QUANTUM SIMULATION
Information scrambling in quantum circuits
Xiao Mi^1 †, Pedram Roushan^1 †, Chris Quintana^1 †, Salvatore Mandrà2,3,
Jeffrey Marshall2,4, Charles Neill^1 , Frank Arute^1 , Kunal Arya^1 , Juan Atalaya^1 , Ryan Babbush^1 ,
Joseph C. Bardin1,5, Rami Barends^1 , Joao Basso^1 , Andreas Bengtsson^1 , Sergio Boixo^1 ,
Alexandre Bourassa1,6, Michael Broughton^1 ,BobB.Buckley^1 ,DavidA.Buell^1 , Brian Burkett^1 ,
Nicholas Bushnell^1 , Zijun Chen^1 , Benjamin Chiaro^1 , Roberto Collins^1 , William Courtney^1 ,
Sean Demura^1 ,AlanR.Derk^1 , Andrew Dunsworth^1 , Daniel Eppens^1 , Catherine Erickson^1 ,
Edward Farhi^1 ,AustinG.Fowler^1 , Brooks Foxen^1 , Craig Gidney^1 , Marissa Giustina^1 ,
Jonathan A. Gross^1 , Matthew P. Harrigan^1 , Sean D. Harrington^1 ,JeremyHilton^1 ,
Alan Ho^1 , Sabrina Hong^1 , Trent Huang^1 , William J. Huggins^1 ,L.B.Ioffe^1 , Sergei V. Isakov^1 ,
Evan Jeffrey^1 , Zhang Jiang^1 , Cody Jones^1 ,DvirKafri^1 , Julian Kelly^1 , Seon Kim^1 ,
Alexei Kitaev1,7,PaulV.Klimov^1 ,AlexanderN.Korotkov1,8, Fedor Kostritsa^1 , David Landhuis^1 ,
Pavel Laptev^1 ,ErikLucero^1 , Orion Martin^1 , Jarrod R. McClean^1 , Trevor McCourt^1 ,
Matt McEwen1,9, Anthony Megrant^1 , Kevin C. Miao^1 , Masoud Mohseni^1 , Shirin Montazeri^1 ,
Wojciech Mruczkiewicz^1 , Josh Mutus^1 , Ofer Naaman^1 , Matthew Neeley^1 , Michael Newman^1 ,
Murphy Yuezhen Niu^1 , Thomas E. O'Brien^1 , Alex Opremcak^1 , Eric Ostby^1 ,BalintPato^1 ,
Andre Petukhov^1 , Nicholas Redd^1 , Nicholas C. Rubin^1 , Daniel Sank^1 , Kevin J. Satzinger^1 ,
Vladimir Shvarts^1 ,DougStrain^1 , Marco Szalay^1 , Matthew D. Trevithick^1 , Benjamin Villalonga^1 ,
Theodore White^1 , Z. Jamie Yao^1 ,PingYeh^1 , Adam Zalcman^1 , Hartmut Neven^1 , Igor Aleiner^1 ,
Kostyantyn Kechedzhi^1 , Vadim Smelyanskiy^1 , Yu Chen^1 *
Interactions in quantum systems can spread initially localized quantum information into the
exponentially many degrees of freedom of the entire system. Understanding this process, known as
quantum scrambling, is key to resolving several open questions in physics. Here, by measuring the
time-dependent evolution and fluctuation of out-of-time-order correlators, we experimentally investigate
the dynamics of quantum scrambling on a 53-qubit quantum processor. We engineer quantum circuits
that distinguish operator spreading and operator entanglement and experimentally observe their
respective signatures. We show that whereas operator spreading is captured by an efficient classical
model, operator entanglement in idealized circuits requires exponentially scaled computational
resources to simulate. These results open the path to studying complex and practically relevant physical
observables with near-term quantum processors.
T
he realization of quantum computers was
motivated by their ability to simulate
dynamical processes that are challeng-
ing for classical computation. A physical
process that fully leverages the computa-
tional power of quantum processors is quantum
scrambling, which describes how interaction
in a quantum system disperses local informa-
tion into the system’s many degrees of freedom
( 1 – 5 ). Quantum scrambling is the underlying
mechanism for the thermalization of isolated
quantum systems ( 6 – 8 ), and accurately model-
ing its dynamics is the key to resolving a number
of open questions, including the fast-scrambling
conjecture for black holes ( 2 , 3 ), the nature of
strange metals ( 9 , 10 ), and many-body local-
ization ( 11 ). Understanding scrambling also
provides a basis for designing algorithms in
quantum benchmarking or machine learning
that would benefit from the efficient explora-
tion of Hilbert spaces ( 12 – 14 ).
A precise formulation of quantum scrambling
is found in the Heisenberg picture, where quan-
tum operators evolve and quantum states are
stationary. Similar to classical chaos, scrambling
manifests itself as a butterfly effect, wherein a
local perturbation is rapidly amplified over time
( 15 , 16 ). More specifically, the perturbation is
realized as an initially local unitary operator^O
acting on one of theNqubitsQb. When the
quantum system undergoes a unitary process
U^,O^acquires a time dependence and becomes
Ot^ðÞ¼U^†O^U^. The resulting^OtðÞcan be ex-
panded asOt^ðÞ¼XnB
i¼ 1 wi^Bi, where^Bi¼
^b 1 ð Þ
i ^b 2 ð Þ
i ...are basis operators consist-
ing of products of single-qubit operators^bjðÞi,
each acting on a different (jth) qubit, andwi
are their coefficients.
Quantum scrambling is enabled by two dif-
ferent mechanisms ( 16 – 20 ): (i) Operator spread-
ing, wherein basis operators are transformed
such that eachB^iinvolves more nonidentitysingle-qubit operators over time and (ii) gen-
eration of operator entanglement ( 21 ), which
is reflected in the growth, in time, of the mini-
mum number of termsnBneeded to expandO^
into products of single-qubit operators, with a
broad distribution of coefficientswi. Indepen-
dent characterizations of these two mechanisms
are essential for a complete understanding of
the nature of quantum scrambling. Additionally,
operator spreading can be effectively mapped to
classical dynamics ( 16 , 18 – 20 , 22 , 23 ), whereas
for the circuits studied in this work, operator
entanglement is a quantum process. As such,
quantum simulation of operator entanglement
dynamics has the potential to deliver quantum
computational advantages in modeling phys-
ical phenomena. However, these two mecha-
nisms are often intertwined, and signatures of
operator entanglement have not been identi-
fied in past experimental studies of quantum
scrambling ( 24 – 28 ).
In this work, we report independent ob-
servations of operator spreading and opera-
tor entanglement in a two-dimensional (2D)
quantum system of 53 superconducting qubits.
Our approach is based on measuring the corre-
lator betweenOt^ðÞand another unitary opera-
torM^, which is a Pauli operator on a different
qubitQ 1CtðÞ¼O^†
ðÞtM^†^
OtðÞM^DE
ð 1 ÞHere,hi...denotes the expectation value over
a particular quantum state.C(t) is known as
the out-of-time-order correlator (OTOC) ( 29 )
and is related to the commutator^OtðÞ;M^
by
Re½ ¼CtðÞ 1 21 ^OtðÞ;M^ †
Ot^ðÞ;M^DE . By
measuringCover quantum evolution with
microscopic differences, operator spreading
may be characterized through the decay of the
average OTOC valueCfrom 1 asOt^ðÞandM^
become noncommuting. In the fully scram-
bled limit, whereOt^ðÞandM^ have random
commutation,Capproaches an exponentially
small limit 22 N^1 1 ( 17 ). If operator entangle-
ment is also generated, the growing number
of contributions toCfrom basis operators^Bi
with similar magnitudes ofwiresults in the
decay of OTOC fluctuationdCas well. This
property will be used as a witness of operator
entanglement for a class of quantum circuits
below.
The experimental scheme, described in Fig. 1A,
utilizes an interferometric protocol that maps
Conto the projectionhi^sy of an ancilla qubit,
Qa( 30 , 31 ). For this work,U^is realized using
quantum circuits comprising random single-
qubit gates and fixed two-qubit gates (Fig. 1B);
this choice was made because of the ease of
controlling different scrambling mechanisms
in such circuits. OTOC measurements are first
conducted on a 1D chain of 21 qubits (Fig. 1C).
We use the qubit at one end of the chain as
Qaand successively choose qubitsQ 2 through
SCIENCEscience.org 17 DECEMBER 2021•VOL 374 ISSUE 6574 1479
(^1) Google Research, Mountain View, CA, USA. (^2) QuAIL, NASA
Ames Research Center, Moffett Field, CA, USA.^3 KBR, Inc.,
Houston, TX, USA.^4 USRA Research Institute for Advanced
Computer Science, Mountain View, CA, USA.^5 Department
of Electrical and Computer Engineering, University of
Massachusetts, Amherst, MA, USA.^6 Pritzker School of
Molecular Engineering, University of Chicago, Chicago, IL,
USA.^7 Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, CA, USA.
(^8) Department of Electrical and Computer Engineering,
University of California, Riverside, CA, USA.^9 Department of
Physics, University of California, Santa Barbara, CA, USA.
*Corresponding author. Email: [email protected] (K.K.);
[email protected] (V.S.); [email protected] (Y.C.)
These authors contributed equally to this work.
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