forNiswap= 251, where the circuit fidelity is
merely 3% ( 36 ). The average OTOC measure-
ments presented here may be used to efficiently
benchmark performances of large quantum
processors owing to the efficient classical
simulation of operator spreading. On the other
hand, the nontrivial computational cost needed
to simulate our experimental results, related
to operator entanglement, indicates that fu-
ture quantum processors may be able to shed
light on certain classically challenging phys-
ical phenomena. Lastly, our experimental pro-
tocols may be readily applied to studying other
quantum dynamics of interest, such as the 2D
XY model or the transverse Ising model ( 41 , 42 ).
We include preliminary experimental mea-
surements of OTOCs in these dynamics in
figs. S3 to S5 ( 36 ). Extending such quantum
simulations to larger systems, where they may
reveal computationally hard features, is a
focus of current research. As the fidelity of
quantum processors continues to increase,
modeling scrambling in quantum gravity and
unconventional quantum phases may become
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ACKNOWLEDGMENTS
P.R. and X.M. acknowledge fruitful discussions with P. Zoller,
B. Vermersch, A. Elben, and M. Knapp.Funding:S.Ma. and J.Ma.
acknowledge support from the NASA Ames Research Center and
support from the NASA Advanced Supercomputing Division for
providing access to the NASA HPC systems, Pleiades and Merope.
S.Ma. and J.Ma. also acknowledge support from the AFRL
Information Directorate under grant no. F4HBKC4162G001. J.Ma. is
partially supported by NAMS contract no. NNA16BD14C. S.Ma. is
also supported by the Prime contract no. 80ARC020D0010 with the
NASA Ames Research Center.Author contributions:V.Sm., K.K.,
X.M., and P.R. devised the experiment. X.M., C.Q., and P.R. executed
the experiment on the Google quantum hardware. X.M., P.R., K.K.,
and Y.C. wrote the manuscript. X.M., S.Ma., J.Ma., and K.K. wrote the
supplementary materials. V.Sm., I.A., X.M., K.K., S.Ma., and J.Ma.
provided theoretical support, analysis techniques, and numerical
computations. I.A. and K.K. developed the Markov process model.
S.Ma. designed and performed the large-scale numerical simulation,
including the algorithms and software development. J.Ma. performed
the noisy numerical simulations. P.R., Y.C., V.Sm., and H.N. led
and coordinated the project. Infrastructure support was provided by
the Google Quantum AI hardware team. The NASA Advanced
Supercomputing Division at NASA Ames provided the infrastructure
to run high-performance computing (HPC) simulations. All authors
contributed to revising the manuscript and the supplementary
materials.Competing interests:The authors declare no competing
interest.Data and materials availability:All experimental and
numerical data in the main text and supplementary materials,
along with the software code for generating quantum circuits,
measurements, population dynamics simulation, and tensor
contraction simulation are available at Zenodo ( 43 ).SUPPLEMENTARY MATERIALS
science.org/doi/10.1126/science.abg5029
Materials and Methods
Supplementary Text
Figs. S1 to S29
References ( 44 – 75 )
9 January 2021; accepted 19 October 2021
Published online 28 October 2021
10.1126/science.abg5029SCIENCEscience.org 17 DECEMBER 2021¥VOL 374 ISSUE 6574 1483
Circuit Instance100806040200
-1 0 1 -0.2 0.0 0.2-0.1 0.0 0.1
CCC11 Cycles
Niswap= 15912 Cycles
Niswap= 20815 Cycles
Niswap= 251AB
Signal
Expt. Error
0.10.01
3
2
SNR 1Niswap160 180 200 220 240 260C10201010100tsim(Hours)200 300 400 500
NiswapNwv= 64
Nwv= 64
Nwv= 40Fig. 4. Classical simulation of quantum scrambling.(A) (Top) Three different circuit configurations, each
having the sameQa(black-outlined unfilled circles) andQ 1 (black filled circles) but differentQb(colored
circles) and number of cycles in^U. The number of iSWAPs in^UandU^
†
that affect classical simulation costs,
Niswap, are indicated for each configuration. (Bottom) Instance-dependent OTOCs measured for each
configuration. The dashed lines are simulation results of idealized circuits using tensor-contraction methods.
Nwvis also indicated for each configuration. (B) (Top) OTOC signal size and experimental error as functions
ofNiswap. (Bottom) SNR as a function ofNiswap. SNR equals the ratio of OTOC signal size to experimental
error.Nwv= 64 for the first three values ofNiswap, andNwv= 40 for the last two values. The reason for
decreasingNwvis to increase the signal size such that it can be resolved with less statistical averaging ( 36 ).
(C) Estimated timetsimneeded to simulateCof a single 53-qubit idealized circuit with a variable number
of iSWAPs,Niswap, on a single CPU core [6 billion floating point operations (Gflop) per second]. Here, the
filled and unfilled circles represent circuit sizes at which experimental data are available or unavailable,
respectively, for comparison with simulation [see text and ( 36 ) for details].
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