E.H., Zhu, GY., Verresen, R. et al. Nishimori transition across the error threshold for constant-depth quantum circuits. Nat. Phys. (2024). https://doi.org/10.1038/s41567-024-02696-6
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DOI: https://doi.org/10.1038/s41567-024-02696-6