Science - USA (2021-12-17)

This is expected because the time-evolved
operatorOt^ðÞis a single Pauli string and
therefore either commutes or anticommutes
withM^. As more non-Clifford gates are in-
troduced into the circuits,Cstarts to assume
intermediate values between ±1 and con-
verges toward 0.
The meanCand fluctuation [i.e., root mean
square (RMS) value]dCofCare then com-
puted from experimental data and plotted
againstNwvin Fig. 3C. We observe different
behaviors forCanddC:Cremains largely
constant and close to 0, confirming that ope-
rator spreading remains unaffected by the
increasing number of non-Clifford gates. Con-
versely,dCdecays from an initial value of
1 and is suppressed by almost two orders of
magnitude asNwvincreases from 0 to 32,
which indicates thatOt^ðÞexpands into a large
number,np, of weakly correlated Pauli strings
with comparable coefficients. This is confirmed
by numerically computed values ofnp, which
increase exponentially withNwv(inset in Fig.
3C). The growth ofnpnecessarily implies sub-
stantial operator entanglement (increase in
nB)aswellasgrowthofmore–commonly used
metrics, such as the operator entanglement
entropy [see section VII of ( 36 ) for a detailed
analysis]. These results demonstrate that the
decay of OTOC fluctuation allows the growth
of operator entanglement to be experimentally
diagnosed.
To determine the accuracy of our measure-
ments, the OTOCs of experimental circuits are
simulated using a Clifford-expansion method

( 36 ) and overlaid on the data in Fig. 3, B and C.

The close agreement between experiment and

simulation motivates us to further explore how

our experimental accuracy changes with clas-

sical simulation complexity. This is done by

systematically increasing the number of iSWAPs

in the quantum circuits,Niswap(here,Niswap

counts only iSWAP gates that lie within the

light cones ofQaandQ 1 ). At the same time,Nwv

is kept at a large value such that the Clifford-

expansion simulation method used in Fig. 3 is

challenging to perform. Instead, we use tensor

contraction to approximately simulateC; these

simulations replicate idealized circuits and do

not take experimental imperfections into ac-

count. Tensor contraction was chosen because

it is the best-performing algorithm for simulat-

ing Sycamore random circuits with low errors

compared with the ideal outcomes ( 36 , 38 – 40 ).

Figure 4A shows representative data for three

circuit configurations with different values of

Niswapalong with the corresponding numer-

ical simulation results. AsNiswapand the com-

putational complexity for tensor-contraction

increase, we observe that the OTOC fluctua-

tion decreases. The agreement between ex-

periment and simulation also degrades—a

result of increased experimental errors, such

as qubit decoherence and imperfect circuit

inversion, that are not taken into account by

the simulation.

To quantify these observations, we define an

ideal OTOC signal as the fluctuationdCcom-

puted from the simulated values ofC.Wealso

define the experimental error as the RMS de-

viation between the simulated and the mea-

sured values ofC. Both quantities are shown

as functions ofNiswapin the upper panel of

Fig. 4B. The ratio of the two is the experimental

signal-to-noise ratio (SNR) forCand is plotted

in the bottom panel of Fig. 4B. The estimated

time to simulate a single value ofCfor idealized

circuits using tensor contraction on a central

processing unit (CPU) core ( 36 ),tsim,isplotted

in Fig. 4C; in estimatingtsim,wehaveonly

counted the time needed to simulateCabove

the experimental error shown in Fig. 4B. For

Niswap> 280, we estimatetsimas the expected

time needed to contract the tensor network of

the quantum circuit a single time. AsNiswap

increases,tsimincreases and the experimental

SNR decreases, reaching SNR≈1 forNiswap=

251 (tsim≈1 hour, or 100 hours for all exper-

imental circuits). Although it is still feasible to

simulate our current results, we estimate that

the classical simulation of idealized circuits

likely becomes intractable whenNiswap> 400,

owing to the exponential scaling oftsim. It is

possible that classical algorithms may be fur-

ther improved if the limited experimental accu-

racy is taken into account. In sections I and VI

of ( 36 ), we present the analysis and simulation

results demonstrating that the observed ex-

perimental errors are consistent with the level

of noise in our device.

We characterize quantum scrambling in a

53-qubit system and find that the accuracy of

quantum processors can be significantly im-

proved through effective error mitigation. For

example, an SNR of ~1 for OTOCs is achieved

1482 17 DECEMBER 2021•VOL 374 ISSUE 6574 science.orgSCIENCE

Circuit Instance

100

80

60

40

20

0

120

BCNwv:^12

Instance-Specific OTOC, C

0 4 8 16 32

1.00

0.10

0.02

OTOC Fluctuation,

δC

Experiment

Simulation

Experiment

Simulation

0.06

0.04

0.02

0.00

-0.02

C

iSWAP

X

x

y

z

A

-1 01 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 01 -1 0 1

12 Cycles

0 5 10 15 20 25 30

Nwv

0 10 20 30

Nwv

106

104

102

100

np

W

+

+

dr

off

il

C

Non-

dr

off

il

C

108

” –

σ

σ

σ

Fig. 3. OTOC fluctuations and signatures of operator entanglement.
(A) Example transformation of a product of Pauli operators (Pauli string) by

different quantum gates. A single Pauli string^I

ðÞ 1

^s

ðÞ 2

z ^s

ðÞ 3

x

^IðÞ^4 ^IðÞ^5 is mapped either

into a different Pauli string by a Clifford gate or a superposition of multiple
Pauli strings (coefficients not shown) by a non-Clifford gate. (B) OTOCs of
individual random circuit instances,C, measured with the number of non-Clifford

gates in^U,Nwv, fixed at different values. Dashed lines are numerical
simulation results. For each circuit, the non-Clifford gates are injected at
random locations within the intersection between the light cones ofQbandQ 1.

The inset shows locations ofQa(black-outlined unfilled circle),Q 1 (black filled

circle), andQb(blue filled circle) as well as the number of circuit cycles

with which the data are taken. Here, and also in Fig. 4, error bars are omitted

because a sufficient number of samples was taken to ensure that the statistical

uncertainty is≤0.01 ( 36 ). (C) The meanC(top) and RMS valuesdC(bottom)

ofCfor differentNwv. Dashed lines are computed from the numerically simulated

values in (B). (Inset) Numerically computed average numbers of Pauli strings

in the time-evolved operatorOt^ðÞ,np, for the experimental circuits. Dashed line is

an exponential fit,np≈ 20 :^96 Nwv.

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