Science - USA (2021-12-17)

(both filled). The transition probabilities are
described by the stochastic matrix

W¼

10 0 0

01 aba b

0 a 1 abb

0

b

3

b

3

1 

2

3

b

0

B

B

B

B

@

1

C

C

C

C

A

ð 2 Þ

wherea¼ 31 sin^4 qandb¼^1312 sin^22 qþ

2sin^2 qÞ,
withqbeing the swap angle of the two-qubit
gate. The average probability of finding a par-
ticle at the site ofQ 1 is then used to estimateC.
We note that the effect of gate errors can also
be included by multiplying all transition prob-
abilities exceptðÞ⋄⋄→⋄⋄ by a constant fac-
tore

16
15 p^2 , wherep 2 is an error rate associated
with each two-qubit cycle ( 36 ). In this classical
picture, the OTOC wavefront corresponds to
the boundary separating the empty region
from the region populated by particles.
The difference in OTOC propagation be-
tween iSWAP and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

iSWAP

p
is then captured by
the dependence ofWonq: After each applica-
tion of an iSWAP gateðÞq¼p= 2 , the particle
occupation changes with the exception ofðÞ⋄⋄.
In particular, any previously empty sites will be
filled, and the region populated by particles
always grows. This leads to the observed max-
imal butterfly velocity. By contrast, the appli-
cation of any

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

iSWAP

p

gateðÞq¼p= 4 can leave

the particle occupation unchanged with a

probability 125 .Ctherefore decays more slowly

in this case, and its broadening is explained

by the fact that the wavefront spreads at a

different velocity for each trial of the Markov

process. The predicted values ofC, plotted as

dashed lines in Fig. 2C, agree well with the ex-

perimental data for iSWAP even when no noise

is added (p 2 = 0). In the case of

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

iSWAP

p

, we

find that population dynamics agree equally well

with experimental data when an empirical error

ratep 2 = 0.012 is included in the simulation.

Unlike operator spreading, an efficient clas-

sical description of operator entanglement is not

known to exist for our system. For noninte-

grable systems in particular, circuit-to-circuit

fluctuation of OTOCs away from asymptotic

limits cannot be modeled by population dy-

namics ( 36 ). Resolving the growth of operator

entanglement is also difficult with a quantum

processor because it is often accompanied

by increased operator spreading ( 16 – 19 ). We

overcome this challenge by gradually adjust-

ing the composition ofU^andU^

†

, realizing a

group of circuits with predominantly Clifford

gatesðiSWAP;

ffiffiffiffiffiffiffi

XT^1

p

or

ffiffiffiffiffiffiffi

YT^1

p

Þ( 37 ) and a small

number of non-Clifford gatesð

ffiffiffiffiffiffiffiffiffi

WT^1

p

ffiffiffiffiffiffiffiffi and

VT^1

p

Þ. For such circuits,Ot^ðÞmay be con-

veniently expanded into products of single-

qubit Pauli operators (Pauli strings) for analysis

of operator entanglement, where numerical

simulation studies demonstrate the same qual-

itative behavior as the minimal product basis^Bi

( 36 ). This is because sufficiently long random

Clifford circuits map a Pauli string into another

weakly correlated Pauli string, and as a result,

this basis cannot be improved by single-qubit

rotations [see section VII of ( 36 )]. The trans-

formation of Pauli strings by representative

gates in our circuits is illustrated in Fig. 3A:

Clifford gates preserve the total number of

Pauli strings but generate operator spreading

given that iSWAP can increase the number of

nonidentity Pauli operators. By contrast, non-

Clifford gates generate operator entanglement

by transforming one single Pauli string into a

superposition of multiple Pauli strings, main-

taining the spatial extent of operator spread-

ing in the process.

These distinctive properties of Clifford and

non-Clifford gates therefore provide us with

a way to independently tune one scrambling

mechanism without affecting the other. We

now focus on operator entanglement and

measure the circuit-to-circuit fluctuation of

OTOCs, as shown in Fig. 3B. Here, the number

ofcircuitcyclesisfixedat12,andthenumber

of non-Clifford gates inU^,Nwv, is successively

changed from 0 to 32. For eachNwv, the indi-

vidual OTOCsCof 130 random circuit instances

are measured using a modified normalized

procedure [fig. S13 ( 36 )]. AtNwv= 0, where

the circuits consist of only Clifford gates, we

see thatCtakesdiscretevaluesof1or−1.

SCIENCEscience.org 17 DECEMBER 2021•VOL 374 ISSUE 6574 1481

6 Cycles 10 Cycles 14 Cycles 18 Cycles 22 Cycles

6 Cycles 8 Cycles 10 Cycles 13 Cycles 15 Cycles

1.0

0.8

0.6

0.4

0.2

0.0

-0.4

-0.2

A

B

C (iSWAP)

C

0 5 10 15 20 25

Cycles

C

0 5 10 15

Cycles

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

C

1.0

0.8

0.6

0.4

0.2

0.0

-0.4

-0.2

C(iSWAP)

Qa

Q 1

Qa

Q 1

1.0

0.8

0.6

0.4

0.0

0.2

—

—

• 
  


• 
  

Fig. 2. OTOC propagation and speed of operator spreading.(A) Spatial
profiles of average OTOCs,C, measured on the full 53-qubit processor.Qa
andQ 1 are indicated by the red arrows. The colors of the other filled circles
representCwith different choices ofQb. The two-qubit gates are iSWAP and
applied between all nearest-neighbor qubits, in the same order as done in ( 38 ).
The dashed lines delineate the light cone ofQ 1. The data are averaged over

38 circuit instances. (B) Similar to (A) but with

ffiffiffiffiffiffiffiffiffiffiffiffiffi

iSWAP

p

as the two-qubit gates.

Here,Cis averaged over 24 circuit instances. (C) Cycle-dependentCfor

four different choices ofQb. The top and bottom panels shows data with iSWAP

and

ffiffiffiffiffiffiffiffiffiffiffiffiffi

iSWAP

p

, respectively, as the two-qubit gates. The colors of the data

points indicate the locations ofQb(inset in bottom panel). Dashed lines show

theoretical predictions based on a classical population dynamics model. A

two-qubit error ratep 2 = 0.012 is included in the prediction for

ffiffiffiffiffiffiffiffiffiffiffiffiffi

iSWAP

p

circuits,

and noiseless simulation can be found in fig. S24 ( 36 ).

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