gate errors, suggesting that continued efforts
are needed to decrease the cycle duration and
improve coherence.
Discussion
Our shallow quantum circuits for realizing
toric code eigenstates and simulating braiding
statistics can be extended to other topologi-
cally ordered states, including string-nets with
non-Abelian anyons ( 40 , 41 ). The quasi-static
protocol that uses controlled Pauli strings to
simulate braiding can be generalized to dy-
namical adiabatic braiding ( 42 ) by using deeper
circuits in future devices. Moreover, the tools
we developed can be readily applied to a wide
class of topologically ordered states generated
on quantum processors. By encoding quan-
tum information in the degenerate ground-
state manifold of the toric code, we provide a
method for studying coherence properties of
logical qubit states. This method could be used
to identify and mitigate noise correlations in
the system, with critical implications for future
error-correction experiments.
Note added in proof: A contemporary work by
G. Semeghiniet al.( 43 ) studies topological spin
liquid states on a Rydberg quantum simulator.REFERENCES AND NOTES- D. C. Tsui, H. L. Stormer, A. C. Gossard,Phys. Rev. Lett. 48 ,
1559 – 1562 (1982). - X. G. Wen,Int. J. Mod. Phys. B 4 , 239–271 (1990).
- A. Y. Kitaev,Ann. Phys. 303 ,2–30 (2003).
- M. H. Freedman,Proc. Natl. Acad. Sci. U.S.A. 95 , 98–101 (1998).
- S. B. Bravyi, A. Y. Kitaev, arXiv:quant-ph/9811052 (1998).
- M. H. Freedman, D. A. Meyer,Found. Comput. Math. 1 , 325–332 (2001).
- E. Dennis, A. Kitaev, A. Landahl, J. Preskill,J. Math. Phys. 43 ,
4452 – 4505 (2002). - R. Raussendorf, J. Harrington,Phys. Rev. Lett. 98 , 190504 (2007).
- A. G. Fowler, M. Mariantoni, J. M. Martinis, A. N. Cleland,
Phys. Rev. A 86 , 032324 (2012).
10. C.-Y. Luet al.,Phys. Rev. Lett. 102 , 030502 (2009).
11. J. K. Pachoset al.,New J. Phys. 11 , 083010 (2009).
12. G. Feng, G. Long, R. Laflamme,Phys. Rev. A 88 , 022305 (2013).
13. A. J. Park, E. McKay, D. Lu, R. Laflamme,New J. Phys. 18 ,
043043 (2016).
14. Y. P. Zhonget al.,Phys. Rev. Lett. 117 , 110501 (2016).
15. H.-N. Daiet al.,Nat. Phys. 13 , 1195–1200 (2017).
16. Z. Luoet al.,Nat. Phys. 14 , 160–165 (2018).
17. C. Songet al.,Phys. Rev. Lett. 121 , 030502 (2018).
18. C. Liuet al.,Optica 6 , 264 (2019).
19. C. K. Andersenet al.,Nat. Phys. 16 , 875–880 (2020).
20. A. Rahmaniet al.,PRX Quantum 1 , 020309 (2020).
21. A. Erhardet al.,Nature 589 , 220–224 (2021).
22. F. Aruteet al.,Nature 574 , 505–510 (2019).
23. S. Bravyi, M. B. Hastings, F. Verstraete,Phys. Rev. Lett. 97 ,
050401 (2006).
24. Materials and methods are available as supplementary materials.
25. A. Kitaev, J. Preskill,Phys. Rev. Lett. 96 , 110404 (2006).
26. M. Levin, X.-G. Wen,Phys. Rev. Lett. 96 , 110405 (2006).
27. A. Kitaev,Ann. Phys. 321 ,2–111 (2006).
28. A. Hamma, R. Ionicioiu, P. Zanardi,Phys.Lett.A 337 , 22–28 (2005).
29. S. J. van Enk, C. W. J. Beenakker,Phys.Rev.Lett. 108 , 110503 (2012).
30. A. Elben, B. Vermersch, M. Dalmonte, J. I. Cirac, P. Zoller,
Phys. Rev. Lett. 120 , 050406 (2018).
31. T. Brydgeset al.,Science 364 , 260–263 (2019).
32. B. Nachman, M. Urbanek, W. A. de Jong, C. W. Bauer,
npj Quantum Inf. 6 , 84 (2020).
33. L. Jianget al.,Nat. Phys. 4 , 482–488 (2008).
34. J. J. Wallman, J. Emerson,Phys. Rev. A 94 , 052325 (2016).
35. S. J. Bealeet al., True-Q Version 2. Zenodo (2020);
doi: 10.5281/zenodo.3945250.
36. C. Horsman, A. G. Fowler, S. Devitt, R. Van Meter,New J. Phys.
14 , 123011 (2012).
37. S. Bravyi, A. Kitaev,Phys. Rev. A 71 , 022316 (2005).
38. D. Gottesman,Quantum Information Processing: From Theory
to Experiment(IOS Press, 2006).
39. Z. Chenet al.,Nature 595 , 383–387 (2021).
40. M. A. Levin, X.-G. Wen,Phys. Rev. B Condens. Matter Mater.
Phys. 71 , 045110 (2005).
41. Y.-J. Liu, K. Shtengel, A. Smith, F. Pollmann, arXiv:2110.02020
[quant-ph] (2021).
42. M.Levin,X.-G.Wen,Phys.Rev.BCondens.Matter 67 , 245316 (2003).
43. G. Semeghiniet al.,Science 374 , 1242–1247 (2021).
44. K. Satzinger, Data for“Realizing topologically ordered states on a
quantum processor”.Zenodo(2021); doi: 10.5281/zenodo.5570481.
ACKNOWLEDGMENTS
We thank B. Bauer, A. Elben, B. Vermersch, and G. Vidal for useful
discussions.Funding:F.P., Y.-J.L., A.S., and M.K. acknowledge
support from the Technical University of Munich–Institute for
Advanced Study, funded by the German Excellence Initiative and
the European Union FP7 under grant agreement 291763; the
Max Planck Gesellschaft (MPG) through the International Max
Planck Research School for Quantum Science and Technology
(IMPRS-QST); the Deutsche Forschungsgemeinschaft (DFG;
German Research Foundation) under Germany’s Excellence
Strategy–EXC– 2111 – 390814868, TRR80, and DFG grant KN1254/
2-1; and from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program
(grant agreements 771537 and 851161). A.S. was supported by a
Research Fellowship from the Royal Commission for the Exhibition
of 1851. C.K. was supported by the Walter Burke Institute for
Theoretical Physics at Caltech, and by the IQIM, an NSF Frontier
center funded by the Gordon and Betty Moore Foundation, the
Packard Foundation, and the Simons Foundation.Author
contributions:A.S., M.K., F.P., K.J.S., Y.-J.L., C.K., and P.R.
designed the experiment. K.J.S. and P.R. performed the
experiment. K.J.S. and Y.-J.L. analyzed the data and wrote the
supplement. Y.-J.L., A.S., C.K., M.K., F.P., and K.J.S. provided
theoretical support and analysis. C.K., K.JS., Y.-J.L., A.S., M.K., F.P.,
and P.R. wrote the manuscript. All authors contributed to revising
the manuscript and supplement. All authors contributed to the
experimental and theoretical infrastructure to enable the
experiment.Competing interests:The authors declare no
competing interests.Data and materials availability:Data and
code used for analysis and simulation are available at ( 44 ).SUPPLEMENTARY MATERIALS
science.org/doi/10.1126/science.abi8378
Supplementary Text
Figs. S1 to S25
References ( 45 – 63 )
2 April 2021; accepted 28 October 2021
10.1126/science.abi8378SCIENCEscience.org 3 DECEMBER 2021¥VOL 374 ISSUE 6572 1241
Fig. 4. Surface code logical qubit states.(A) Measured parity values for surface code logical qubit
statesj i¼TL j iþ (^0) L eip=^4 ji (^1) L=
ffiffiffi
2
p
on 5-by-5 and 3-by-3 qubit arrays. Logical operatorsZLandXLspan
across each array. (B) Logical measurement with error correction. We measured a 25-qubit bitstring inXor
Zbasis and evaluated the local parities of the same basis. Negative parities indicate an error. We flipped the
circled qubits to restore positive parities. (C) Experimental logical qubit tomography immediately after
state injection for 128 states (sweeping the initial state of the center qubitaji 0 þbji 1 ), plotted in the
Bloch sphere (5 by 5). The ideal states lie on five planes:x= 0 (yellow),y= 0 (purple),z¼ 1 =
ffiffiffi
2
p
(red),z=0
(blue), andz¼ 1 =
ffiffiffi
2
p
(green). Mean Bloch vector length, 0.6 ± 0.1 (1s). (D) We prepared logical states,
waited for a timet, and then performed a logical measurement. We compared logical measurements
for 1jiL (red) andjiþL (purple), for both 5-by-5 (5-sided markers) and 3-by-3 (3-sided markers) states.
Stars indicate raw measurements. Open symbols indicate corrected measurements. Solid symbols indicate
corrected measurements with dynamical decoupling during the wait time (jiþL only). Each logical
measurement used 10^4 repetitions.
RESEARCH | RESEARCH ARTICLES