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Preparing random states and benchmarking with many-body quantum chaos


Producing quantum states at random has become increasingly important in modern quantum science, with applications being both theoretical and practical. In particular, ensembles of such randomly distributed, but pure, quantum states underlie our understanding of complexity in quantum circuits1 and black holes2, and have been used for benchmarking quantum devices3,4 in tests of quantum advantage5,6. However, creating random ensembles has necessitated a high degree of spatio-temporal control7,8,9,10,11,12 placing such studies out of reach for a wide class of quantum systems. Here we solve this problem by predicting and experimentally observing the emergence of random state ensembles naturally under time-independent Hamiltonian dynamics, which we use to implement an efficient, widely applicable benchmarking protocol. The observed random ensembles emerge from projective measurements and are intimately linked to universal correlations built up between subsystems of a larger quantum system, offering new insights into quantum thermalization13. Predicated on this discovery, we develop a fidelity estimation scheme, which we demonstrate for a Rydberg quantum simulator with up to 25 atoms using fewer than 104 experimental samples. This method has broad applicability, as we demonstrate for Hamiltonian parameter estimation, target-state generation benchmarking, and comparison of analogue and digital quantum devices. Our work has implications for understanding randomness in quantum dynamics14 and enables applications of this concept in a much wider context4,5,9,10,15,16,17,18,19,20.

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Fig. 1: Random pure state ensembles from Hamiltonian dynamics.
Fig. 2: Experimental signatures of random pure state ensembles.
Fig. 3: Development of emergent randomness.
Fig. 4: Fidelity estimation of an analogue Rydberg quantum simulator.
Fig. 5: Hamiltonian learning and target-state benchmarking.

Data availability

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

Code availability

The code that supports the findings of this study is available from the corresponding authors upon reasonable request. 


  1. Brandão, F. G. S. L., Chemissany, W., Hunter-Jones, N., Kueng, R. & Preskill, J. Models of quantum complexity growth. PRX Quantum 2, 30316 (2021).

    Article  Google Scholar 

  2. Hayden, P. & Preskill, J. Black holes as mirrors: quantum information in random subsystems. J. High. Energy Phys. 2007, 120 (2007).

    Article  MathSciNet  Google Scholar 

  3. Neill, C. et al. A blueprint for demonstrating quantum supremacy with superconducting qubits. Science 360, 195–199 (2018).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  4. Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 32328 (2019).

    Article  ADS  CAS  Google Scholar 

  5. Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019).

    Article  ADS  CAS  Google Scholar 

  6. Wu, Y. et al. Strong quantum computational advantage using a superconducting quantum processor. Phys. Rev. Lett. 127, 180501 (2021).

    Article  ADS  CAS  Google Scholar 

  7. Emerson, J., Weinstein, Y. S., Saraceno, M., Lloyd, S. & Cory, D. G. Pseudo-random unitary operators for quantum information processing. Science 302, 2098–2100 (2003).

    Article  ADS  MathSciNet  CAS  MATH  Google Scholar 

  8. Harrow, A. W. & Low, R. A. Random quantum circuits are approximate 2-designs. Commun. Math. Phys. 291, 257–302 (2009).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Dankert, C., Cleve, R., Emerson, J. & Livine, E. Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A 80, 12304 (2009).

    Article  ADS  Google Scholar 

  10. Brandão, F. G. S. L., Harrow, A. W. & Horodecki, M. Local random quantum circuits are approximate polynomial-designs. Commun. Math. Phys. 346, 397–434 (2016).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Ohliger, M., Nesme, V. & Eisert, J. Efficient and feasible state tomography of quantum many-body systems. New. J. Phys. 15, 15024 (2013).

    Article  MATH  Google Scholar 

  12. Nakata, Y., Hirche, C., Koashi, M. & Winter, A. Efficient quantum pseudorandomness with nearly time-independent Hamiltonian dynamics. Phys. Rev. X 7, 21006 (2017).

    Google Scholar 

  13. Kaufman, A. M. et al. Quantum thermalization through entanglement in an isolated many-body system. Science 353, 794–800 (2016).

    Article  ADS  CAS  Google Scholar 

  14. Cotler, J. et al. Emergent quantum state designs from individual many-body wavefunctions. Preprint at (2021).

  15. Boixo, S. et al. Characterizing quantum supremacy in near-term devices. Nat. Phys. 14, 595–600 (2018).

    Article  CAS  Google Scholar 

  16. Bouland, A., Fefferman, B., Nirkhe, C. & Vazirani, U. On the complexity and verification of quantum random circuit sampling. Nat. Phys. 15, 159–163 (2019).

    Article  CAS  MATH  Google Scholar 

  17. Haferkamp, J. et al. Closing gaps of a quantum advantage with short-time Hamiltonian dynamics. Phys. Rev. Lett. 125, 250501 (2020).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  18. Brydges, T. et al. Probing Rényi entanglement entropy via randomized measurements. Science 364, 260–263 (2019).

    Article  ADS  CAS  Google Scholar 

  19. Elben, A. et al. Cross-platform verification of intermediate scale quantum devices. Phys. Rev. Lett. 124, 10504 (2020).

    Article  ADS  CAS  Google Scholar 

  20. Huang, H. Y., Kueng, R. & Preskill, J. Predicting many properties of a quantum system from very few measurements. Nat. Phys. 16, 1050–1057 (2020).

    Article  CAS  Google Scholar 

  21. Harrow, A. W. The church of the symmetric subspace. Preprint at; (2013).

  22. Jurcevic, P. et al. Demonstration of quantum volume 64 on a superconducting quantum computing system. Quantum Sci. Technol. 6, 025020 (2021).

    Article  ADS  Google Scholar 

  23. Piroli, L., Sünderhauf, C. & Qi, X. L. A random unitary circuit model for black hole evaporation. J. High. Energy Phys. 2020, 63 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  24. Bernien, H. et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature 551, 579–584 (2017).

    Article  ADS  CAS  Google Scholar 

  25. Browaeys, A. & Lahaye, T. Many-body physics with individually controlled Rydberg atoms. Nat. Phys. 16, 132–142 (2020).

    Article  CAS  Google Scholar 

  26. Madjarov, I. S. et al. High-fidelity entanglement and detection of alkaline-earth Rydberg atoms. Nat. Phys. 16, 857–861 (2020).

    Article  CAS  Google Scholar 

  27. Popp, M., Verstraete, F., Martín-Delgado, M. A. & Cirac, J. I. Localizable entanglement. Phys. Rev. A 71, 42306 (2005).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Goldstein, S., Lebowitz, J. L., Mastrodonato, C., Tumulka, R. & Zanghì, N. Universal probability distribution for the wave function of a quantum system entangled with its environment. Commun. Math. Phys. 342, 965–988 (2016).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Norcia, M. A., Young, A. W. & Kaufman, A. M. Microscopic control and detection of ultracold strontium in optical-tweezer arrays. Phys. Rev. X 8, 41054 (2018).

    Google Scholar 

  30. Cooper, A. et al. Alkaline-Earth atoms in optical tweezers. Phys. Rev. X 8, 41055 (2018).

    CAS  Google Scholar 

  31. Saskin, S., Wilson, J. T., Grinkemeyer, B. & Thompson, J. D. Narrow-line cooling and imaging of ytterbium atoms in an optical tweezer array. Phys. Rev. Lett. 122, 143002 (2019).

    Article  ADS  CAS  Google Scholar 

  32. Covey, J. P., Madjarov, I. S., Cooper, A. & Endres, M. 2000-times repeated imaging of strontium atoms in clock-magic tweezer arrays. Phys. Rev. Lett. 122, 173201 (2019).

    Article  ADS  CAS  Google Scholar 

  33. Rigol, M., Dunjko, V. & Olshanii, M. Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854–858 (2008).

    Article  ADS  CAS  Google Scholar 

  34. Abanin, D. A., Altman, E., Bloch, I. & Serbyn, M. Colloquium: many-body localization, thermalization,and entanglement. Rev. Mod. Phys. 91, 21001 (2019).

    Article  MathSciNet  CAS  Google Scholar 

  35. Ueda, M. Quantum equilibration, thermalization and prethermalization in ultracold atoms. Nat. Rev. Phys. 2, 669–681 (2020).

    Article  CAS  Google Scholar 

  36. Popescu, S., Short, A. J. & Winter, A. Entanglement and the foundations of statistical mechanics. Nat. Phys. 2, 754–758 (2006).

    Article  CAS  Google Scholar 

  37. del Rio, L., Hutter, A., Renner, R. & Wehner, S. Relative thermalization. Phys. Rev. E 94, 22104 (2016).

    Article  Google Scholar 

  38. Porter, C. E. & Thomas, R. G. Fluctuations of nuclear reaction widths. Phys. Rev. 104, 483–491 (1956).

    Article  ADS  CAS  Google Scholar 

  39. Ambainis, A. and Emerson, J. Quantum t-designs: t-wise independence in the quantum world. In Proceedings Twenty-Second Annual IEEE Conference on Computational Complexity 129–140 (IEEE, 2007).

  40. Khemani, V., Vishwanath, A. & Huse, D. A. Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws. Phys. Rev. X 8, 31057 (2018).

    CAS  Google Scholar 

  41. Flammia, S. T. & Liu, Y. K. Direct fidelity estimation from few Pauli measurements. Phys. Rev. Lett. 106, 230501 (2011).

    Article  ADS  Google Scholar 

  42. Slagle, K. et al. Microscopic characterization of Ising conformal field theory in Rydberg chains. Phys. Rev. B 104, 235109 (2021).

    Article  ADS  CAS  Google Scholar 

  43. Cotler, J., Hunter-Jones, N. & Ranard, D. Fluctuations of subsystem entropies at late times. Phys. Rev. A 105, 022416 (2022).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  44. Nandkishore, R. & Huse, D. A. Many-body localization and thermalization in quantum statistical mechanics. Annu. Rev. Condens. Matter Phys. 6, 15–38 (2015).

    Article  ADS  Google Scholar 

  45. Turner, C. J., Michailidis, A. A., Abanin, D. A., Serbyn, M. & Papić, Z. Weak ergodicity breaking from quantum many-body scars. Nat. Phys. 14, 745–749 (2018).

    Article  CAS  Google Scholar 

  46. Monroe, C. et al. Programmable quantum simulations of spin systems with trapped ions. Rev. Mod. Phys. 93, 025001 (2021).

    Article  ADS  MathSciNet  CAS  Google Scholar 

  47. Zhong, H. S. et al. Quantum computational advantage using photons. Science 370, 1460–1463 (2020).

    Article  ADS  CAS  Google Scholar 

  48. Gross, C. & Bloch, I. Quantum simulations with ultracold atoms in optical lattices. Science 357, 995–1001 (2017).

    Article  ADS  CAS  Google Scholar 

  49. Giovannetti, V. Quantum-enhanced measurements: beating the standard quantum limit. Science 306, 1330–1336 (2004).

    Article  ADS  CAS  Google Scholar 

  50. Mark, D. K., Choi, J., Shaw, A. L., Endres, M. and Choi, S. Benchmarking quantum simulators using quantum chaos. Preprint at (2022).

  51. Madjarov, I. S. et al. An atomic-array optical clock with single-atom readout. Phys. Rev. X 9, 41052 (2019).

    CAS  Google Scholar 

  52. Barredo, D., de Leseleuc, S., Lienhard, V., Lahaye, T. & Browaeys, A. An atom-by-atom assembler of defect-free arbitrary two-dimensional atomic arrays. Science 354, 1021–1023 (2016).

    Article  ADS  CAS  Google Scholar 

  53. Endres, M. et al. Atom-by-atom assembly of defect-free one-dimensional cold atom arrays. Science 354, 1024–1027 (2016).

    Article  ADS  CAS  Google Scholar 

  54. Page, D. N. Average entropy of a subsystem. Phys. Rev. Lett. 71, 1291–1294 (1993).

    Article  ADS  MathSciNet  CAS  MATH  Google Scholar 

  55. de Léséleuc, S. et al. Observation of a symmetry-protected topological phase of interacting bosons with Rydberg atoms. Science 365, 775–780 (2019).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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We acknowledge experimental help from P. Scholl during the revision of this manuscript, as well as discussions with A. Deshpande and A. Gorshkov. We acknowledge funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF grant no. PHY-1733907), the NSF CAREER award (no. 1753386), the AFOSR YIP (no. FA9550-19-1-0044), the DARPA ONISQ programme (no. W911NF2010021), the Army Research Office MURI program (no. W911NF2010136), the NSF QLCI program (no. 2016245), the DOE (grant no. DE-SC0021951), the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator (grant no. DE-AC02-05CH11231) and F. Blum. J.C. acknowledges support from the IQIM postdoctoral fellowship. A.L.S. acknowledges support from the Eddleman Quantum graduate fellowship. R.F. acknowledges support from the Troesh postdoctoral fellowship. J.P.C. acknowledges support from the PMA Prize postdoctoral fellowship. H.P. acknowledges support by the Gordon and Betty Moore Foundation. H.-Y.H. is supported by the J. Yang & Family Foundation. A.K. acknowledges funding from the Harvard Quantum Initiative (HQI) graduate fellowship. J.S.C. is supported by a Junior Fellowship from the Harvard Society of Fellows and the US Department of Energy under grant contract no. DE-SC0012567. S.C. acknowledges support from the Miller Institute for Basic Research in Science.

Author information

Authors and Affiliations



J.C., A.L.S, S.C. and M.E. conceived the idea and experiment. J.C. and A.L.S. performed the experiments and data analysis. J.C., A.L.S., J.S.C., D.K.M., H.-Y.H., H.P., F.G.S.L.B., S.C. and M.E. contributed to the underlying theory. J.C., A.L.S., I.S.M., X.X., R.F., J.P.C. and A.K. contributed to building the experimental set-up and data taking. J.C., A.L.S., S.C. and M.E. wrote the manuscript with input from all authors. S.C. and M.E. supervised this project.

Corresponding authors

Correspondence to Soonwon Choi or Manuel Endres.

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

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Extended data figures and tables

Extended Data Fig. 1 Experimental system and parameter feedback.

a, Illustration of a Rydberg quantum simulator consisting of strontium-88 atoms trapped in optical tweezers (red funnels). All atoms are driven by a global transverse control field (purple horizontal beam) at a Rabi frequency Ω and a detuning Δ (right panel). The interaction strength is given as \({C}_{6}/{R}_{ij}^{6}\) with an interaction constant C6 and atomic separations Rij between two atoms at site i and j. b, Schematic of the experimental feedback scheme. We automatically interleave data taking with feedback to global control parameters and systematic variables through a home-built control architecture (Methods); in particular, we feedback to the clock laser frequency (to maintain optimal state preparation fidelity), the Rydberg laser alignment, the Rydberg detuning Δ, and the Rabi frequency Ω. c, Example of the interleaved automatic Rabi frequency stabilization over the course of ≈ 20 hours with no human intervention. Feedback is comprised of performing single-atom Rabi oscillations, fitting the observed Rabi frequency, and updating the laser amplitude, rather than simply stabilizing the laser amplitude against a photodiode reference. While the Rabi frequency setpoint (orange squares) changes over the course of the sequence (due to long-time instabilities like temperature drifts), the measured Rabi frequency (blue circles) stays constant to within < 0.3%, with a standard deviation of 0.15%. This same stability is seen over the course of multiple days with nearly continuous experimental uptime.

Extended Data Fig. 2 Universality of moments of the projected ensemble.

kth moments of the conditional probability distributions in Fig. 2b,d, evaluated at intermediate time (Ωt/2π = 2.3) and for a variety of choices of subsystems (see panel on the right); we find a universal convergence to ≈ k!, independent of subsystem choice, suggesting that a subsystem’s projected ensemble converges to the uniform random ensemble irrespective of the details of placement, or connectivity. Error bars are the standard deviation over temporal fluctuations in moments near the evaluated time, as shown in Fig. 3a.

Extended Data Fig. 3 Emergent randomness and benchmarking in other quantum systems.

a, Fidelity estimation for the case of a trapped ion quantum simulator governed by chaotic Hamiltonian evolution (left) and a quantum computer implementing a random unitary circuit (RUC) (right); see Supplementary Information for simulation details. In both cases, we plot the many-body fidelity (dashed line), as well as our fidelity estimator, Fc (solid line); for the RUC case we also plot the more conventional linear cross-entropy-benchmark, FXEB5 (dotted line). We find that Fc approximates the fidelity at much earlier times than FXEB. b, Numerically computed trace distances between the projected ensemble of a two-qubit subsystem and the corresponding k-design. Results are shown for multiple different total system sizes: 10, 13, 16 for the trapped ion case, and 10, 12, 14, 16 for the RUC case, with darker colors corresponding to larger total system sizes.

Extended Data Fig. 4 Detecting errors during quantum evolution.

a, Schematic of noisy time evolution with an error occurring at time terr. The influence of the local error propagates outward, affecting the measurement outcomes non-locally at a later time. b, Errors during evolution can be detected by correlating the measurement outcomes with an error-free, ideal evolution case. We numerically tested this by applying a local, instantaneous phase error to the middle qubit of an N = 16 atom Rydberg simulator at time Ωterr/2π ≈ 1. The proposed fidelity estimator, Fc (solid line), accurately approximates the many-body overlap (dashed line) between states produced with and without errors, after a slightly delayed time. Inset: Conditional probability distributions in A before (blue) and after (red) the error, showing decorrelation.

Extended Data Fig. 5 Finite sampling analysis for Fc.

a, Statistical fluctuations of the fidelity estimator, Fc, at N = 13 (dark purple) and N = 22 (light purple), computed both using our ab initio error model (solid lines) and experiment (markers) evaluated with a finite number of M bitstring samples. Data are consistent with a \(1/\sqrt{M}\) scaling, shown here as a guide to the eye (grey dashed line) b, Sample complexity of the fidelity estimator, evaluated at the N-dependent entanglement saturation time for the error model (blue crosses), and for the experimental data in Fig. 4d (red circles). A fit to the experimental data (dashed line) with functional form \(\sigma ({F}_{c})\sqrt{M}={a}^{N}\) yields an estimate of a = 1.037(2) (a similar fit to the error model yields an estimate of a = 1.039(2)).

Extended Data Fig. 6 Predicting fidelity scaling.

a, We use our ab initio error model (which includes state preparation errors) to predict the fidelity decay rate as a function of system size. For various system sizes we plot the model fidelity (solid lines), as well as fits to exponential decay with an unconstrained value at t = 0 (dashed lines), which we see are consistent with the time-dependent fidelity. b, For the range of system sizes for which our error model is readily calculable, we see the fidelity decay rate normalized by the Rabi frequency, γ(N)Ω/2π (markers), is consistent with a linear function of system size (red line). The shaded region comes from uncertainty in the fit parameters.

Extended Data Fig. 7 Comparison to digital quantum devices executing random circuits.

a, Numerical simulations of a one-dimensional digital quantum device implementing a random unitary circuit (RUC). Two different digital gate implementations are tested: a configuration based on the gate-set used in ref. 5 (bottom), and a configuration where each cycle is composed of parallel two-qubit SU(4) gates (top)4. Cross markers indicate when the half-chain entanglement entropy saturates. b, Due to the Rydberg blockade mechanism, as well as symmetries of the Rydberg Hamiltonian (Supplementary Information), an equal number of atoms in the Rydberg simulator, N, and qubits in the RUC, NRUC, will not saturate to the same half-chain entanglement entropy. However, we can still find an equivalence by plotting the saturated entanglement entropy for the RUC (blue crosses for the SU(4) gate-set, open red squares for gate-set from ref. 5) and for the Rydberg simulator (grey markers) as a function of their respective system sizes. We fit the results for the Rydberg simulator (black line), and plot the analytic prediction for the RUC54 (purple line), from which we can write an equivalent NRUC as a function of N, in the sense of maximum achievable entanglement entropy (Methods). c, For a given N (and equivalent NRUC), we plot the SPAM-corrected, two-qubit cycle fidelity for an equivalently-sized RUC to match the evolution fidelity of our Rydberg simulator at the time/depth when entanglement saturates. Red lines, markers and crosses are for the gate-set of ref. 5, while blue are for the SU(4) gate-set. Shaded regions come from the error on fitting the various N-dependent parameters which enter this calculation (Methods).

Extended Data Fig. 8 Applications to target state benchmarking.

a, Benchmarking of a one-dimensional cluster state, b, a pure Haar-random state benchmarked in a two-dimensional square Rydberg atom array, and c, a symmetry-protected topological (SPT) ground state prepared in a Rydberg ladder array realizing the Su-Schrieffer-Heeger topological model55. In a, CZ denotes a controlled-Z gate and \(| \,+\,\rangle =\frac{| \,0\rangle +| \,1\rangle }{\sqrt{2}}\). In b, RB denotes the Rydberg blockade radius within which more than a single excitation is not allowed24,25,26. In c, J and \({J}^{{\prime} }\) are the alternating coupling strengths of a two-leg ladder array, respectively. In all cases, N = 16 qubits are used, and imperfect quantum states are prepared via phase rotations such that the many-body fidelity overlap becomes 0.5 (red dashed line). Additionally, chaotic evolution is performed such that the initial state is at infinite effective temperature to apply our Fc formalism (blue solid lines) (Methods).

Supplementary information

Supplementary Information

Supplementary Sections 1–8 and references.

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Choi, J., Shaw, A.L., Madjarov, I.S. et al. Preparing random states and benchmarking with many-body quantum chaos. Nature 613, 468–473 (2023).

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