Q 20 asQb. A two-qubit gate (iSWAP) is applied
to each pair (Qj,Qj+1), wherej= 0,2,4...in odd
circuit cycles andj= 1,3,5...in even circuit
cycles.
In the left panel of Fig. 1C, experimental
values ofhis^yare shown for different numbers
of cycles inU^. Here, we have averaged the data
over different circuit instances to initially
focus on operator spreading. A sharp decrease
is seen in eachhi^sywhen the number of circuit
cycles has first exceeded the number of qubits
betweenQbandQ 1 —the minimum time re-
quired forM^and the time-evolved operator
Ot^ðÞto overlap and become noncommuting.
This propagating behavior, although indica-
tive of operator spreading, is also complicated
by errors in the quantum circuits, such as qubit
decoherence or any mismatch betweenU^and
U^†. These noisy effects are evident from the
decay inhis^ywhileOt^ðÞandM^still commute.
To clearly distinguish noise from scrambling,
we also measurehi^sy in the absence of^O
(shown ashi^syIin the left panel of Fig. 1C) ( 32 ).
This control experiment realizes an identity
operation, and the decay ofhi^syIis induced
purely by noise ( 25 ). As such, the difference
betweenhis^yIand eachhis^y in Fig. 1C after
the overlap ofOt^ðÞandM^constitutes an un-
ambiguous signature of operator spreading in
our experiment.
In addition to verifying operator spread-
ing,hi^syIapproximates circuit fidelity ( 25 )
and therefore enables the reconstruction of
ideal circuit dynamics when used to normalize
eachhi^sy( 32 ). We test this error-mitigation
protocol and plot the ratioC¼his^y=hi^syIin
therightpanelofFig.1C(thedatafortherest
of this work are shown after similar normal-
ization procedures).Cexhibits nearly noise-
free features of operator spreading: For each
location ofQb,Cretains values of ~1 whenOt^ðÞ
andM^are nonoverlapping and commute witheach other. AfterOt^ðÞandM^have overlapped
andceasedtocommute,Cconverges to much
smaller values. Additionally, we observe that
the time evolution ofCfor eachQbresembles
a ballistically propagating wave. The front of
each wave coincides with the edge of the
light cone associated withQ 1 —i.e., the set
of qubits that have been entangled withQ 1.
This profile is attributed to the iSWAP gates
used in these circuits, which spread single-
qubitoperatorsatthesamerateastheir
light cones expand ( 33 , 34 ). For generic quan-
tum circuits, the spreading velocity (also
known as the butterfly velocity) is typically
slower. Using the full 2D system, we next
demonstrate how the evolution ofCmay be
used to diagnose the butterfly velocity of ope-
rator spreading.
In Fig. 2A, the spatial distribution ofCis
shown for five different numbers of cycles in
U^, with iSWAP still being the two-qubit gate.
We see that the number of qubits withC< 1
rapidly increases with the number of cycles,
consistent with the spatial spread of the time-
evolved operatorOt^ðÞ. Moreover, for each cir-
cuit cycle, the values ofCabruptly change
acrosstheedgeofthelightconeassociated
withQ 1 (dashed lines in Fig. 2A). By contrast,
the spatiotemporal evolution ofCshown in
Fig. 2B is significantly different. Here, the
iSWAP gates are replaced withffiffiffiffiffiffiffiffiffiffiffiffiffiffi
iSWAPp
gates,
and the decay ofCis slower. Qubits far from
Q 1 retainCclose to 1 even after 22 cycles. The
sharp, step-like spatial transition seen with
iSWAP is also absent forffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
iSWAPp. Instead,C
changes in a gradual fashion asQbmoves fur-
ther away fromQ 1.
The different OTOC behaviors can alterna-
tively be seen in the full temporal evolution of
four specific qubits (Fig. 2C). For iSWAP, the
shape of the OTOC wavefront remains sharp
and relatively insensitive to the location ofQb,
similar to the 1D example in Fig. 1C. On the
other hand, the wavefront propagates more
slowly for
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
iSWAPp
and also broadens as the
distance betweenQbandQ 1 increases. As a
result, more circuit cycles are required before
Creaches 0 forffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
iSWAPp. The wavefront be-
havior seen with
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
iSWAPp
is similar to that ob-
served for generic quantum circuits analyzed in
past works ( 18 , 19 , 35 ).
The observed features of average OTOCs
are quantitatively understood by mapping
operator spreading to a classical Markov pro-
cess involving population dynamics ( 36 ). In
this model, the 2D qubit lattice is populated by
fictitious particles representing two copies of
a single-qubit operator. The initial state of the
entire system is a single particle at the site of
Qb. Whenever a two-qubit gate is applied to
two neighboring lattice sites, their particle
occupation changes between four possible
states:⋄⋄(both empty),⋄♦(left empty, right
filled),♦⋄(right empty, left filled), and♦♦1480 17 DECEMBER 2021•VOL 374 ISSUE 6574 science.orgSCIENCE
Fig. 1. OTOC measurement protocol.(A) Experimental scheme: A quantum circuit^Uand its inverse^U
are successively applied to a quantum system (qubitsQ 1 throughQN), with a local operatorO^¼^sxðÞb
in between (butterfly symbol).Qais entangled withQ 1 through controlled-phase (CZ) gates, and its projection
to theyaxis of the Bloch sphere ^sy
is measured at the end. The unitary operatorM^iss^zðÞ^1 in this
experiment. +Xand +Ydenote single-qubit states along thexandyaxes, respectively, of the Bloch
sphere. (B) The structure of^Uconsists ofKcycles: Each cycle includes one layer of single-qubit gates
randomly chosen fromffiffiffiffiffiffi
XT^1p
;ffiffiffiffiffiffi
YT^1p
;ffiffiffiffiffiffiffiffi
WT^1p
;andffiffiffiffiffiffi
VT^1p
and one layer of two-qubit entangling gates (EGs).
Here,W¼Xpþffiffi 2 YandV¼Xp ffiffi 2 Y.(C) (Left) The filled circles represent cycle-dependent ^sy
measured withQb
successively chosen fromQ 2 throughQN. Here,N= 20. The open gray circles represent^sy
I, which is ^sy
without applyingO^. The data are averaged over 60 random circuit instances. (Right) Average OTOCsC
for differentQb, obtained by dividing the corresponding ^sy
by ^sy
I.
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