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# Preparing random states and benchmarking with many-body quantum chaos

## Abstract

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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## 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.

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## Acknowledgements

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

### Contributions

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.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

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). https://doi.org/10.1038/s41586-022-05442-1

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• DOI: https://doi.org/10.1038/s41586-022-05442-1