- K. K. Brown, J. B. Spinelli, J. M. Asara, A. Toker,Cancer Discov.
7 , 391–399 (2017). - J. Rohlena, L.-F. Dong, S. J. Ralph, J. Neuzil,Antioxid. Redox
Signal. 15 , 2951–2974 (2011). - A. N. Lau, M. G. Vander Heiden,Annu. Rev. Cancer Biol. 4 ,
17 – 40 (2020).
ACKNOWLEDGMENTS
We thank all members of the Sabatini lab for their thoughtful
comments, especially G. Frenkel, A. Armani, and N. Kory for
thoughtful feedback on the manuscript and experiments. We thank
the Whitehead Institute Metabolite Profiling core for their
assistance with experimental design and data interpretation and
the Keck Imaging facility for help with microscopy experiments.
Funding:This work was funded by an R01 application granted
to D.M.S. (R01CA219859). J.B.S. is funded by the NCI F99/K00
predoctoral to postdoctoral transition fellowship (K00CA234839)
and P.C.R. by a predoctoral fellowship from the NCI (F31
CA254162-01). H.G.S. is funded by the DFG (Deutsche
Forschungsgemeinschaft; SP 1897/1-1) and is a Hope Funds for
Cancer Research Fellow supported by the Hope Funds for Cancer
Research (HFCR-20-03-01). A.M.P. is funded by a William N.
and Bernice E. Bumpus Fellowship, A.L.C. by an NIH F31
predoctoral fellowship (F31 5F31DK113665), K.J.C. by an MIT
School of Science Fellowship in Cancer Research and an NSF
fellowship (2016197106), and N.S.C. by NIH grants (R35CA197532
and 5P01AG049665). D.M.S. is formerly an investigator of the
Howard Hughes Medical Institute.Author contributions:J.B.S.
initiated the project, designed and analyzed most experiments, and
interpreted experimental results with guidance from D.M.S.;
J.B.S., P.C.R., T.Z., and A.H. purified mitochondria and performed
biochemical assays. J.B.S., J.M.R., H.-G.S., A.M.P., and A.L.C
performed mouse experiments. J.L.M. performed proliferation
experiments. J.B.S. and K.J.C. performed fluorescence-activated
cell sorting experiments. J.B.S., T.K., J.L.M., and C.A.L. designed
and analyzed LC-MS experiments. N.S.C. guided experimental
design and data interpretation. J.B.S. and D.M.S. wrote the
manuscript and acquired funding.Competing interests:Allauthors declare that they have no competing interests.Data and
materials availability:All data are available in the manuscript
or the supplementary materials. To ensure sustainable access to
data and materials associated with this study, the institution has
committed to assuring long-term access and has designated
[email protected] as a contact. Access to reagents not found
on Addgene will be facilitated by [email protected].SUPPLEMENTARY MATERIALS
science.org/doi/10.1126/science.abi7495
Materials and Methods
Figs. S1 to S12
References ( 60 – 64 )
MDAR Reproducibility Checklist28 March 2021; accepted 2 November 2021
10.1126/science.abi7495TOPOLOGICAL MATTER
Realizing topologically ordered states on a
quantum processor
K. J. Satzinger^1 , Y.-J Liu2,3, A. Smith2,4,5, C. Knapp6,7 , M. Newman^1 , C. Jones^1 , Z. Chen^1 , C. Quintana^1 ,
X. Mi^1 , A. Dunsworth^1 , C. Gidney^1 , I. Aleiner^1 , F. Arute^1 , K. Arya^1 , J. Atalaya^1 , R. Babbush^1 ,
J. C. Bardin1,8, R. Barends^1 , J. Basso^1 , A. Bengtsson^1 , A. Bilmes^1 , M. Broughton^1 , B. B. Buckley^1 ,
D. A. Buell^1 , B. Burkett^1 , N. Bushnell^1 , B. Chiaro^1 , R. Collins^1 , W. Courtney^1 , S. Demura^1 , A. R. Derk^1 ,
D. Eppens^1 , C. Erickson^1 , L. Faoro^9 , E. Farhi^1 , A. G. Fowler^1 , B. Foxen^1 , M. Giustina^1 , A. Greene1,10,
J. A. Gross^1 , M. P. Harrigan^1 , S. D. Harrington^1 , J. Hilton^1 , S. Hong^1 , T. Huang^1 , W. J. Huggins^1 ,
L. B. Ioffe^1 , S. V. Isakov^1 , E. Jeffrey^1 , Z. Jiang^1 , D. Kafri^1 , K. Kechedzhi^1 , T. Khattar^1 , S. Kim^1 ,
P. V. Klimov^1 , A. N. Korotkov1,11, F. Kostritsa^1 , D. Landhuis^1 , P. Laptev^1 , A. Locharla^1 , E. Lucero^1 ,
O. Martin^1 , J. R. McClean^1 , M. McEwen1,12, K. C. Miao^1 , M. Mohseni^1 , S. Montazeri^1 , W. Mruczkiewicz^1 ,
J. Mutus^1 , O. Naaman^1 , M. Neeley^1 , C. Neill^1 ,M.Y.Niu^1 , T. E. OÕBrien^1 , A. Opremcak^1 , B. Pató^1 ,
A. Petukhov^1 , N. C. Rubin^1 , D. Sank^1 , V. Shvarts^1 , D. Strain^1 , M. Szalay^1 , B. Villalonga^1 , T. C. White^1 ,
Z. Yao^1 , P. Ye h^1 , J. Yoo^1 , A. Zalcman^1 , H. Neven^1 , S. Boixo^1 , A. Megrant^1 , Y. Chen^1 , J. Kelly^1 ,
V. Smelyanskiy^1 , A. Kitaev1,6,7, M. Knap2,3,13, F. Pollmann2,3, P. Roushan^1 *
The discovery of topological order has revised the understanding of quantum matter and provided the
theoretical foundation for many quantum error–correcting codes. Realizing topologically ordered states
has proven to be challenging in both condensed matter and synthetic quantum systems. We prepared the
ground state of the toric code Hamiltonian using an efficient quantum circuit on a superconducting
quantum processor. We measured a topological entanglement entropy near the expected value of–ln2 and
simulated anyon interferometry to extract the braiding statistics of the emergent excitations. Furthermore,
we investigated key aspects of the surface code, including logical state injection and the decay of the
nonlocal order parameter. Our results demonstrate the potential for quantum processors to provide
insights into topological quantum matter and quantum error correction.
D
ifferent phases of matter can commonly
be distinguished in terms of spontane-
ous symmetry breaking and local order
parameters. However, several exotic
quantum phases have been discovered
in recent decades that defy this simple classi-
fication, instead exhibiting topological order
( 1 , 2 ). These phases are characterized by their
long-range quantum entanglement and the
emergence of quasiparticles with anyonic ex-
change statistics. Moreover, they have energet-
ically gapped ground states with degeneracies
that depend on their boundary conditions.
The nonlocal nature of these states makesthem particularly attractive platforms for
fault-tolerant quantum computation because
quantum information encoded in locally in-
distinguishable ground states is robust to local
perturbations ( 3 , 4 ). This is the underlying prin-
ciple of topological quantum error–correcting
codes, in which the logical codespace corre-
sponds to the degenerate ground-state sub-
space of a lattice model ( 5 – 7 ).
An archetypical topological two-dimensional
(2D) lattice model is the toric code, which ex-
hibits so-calledℤ 2 topological order ( 3 ). The
realization of the toric code on a plane—the
surface code—has emerged as one of the most
promising stabilizer codes for quantum error
correction owing to its amenable physical re-
quirements ( 8 , 9 ). Given both its inherent
richness and applications in quantum com-
puting, experimentally realizingℤ 2 topological
order has sparked extensive interest, resulting
in several proposals and experimental studies
with comparatively small-scale synthetic quan-
tum systems ( 10 – 21 ). Despite these efforts,
the experimental realization of topologically
ordered states remains a major challenge, re-
quiring the generation of long-range entan-
glement. This can be achieved by identifying
suitable quantum systems with topologically
ordered ground states or by constructing a
topologically ordered state in an engineered
quantum system. Probing the nonlocal topo-
logical properties of such a state on an array of
qubits requires high-fidelity gates and a suffi-
ciently large 2D lattice.
In this work, we developed an efficient quan-
tum circuit to prepare the toric code ground
state on a lattice of 31 superconducting qu-
bits. We then experimentally established theSCIENCEscience.org 3 DECEMBER 2021•VOL 374 ISSUE 6572 1237
(^1) Google Quantum AI, Mountain View, CA, USA. (^2) Department of Physics, Technical University of Munich, 85748 Garching, Germany. (^3) Munich Center for Quantum Science and Technology
(MCQST), Schellingstraße 4, 80799 München, Germany.^4 School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK.^5 Centre for the Mathematics and Theoretical
Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, UK.^6 Department of Physics and Institute for Quantum Information and Matter, California Institute
of Technology, Pasadena, CA, USA.^7 Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA, USA.^8 Department of Electrical and Computer Engineering,
University of Massachusetts, Amherst, MA, USA.^9 Laboratoire de Physique Theorique et Hautes Energies, Sorbonne Université, 75005 Paris, France.^10 Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA.^11 Department of Electrical and Computer Engineering, University of California, Riverside, CA, USA.^12 Department of Physics,
University of California, Santa Barbara, CA, USA.^13 Institute for Advanced Study, Technical University of Munich, 85748 Garching, Germany.
*Corresponding author. Email: [email protected] (F.P); [email protected] (P.R.); [email protected] (K.J.S.)
†Present address: Station Q, Microsoft, Santa Barbara, CA, USA.
RESEARCH | RESEARCH ARTICLES