Abstract

Anyons are quasiparticles that obey fractional statistics, producing a phase intermediate to bosons and fermions under particle interchange. Anyons form the basis for topological quantum computation and error correction, where the topological aspect of anyonic braiding is one of the important features that gives rise to fault tolerance. A central model that exhibits anyons is Kitaev’s toric code, where the excited states involve anyons that exhibit topological behavior. Here, we experimentally create the ground state and anyonic excitations of a nine-qubit planar code using eight photons. Stabilizer measurements of the states of the planar code are performed to directly reveal the locations of the anyons in the system. We further perform braiding operations on the anyons, which gives rise to a topologically path-independent phase. Our work provides a platform for simulating the braiding operations with linear optics, opening up the possibility of further exploring the features of anyonic statistics.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

C. Song, D. Xu, P. Zhang, J. Wang, Q. Guo, W. Liu, K. Xu, H. Deng, K. Huang, D. Zheng, and S. B. Zheng, “Demonstration of topological robustness of anyonic braiding statistics with a superconducting quantum circuit,” Phys. Rev. Lett. 121, 030502 (2018).
[Crossref]

2017 (4)

J. R. Wootton, “Demonstrating non-abelian braiding of surface code defects in a five qubit experiment,” Quantum Sci. Technol. 2, 015006 (2017).
[Crossref]

Y.-M. He, J. Liu, S. Maier, M. Emmerling, S. Gerhardt, M. Davanco, K. Srinivasan, C. Schneider, and S. Höfling, “Deterministic implementation of a bright, on-demand single-photon source with near-unity indistinguishability via quantum dot imaging,” Optica 4, 802–808(2017).
[Crossref]

N. M. Linke, M. Gutierrez, K. A. Landsman, C. Figgatt, S. Debnath, K. R. Brown, and C. Monroe, “Fault-tolerant quantum error detection,” Sci. Adv. 3, e1701074 (2017).
[Crossref]

H.-N. Dai, B. Yang, A. Reingruber, H. Sun, X.-F. Xu, Y.-A. Chen, Z.-S. Yuan, and J.-W. Pan, “Four-body ring-exchange interactions and anyonic statistics within a minimal toric-code Hamiltonian,” Nat. Phys. 13, 1195–1200 (2017).
[Crossref]

2016 (4)

A. J. Park, E. McKay, D. Lu, and R. Laflamme, “Simulation of anyonic statistics and its topological path independence using a seven-qubit quantum simulator,” New J. Phys. 18, 043043 (2016).
[Crossref]

M. Takita, A. D. Córcoles, E. Magesan, B. Abdo, M. Brink, A. Cross, J. M. Chow, and J. M. Gambetta, “Demonstration of weight-four parity measurements in the surface code architecture,” Phys. Rev. Lett. 117, 210505 (2016).
[Crossref]

Y. Zhong, D. Xu, P. Wang, C. Song, Q. Guo, W. Liu, K. Xu, B. Xia, C.-Y. Lu, S. Han, and J. W. Pan, “Emulating anyonic fractional statistical behavior in a superconducting quantum circuit,” Phys. Rev. Lett. 117, 110501 (2016).
[Crossref]

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, and H. Lu, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref]

2015 (1)

2014 (1)

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, and Y. Chen, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature 508, 500–503 (2014).
[Crossref]

2012 (1)

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324 (2012).
[Crossref]

2009 (2)

C.-Y. Lu, W.-B. Gao, O. Gühne, X.-Q. Zhou, Z.-B. Chen, and J.-W. Pan, “Demonstrating anyonic fractional statistics with a six-qubit quantum simulator,” Phys. Rev. Lett. 102, 030502 (2009).
[Crossref]

J. Pachos, W. Wieczorek, C. Schmid, N. Kiesel, R. Pohlner, and H. Weinfurter, “Revealing anyonic features in a toric code quantum simulation,” New J. Phys. 11, 083010 (2009).
[Crossref]

2008 (1)

C. Nayak, “Non-abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
[Crossref]

2007 (4)

R. Raussendorf and J. Harrington, “Fault-tolerant quantum computation with high threshold in two dimensions,” Phys. Rev. Lett. 98, 190504 (2007).
[Crossref]

R. Raussendorf, J. Harrington, and K. Goyal, “Topological fault-tolerance in cluster state quantum computation,” New J. Phys. 9, 199 (2007).
[Crossref]

Y.-J. Han, R. Raussendorf, and L.-M. Duan, “Scheme for demonstration of fractional statistics of anyons in an exactly solvable model,” Phys. Rev. Lett. 98, 150404 (2007).
[Crossref]

C.-Y. Lu, X.-Q. Zhou, O. Gühne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91–95 (2007).
[Crossref]

2005 (2)

E.-A. Kim, M. Lawler, S. Vishveshwara, and E. Fradkin, “Signatures of fractional statistics in noise experiments in quantum Hall fluids,” Phys. Rev. Lett. 95, 176402 (2005).
[Crossref]

S. D. Sarma, M. Freedman, and C. Nayak, “Topologically protected qubits from a possible non-abelian fractional quantum Hall state,” Phys. Rev. Lett. 94, 166802 (2005).
[Crossref]

2004 (1)

M. Hein, J. Eisert, and H. J. Briegel, “Multiparty entanglement in graph states,” Phys. Rev. A 69, 062311 (2004).
[Crossref]

2003 (2)

R. Raussendorf, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68, 022312 (2003).
[Crossref]

A. Y. Kitaev, “Fault-tolerant quantum computation by anyons,” Ann. Phys. 303, 2–30 (2003).
[Crossref]

2001 (1)

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188–5191 (2001).
[Crossref]

1982 (2)

F. Wilczek, “Magnetic flux, angular momentum, and statistics,” Phys. Rev. Lett. 48, 1144–1146 (1982).
[Crossref]

F. Wilczek, “Quantum mechanics of fractional-spin particles,” Phys. Rev. Lett. 49, 957–959 (1982).
[Crossref]

Abdo, B.

M. Takita, A. D. Córcoles, E. Magesan, B. Abdo, M. Brink, A. Cross, J. M. Chow, and J. M. Gambetta, “Demonstration of weight-four parity measurements in the surface code architecture,” Phys. Rev. Lett. 117, 210505 (2016).
[Crossref]

Anderson, J. T.

A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault-tolerant quantum computing with color codes,” arXiv:1108.5738 (2011).

Barends, R.

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, and Y. Chen, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature 508, 500–503 (2014).
[Crossref]

Briegel, H. J.

M. Hein, J. Eisert, and H. J. Briegel, “Multiparty entanglement in graph states,” Phys. Rev. A 69, 062311 (2004).
[Crossref]

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188–5191 (2001).
[Crossref]

Brink, M.

M. Takita, A. D. Córcoles, E. Magesan, B. Abdo, M. Brink, A. Cross, J. M. Chow, and J. M. Gambetta, “Demonstration of weight-four parity measurements in the surface code architecture,” Phys. Rev. Lett. 117, 210505 (2016).
[Crossref]

Brown, K. R.

N. M. Linke, M. Gutierrez, K. A. Landsman, C. Figgatt, S. Debnath, K. R. Brown, and C. Monroe, “Fault-tolerant quantum error detection,” Sci. Adv. 3, e1701074 (2017).
[Crossref]

Campbell, B.

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, and Y. Chen, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature 508, 500–503 (2014).
[Crossref]

Chen, C.

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, and H. Lu, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref]

Chen, L.-K.

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, and H. Lu, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref]

Chen, Y.

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, and Y. Chen, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature 508, 500–503 (2014).
[Crossref]

Chen, Y.-A.

H.-N. Dai, B. Yang, A. Reingruber, H. Sun, X.-F. Xu, Y.-A. Chen, Z.-S. Yuan, and J.-W. Pan, “Four-body ring-exchange interactions and anyonic statistics within a minimal toric-code Hamiltonian,” Nat. Phys. 13, 1195–1200 (2017).
[Crossref]

Chen, Z.-B.

C.-Y. Lu, W.-B. Gao, O. Gühne, X.-Q. Zhou, Z.-B. Chen, and J.-W. Pan, “Demonstrating anyonic fractional statistics with a six-qubit quantum simulator,” Phys. Rev. Lett. 102, 030502 (2009).
[Crossref]

Chow, J. M.

M. Takita, A. D. Córcoles, E. Magesan, B. Abdo, M. Brink, A. Cross, J. M. Chow, and J. M. Gambetta, “Demonstration of weight-four parity measurements in the surface code architecture,” Phys. Rev. Lett. 117, 210505 (2016).
[Crossref]

Christensen, B. G.

Cleland, A. N.

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324 (2012).
[Crossref]

Córcoles, A. D.

M. Takita, A. D. Córcoles, E. Magesan, B. Abdo, M. Brink, A. Cross, J. M. Chow, and J. M. Gambetta, “Demonstration of weight-four parity measurements in the surface code architecture,” Phys. Rev. Lett. 117, 210505 (2016).
[Crossref]

Cross, A.

M. Takita, A. D. Córcoles, E. Magesan, B. Abdo, M. Brink, A. Cross, J. M. Chow, and J. M. Gambetta, “Demonstration of weight-four parity measurements in the surface code architecture,” Phys. Rev. Lett. 117, 210505 (2016).
[Crossref]

Dai, H.-N.

H.-N. Dai, B. Yang, A. Reingruber, H. Sun, X.-F. Xu, Y.-A. Chen, Z.-S. Yuan, and J.-W. Pan, “Four-body ring-exchange interactions and anyonic statistics within a minimal toric-code Hamiltonian,” Nat. Phys. 13, 1195–1200 (2017).
[Crossref]

Davanco, M.

Debnath, S.

N. M. Linke, M. Gutierrez, K. A. Landsman, C. Figgatt, S. Debnath, K. R. Brown, and C. Monroe, “Fault-tolerant quantum error detection,” Sci. Adv. 3, e1701074 (2017).
[Crossref]

Deng, H.

C. Song, D. Xu, P. Zhang, J. Wang, Q. Guo, W. Liu, K. Xu, H. Deng, K. Huang, D. Zheng, and S. B. Zheng, “Demonstration of topological robustness of anyonic braiding statistics with a superconducting quantum circuit,” Phys. Rev. Lett. 121, 030502 (2018).
[Crossref]

Duan, L.-M.

Y.-J. Han, R. Raussendorf, and L.-M. Duan, “Scheme for demonstration of fractional statistics of anyons in an exactly solvable model,” Phys. Rev. Lett. 98, 150404 (2007).
[Crossref]

Eisert, J.

M. Hein, J. Eisert, and H. J. Briegel, “Multiparty entanglement in graph states,” Phys. Rev. A 69, 062311 (2004).
[Crossref]

Emmerling, M.

Figgatt, C.

N. M. Linke, M. Gutierrez, K. A. Landsman, C. Figgatt, S. Debnath, K. R. Brown, and C. Monroe, “Fault-tolerant quantum error detection,” Sci. Adv. 3, e1701074 (2017).
[Crossref]

Fowler, A. G.

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, and Y. Chen, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature 508, 500–503 (2014).
[Crossref]

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324 (2012).
[Crossref]

Fradkin, E.

E.-A. Kim, M. Lawler, S. Vishveshwara, and E. Fradkin, “Signatures of fractional statistics in noise experiments in quantum Hall fluids,” Phys. Rev. Lett. 95, 176402 (2005).
[Crossref]

Freedman, M.

S. D. Sarma, M. Freedman, and C. Nayak, “Topologically protected qubits from a possible non-abelian fractional quantum Hall state,” Phys. Rev. Lett. 94, 166802 (2005).
[Crossref]

Gambetta, J. M.

M. Takita, A. D. Córcoles, E. Magesan, B. Abdo, M. Brink, A. Cross, J. M. Chow, and J. M. Gambetta, “Demonstration of weight-four parity measurements in the surface code architecture,” Phys. Rev. Lett. 117, 210505 (2016).
[Crossref]

Gao, W.-B.

C.-Y. Lu, W.-B. Gao, O. Gühne, X.-Q. Zhou, Z.-B. Chen, and J.-W. Pan, “Demonstrating anyonic fractional statistics with a six-qubit quantum simulator,” Phys. Rev. Lett. 102, 030502 (2009).
[Crossref]

C.-Y. Lu, X.-Q. Zhou, O. Gühne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91–95 (2007).
[Crossref]

Gerhardt, S.

Goebel, A.

C.-Y. Lu, X.-Q. Zhou, O. Gühne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91–95 (2007).
[Crossref]

Goyal, K.

R. Raussendorf, J. Harrington, and K. Goyal, “Topological fault-tolerance in cluster state quantum computation,” New J. Phys. 9, 199 (2007).
[Crossref]

Gühne, O.

C.-Y. Lu, W.-B. Gao, O. Gühne, X.-Q. Zhou, Z.-B. Chen, and J.-W. Pan, “Demonstrating anyonic fractional statistics with a six-qubit quantum simulator,” Phys. Rev. Lett. 102, 030502 (2009).
[Crossref]

C.-Y. Lu, X.-Q. Zhou, O. Gühne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91–95 (2007).
[Crossref]

Guo, Q.

C. Song, D. Xu, P. Zhang, J. Wang, Q. Guo, W. Liu, K. Xu, H. Deng, K. Huang, D. Zheng, and S. B. Zheng, “Demonstration of topological robustness of anyonic braiding statistics with a superconducting quantum circuit,” Phys. Rev. Lett. 121, 030502 (2018).
[Crossref]

Y. Zhong, D. Xu, P. Wang, C. Song, Q. Guo, W. Liu, K. Xu, B. Xia, C.-Y. Lu, S. Han, and J. W. Pan, “Emulating anyonic fractional statistical behavior in a superconducting quantum circuit,” Phys. Rev. Lett. 117, 110501 (2016).
[Crossref]

Gutierrez, M.

N. M. Linke, M. Gutierrez, K. A. Landsman, C. Figgatt, S. Debnath, K. R. Brown, and C. Monroe, “Fault-tolerant quantum error detection,” Sci. Adv. 3, e1701074 (2017).
[Crossref]

Han, S.

Y. Zhong, D. Xu, P. Wang, C. Song, Q. Guo, W. Liu, K. Xu, B. Xia, C.-Y. Lu, S. Han, and J. W. Pan, “Emulating anyonic fractional statistical behavior in a superconducting quantum circuit,” Phys. Rev. Lett. 117, 110501 (2016).
[Crossref]

Han, Y.-J.

Y.-J. Han, R. Raussendorf, and L.-M. Duan, “Scheme for demonstration of fractional statistics of anyons in an exactly solvable model,” Phys. Rev. Lett. 98, 150404 (2007).
[Crossref]

Harrington, J.

R. Raussendorf and J. Harrington, “Fault-tolerant quantum computation with high threshold in two dimensions,” Phys. Rev. Lett. 98, 190504 (2007).
[Crossref]

R. Raussendorf, J. Harrington, and K. Goyal, “Topological fault-tolerance in cluster state quantum computation,” New J. Phys. 9, 199 (2007).
[Crossref]

He, Y.-M.

Hein, M.

M. Hein, J. Eisert, and H. J. Briegel, “Multiparty entanglement in graph states,” Phys. Rev. A 69, 062311 (2004).
[Crossref]

Höfling, S.

Huang, H.-L.

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, and H. Lu, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref]

Huang, K.

C. Song, D. Xu, P. Zhang, J. Wang, Q. Guo, W. Liu, K. Xu, H. Deng, K. Huang, D. Zheng, and S. B. Zheng, “Demonstration of topological robustness of anyonic braiding statistics with a superconducting quantum circuit,” Phys. Rev. Lett. 121, 030502 (2018).
[Crossref]

Jeffrey, E.

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, and Y. Chen, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature 508, 500–503 (2014).
[Crossref]

Kaneda, F.

Kelly, J.

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, and Y. Chen, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature 508, 500–503 (2014).
[Crossref]

Kiesel, N.

J. Pachos, W. Wieczorek, C. Schmid, N. Kiesel, R. Pohlner, and H. Weinfurter, “Revealing anyonic features in a toric code quantum simulation,” New J. Phys. 11, 083010 (2009).
[Crossref]

Kim, E.-A.

E.-A. Kim, M. Lawler, S. Vishveshwara, and E. Fradkin, “Signatures of fractional statistics in noise experiments in quantum Hall fluids,” Phys. Rev. Lett. 95, 176402 (2005).
[Crossref]

Kitaev, A. Y.

A. Y. Kitaev, “Fault-tolerant quantum computation by anyons,” Ann. Phys. 303, 2–30 (2003).
[Crossref]

Kwiat, P. G.

Laflamme, R.

A. J. Park, E. McKay, D. Lu, and R. Laflamme, “Simulation of anyonic statistics and its topological path independence using a seven-qubit quantum simulator,” New J. Phys. 18, 043043 (2016).
[Crossref]

Landahl, A. J.

A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault-tolerant quantum computing with color codes,” arXiv:1108.5738 (2011).

Landsman, K. A.

N. M. Linke, M. Gutierrez, K. A. Landsman, C. Figgatt, S. Debnath, K. R. Brown, and C. Monroe, “Fault-tolerant quantum error detection,” Sci. Adv. 3, e1701074 (2017).
[Crossref]

Lawler, M.

E.-A. Kim, M. Lawler, S. Vishveshwara, and E. Fradkin, “Signatures of fractional statistics in noise experiments in quantum Hall fluids,” Phys. Rev. Lett. 95, 176402 (2005).
[Crossref]

Li, W.

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, and H. Lu, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref]

Li, Z.-D.

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, and H. Lu, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref]

Linke, N. M.

N. M. Linke, M. Gutierrez, K. A. Landsman, C. Figgatt, S. Debnath, K. R. Brown, and C. Monroe, “Fault-tolerant quantum error detection,” Sci. Adv. 3, e1701074 (2017).
[Crossref]

Liu, C.

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, and H. Lu, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref]

Liu, J.

Liu, W.

C. Song, D. Xu, P. Zhang, J. Wang, Q. Guo, W. Liu, K. Xu, H. Deng, K. Huang, D. Zheng, and S. B. Zheng, “Demonstration of topological robustness of anyonic braiding statistics with a superconducting quantum circuit,” Phys. Rev. Lett. 121, 030502 (2018).
[Crossref]

Y. Zhong, D. Xu, P. Wang, C. Song, Q. Guo, W. Liu, K. Xu, B. Xia, C.-Y. Lu, S. Han, and J. W. Pan, “Emulating anyonic fractional statistical behavior in a superconducting quantum circuit,” Phys. Rev. Lett. 117, 110501 (2016).
[Crossref]

Lu, C.-Y.

Y. Zhong, D. Xu, P. Wang, C. Song, Q. Guo, W. Liu, K. Xu, B. Xia, C.-Y. Lu, S. Han, and J. W. Pan, “Emulating anyonic fractional statistical behavior in a superconducting quantum circuit,” Phys. Rev. Lett. 117, 110501 (2016).
[Crossref]

C.-Y. Lu, W.-B. Gao, O. Gühne, X.-Q. Zhou, Z.-B. Chen, and J.-W. Pan, “Demonstrating anyonic fractional statistics with a six-qubit quantum simulator,” Phys. Rev. Lett. 102, 030502 (2009).
[Crossref]

C.-Y. Lu, X.-Q. Zhou, O. Gühne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91–95 (2007).
[Crossref]

Lu, D.

A. J. Park, E. McKay, D. Lu, and R. Laflamme, “Simulation of anyonic statistics and its topological path independence using a seven-qubit quantum simulator,” New J. Phys. 18, 043043 (2016).
[Crossref]

Lu, H.

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, and H. Lu, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref]

Luo, Y.-H.

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, and H. Lu, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref]

Magesan, E.

M. Takita, A. D. Córcoles, E. Magesan, B. Abdo, M. Brink, A. Cross, J. M. Chow, and J. M. Gambetta, “Demonstration of weight-four parity measurements in the surface code architecture,” Phys. Rev. Lett. 117, 210505 (2016).
[Crossref]

Maier, S.

Mariantoni, M.

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324 (2012).
[Crossref]

Martinis, J. M.

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324 (2012).
[Crossref]

McCusker, K. T.

McKay, E.

A. J. Park, E. McKay, D. Lu, and R. Laflamme, “Simulation of anyonic statistics and its topological path independence using a seven-qubit quantum simulator,” New J. Phys. 18, 043043 (2016).
[Crossref]

Megrant, A.

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, and Y. Chen, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature 508, 500–503 (2014).
[Crossref]

Monroe, C.

N. M. Linke, M. Gutierrez, K. A. Landsman, C. Figgatt, S. Debnath, K. R. Brown, and C. Monroe, “Fault-tolerant quantum error detection,” Sci. Adv. 3, e1701074 (2017).
[Crossref]

Mutus, J.

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, and Y. Chen, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature 508, 500–503 (2014).
[Crossref]

Nayak, C.

C. Nayak, “Non-abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
[Crossref]

S. D. Sarma, M. Freedman, and C. Nayak, “Topologically protected qubits from a possible non-abelian fractional quantum Hall state,” Phys. Rev. Lett. 94, 166802 (2005).
[Crossref]

Pachos, J.

J. Pachos, W. Wieczorek, C. Schmid, N. Kiesel, R. Pohlner, and H. Weinfurter, “Revealing anyonic features in a toric code quantum simulation,” New J. Phys. 11, 083010 (2009).
[Crossref]

Pan, J. W.

Y. Zhong, D. Xu, P. Wang, C. Song, Q. Guo, W. Liu, K. Xu, B. Xia, C.-Y. Lu, S. Han, and J. W. Pan, “Emulating anyonic fractional statistical behavior in a superconducting quantum circuit,” Phys. Rev. Lett. 117, 110501 (2016).
[Crossref]

Pan, J.-W.

H.-N. Dai, B. Yang, A. Reingruber, H. Sun, X.-F. Xu, Y.-A. Chen, Z.-S. Yuan, and J.-W. Pan, “Four-body ring-exchange interactions and anyonic statistics within a minimal toric-code Hamiltonian,” Nat. Phys. 13, 1195–1200 (2017).
[Crossref]

C.-Y. Lu, W.-B. Gao, O. Gühne, X.-Q. Zhou, Z.-B. Chen, and J.-W. Pan, “Demonstrating anyonic fractional statistics with a six-qubit quantum simulator,” Phys. Rev. Lett. 102, 030502 (2009).
[Crossref]

C.-Y. Lu, X.-Q. Zhou, O. Gühne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91–95 (2007).
[Crossref]

Park, A. J.

A. J. Park, E. McKay, D. Lu, and R. Laflamme, “Simulation of anyonic statistics and its topological path independence using a seven-qubit quantum simulator,” New J. Phys. 18, 043043 (2016).
[Crossref]

Park, H. S.

Pohlner, R.

J. Pachos, W. Wieczorek, C. Schmid, N. Kiesel, R. Pohlner, and H. Weinfurter, “Revealing anyonic features in a toric code quantum simulation,” New J. Phys. 11, 083010 (2009).
[Crossref]

Raussendorf, R.

Y.-J. Han, R. Raussendorf, and L.-M. Duan, “Scheme for demonstration of fractional statistics of anyons in an exactly solvable model,” Phys. Rev. Lett. 98, 150404 (2007).
[Crossref]

R. Raussendorf and J. Harrington, “Fault-tolerant quantum computation with high threshold in two dimensions,” Phys. Rev. Lett. 98, 190504 (2007).
[Crossref]

R. Raussendorf, J. Harrington, and K. Goyal, “Topological fault-tolerance in cluster state quantum computation,” New J. Phys. 9, 199 (2007).
[Crossref]

R. Raussendorf, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68, 022312 (2003).
[Crossref]

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188–5191 (2001).
[Crossref]

Reingruber, A.

H.-N. Dai, B. Yang, A. Reingruber, H. Sun, X.-F. Xu, Y.-A. Chen, Z.-S. Yuan, and J.-W. Pan, “Four-body ring-exchange interactions and anyonic statistics within a minimal toric-code Hamiltonian,” Nat. Phys. 13, 1195–1200 (2017).
[Crossref]

Rice, P. R.

A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault-tolerant quantum computing with color codes,” arXiv:1108.5738 (2011).

Sank, D.

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, and Y. Chen, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature 508, 500–503 (2014).
[Crossref]

Sarma, S. D.

S. D. Sarma, M. Freedman, and C. Nayak, “Topologically protected qubits from a possible non-abelian fractional quantum Hall state,” Phys. Rev. Lett. 94, 166802 (2005).
[Crossref]

Schmid, C.

J. Pachos, W. Wieczorek, C. Schmid, N. Kiesel, R. Pohlner, and H. Weinfurter, “Revealing anyonic features in a toric code quantum simulation,” New J. Phys. 11, 083010 (2009).
[Crossref]

Schneider, C.

Song, C.

C. Song, D. Xu, P. Zhang, J. Wang, Q. Guo, W. Liu, K. Xu, H. Deng, K. Huang, D. Zheng, and S. B. Zheng, “Demonstration of topological robustness of anyonic braiding statistics with a superconducting quantum circuit,” Phys. Rev. Lett. 121, 030502 (2018).
[Crossref]

Y. Zhong, D. Xu, P. Wang, C. Song, Q. Guo, W. Liu, K. Xu, B. Xia, C.-Y. Lu, S. Han, and J. W. Pan, “Emulating anyonic fractional statistical behavior in a superconducting quantum circuit,” Phys. Rev. Lett. 117, 110501 (2016).
[Crossref]

Srinivasan, K.

Su, Z.-E.

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, and H. Lu, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref]

Sun, H.

H.-N. Dai, B. Yang, A. Reingruber, H. Sun, X.-F. Xu, Y.-A. Chen, Z.-S. Yuan, and J.-W. Pan, “Four-body ring-exchange interactions and anyonic statistics within a minimal toric-code Hamiltonian,” Nat. Phys. 13, 1195–1200 (2017).
[Crossref]

Takita, M.

M. Takita, A. D. Córcoles, E. Magesan, B. Abdo, M. Brink, A. Cross, J. M. Chow, and J. M. Gambetta, “Demonstration of weight-four parity measurements in the surface code architecture,” Phys. Rev. Lett. 117, 210505 (2016).
[Crossref]

Veitia, A.

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, and Y. Chen, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature 508, 500–503 (2014).
[Crossref]

Vishveshwara, S.

E.-A. Kim, M. Lawler, S. Vishveshwara, and E. Fradkin, “Signatures of fractional statistics in noise experiments in quantum Hall fluids,” Phys. Rev. Lett. 95, 176402 (2005).
[Crossref]

Vuillot, C.

C. Vuillot, “Is error detection helpful on IBM 5q chips?” arXiv:1705.08957 (2017).

Wang, J.

C. Song, D. Xu, P. Zhang, J. Wang, Q. Guo, W. Liu, K. Xu, H. Deng, K. Huang, D. Zheng, and S. B. Zheng, “Demonstration of topological robustness of anyonic braiding statistics with a superconducting quantum circuit,” Phys. Rev. Lett. 121, 030502 (2018).
[Crossref]

Wang, P.

Y. Zhong, D. Xu, P. Wang, C. Song, Q. Guo, W. Liu, K. Xu, B. Xia, C.-Y. Lu, S. Han, and J. W. Pan, “Emulating anyonic fractional statistical behavior in a superconducting quantum circuit,” Phys. Rev. Lett. 117, 110501 (2016).
[Crossref]

Wang, X.-L.

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, and H. Lu, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref]

Weinfurter, H.

J. Pachos, W. Wieczorek, C. Schmid, N. Kiesel, R. Pohlner, and H. Weinfurter, “Revealing anyonic features in a toric code quantum simulation,” New J. Phys. 11, 083010 (2009).
[Crossref]

White, T. C.

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, and Y. Chen, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature 508, 500–503 (2014).
[Crossref]

Wieczorek, W.

J. Pachos, W. Wieczorek, C. Schmid, N. Kiesel, R. Pohlner, and H. Weinfurter, “Revealing anyonic features in a toric code quantum simulation,” New J. Phys. 11, 083010 (2009).
[Crossref]

Wilczek, F.

F. Wilczek, “Quantum mechanics of fractional-spin particles,” Phys. Rev. Lett. 49, 957–959 (1982).
[Crossref]

F. Wilczek, “Magnetic flux, angular momentum, and statistics,” Phys. Rev. Lett. 48, 1144–1146 (1982).
[Crossref]

Wong, J. J.

Wootton, J. R.

J. R. Wootton, “Demonstrating non-abelian braiding of surface code defects in a five qubit experiment,” Quantum Sci. Technol. 2, 015006 (2017).
[Crossref]

Wu, D.

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, and H. Lu, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref]

Xia, B.

Y. Zhong, D. Xu, P. Wang, C. Song, Q. Guo, W. Liu, K. Xu, B. Xia, C.-Y. Lu, S. Han, and J. W. Pan, “Emulating anyonic fractional statistical behavior in a superconducting quantum circuit,” Phys. Rev. Lett. 117, 110501 (2016).
[Crossref]

Xu, D.

C. Song, D. Xu, P. Zhang, J. Wang, Q. Guo, W. Liu, K. Xu, H. Deng, K. Huang, D. Zheng, and S. B. Zheng, “Demonstration of topological robustness of anyonic braiding statistics with a superconducting quantum circuit,” Phys. Rev. Lett. 121, 030502 (2018).
[Crossref]

Y. Zhong, D. Xu, P. Wang, C. Song, Q. Guo, W. Liu, K. Xu, B. Xia, C.-Y. Lu, S. Han, and J. W. Pan, “Emulating anyonic fractional statistical behavior in a superconducting quantum circuit,” Phys. Rev. Lett. 117, 110501 (2016).
[Crossref]

Xu, K.

C. Song, D. Xu, P. Zhang, J. Wang, Q. Guo, W. Liu, K. Xu, H. Deng, K. Huang, D. Zheng, and S. B. Zheng, “Demonstration of topological robustness of anyonic braiding statistics with a superconducting quantum circuit,” Phys. Rev. Lett. 121, 030502 (2018).
[Crossref]

Y. Zhong, D. Xu, P. Wang, C. Song, Q. Guo, W. Liu, K. Xu, B. Xia, C.-Y. Lu, S. Han, and J. W. Pan, “Emulating anyonic fractional statistical behavior in a superconducting quantum circuit,” Phys. Rev. Lett. 117, 110501 (2016).
[Crossref]

Xu, X.-F.

H.-N. Dai, B. Yang, A. Reingruber, H. Sun, X.-F. Xu, Y.-A. Chen, Z.-S. Yuan, and J.-W. Pan, “Four-body ring-exchange interactions and anyonic statistics within a minimal toric-code Hamiltonian,” Nat. Phys. 13, 1195–1200 (2017).
[Crossref]

Yang, B.

H.-N. Dai, B. Yang, A. Reingruber, H. Sun, X.-F. Xu, Y.-A. Chen, Z.-S. Yuan, and J.-W. Pan, “Four-body ring-exchange interactions and anyonic statistics within a minimal toric-code Hamiltonian,” Nat. Phys. 13, 1195–1200 (2017).
[Crossref]

Yang, T.

C.-Y. Lu, X.-Q. Zhou, O. Gühne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91–95 (2007).
[Crossref]

Yuan, Z.-S.

H.-N. Dai, B. Yang, A. Reingruber, H. Sun, X.-F. Xu, Y.-A. Chen, Z.-S. Yuan, and J.-W. Pan, “Four-body ring-exchange interactions and anyonic statistics within a minimal toric-code Hamiltonian,” Nat. Phys. 13, 1195–1200 (2017).
[Crossref]

C.-Y. Lu, X.-Q. Zhou, O. Gühne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91–95 (2007).
[Crossref]

Zhang, J.

C.-Y. Lu, X.-Q. Zhou, O. Gühne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91–95 (2007).
[Crossref]

Zhang, P.

C. Song, D. Xu, P. Zhang, J. Wang, Q. Guo, W. Liu, K. Xu, H. Deng, K. Huang, D. Zheng, and S. B. Zheng, “Demonstration of topological robustness of anyonic braiding statistics with a superconducting quantum circuit,” Phys. Rev. Lett. 121, 030502 (2018).
[Crossref]

Zheng, D.

C. Song, D. Xu, P. Zhang, J. Wang, Q. Guo, W. Liu, K. Xu, H. Deng, K. Huang, D. Zheng, and S. B. Zheng, “Demonstration of topological robustness of anyonic braiding statistics with a superconducting quantum circuit,” Phys. Rev. Lett. 121, 030502 (2018).
[Crossref]

Zheng, S. B.

C. Song, D. Xu, P. Zhang, J. Wang, Q. Guo, W. Liu, K. Xu, H. Deng, K. Huang, D. Zheng, and S. B. Zheng, “Demonstration of topological robustness of anyonic braiding statistics with a superconducting quantum circuit,” Phys. Rev. Lett. 121, 030502 (2018).
[Crossref]

Zhong, Y.

Y. Zhong, D. Xu, P. Wang, C. Song, Q. Guo, W. Liu, K. Xu, B. Xia, C.-Y. Lu, S. Han, and J. W. Pan, “Emulating anyonic fractional statistical behavior in a superconducting quantum circuit,” Phys. Rev. Lett. 117, 110501 (2016).
[Crossref]

Zhou, X.-Q.

C.-Y. Lu, W.-B. Gao, O. Gühne, X.-Q. Zhou, Z.-B. Chen, and J.-W. Pan, “Demonstrating anyonic fractional statistics with a six-qubit quantum simulator,” Phys. Rev. Lett. 102, 030502 (2009).
[Crossref]

C.-Y. Lu, X.-Q. Zhou, O. Gühne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91–95 (2007).
[Crossref]

Ann. Phys. (1)

A. Y. Kitaev, “Fault-tolerant quantum computation by anyons,” Ann. Phys. 303, 2–30 (2003).
[Crossref]

Nat. Phys. (2)

H.-N. Dai, B. Yang, A. Reingruber, H. Sun, X.-F. Xu, Y.-A. Chen, Z.-S. Yuan, and J.-W. Pan, “Four-body ring-exchange interactions and anyonic statistics within a minimal toric-code Hamiltonian,” Nat. Phys. 13, 1195–1200 (2017).
[Crossref]

C.-Y. Lu, X.-Q. Zhou, O. Gühne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nat. Phys. 3, 91–95 (2007).
[Crossref]

Nature (1)

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, and Y. Chen, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature 508, 500–503 (2014).
[Crossref]

New J. Phys. (3)

A. J. Park, E. McKay, D. Lu, and R. Laflamme, “Simulation of anyonic statistics and its topological path independence using a seven-qubit quantum simulator,” New J. Phys. 18, 043043 (2016).
[Crossref]

R. Raussendorf, J. Harrington, and K. Goyal, “Topological fault-tolerance in cluster state quantum computation,” New J. Phys. 9, 199 (2007).
[Crossref]

J. Pachos, W. Wieczorek, C. Schmid, N. Kiesel, R. Pohlner, and H. Weinfurter, “Revealing anyonic features in a toric code quantum simulation,” New J. Phys. 11, 083010 (2009).
[Crossref]

Optica (2)

Phys. Rev. A (3)

M. Hein, J. Eisert, and H. J. Briegel, “Multiparty entanglement in graph states,” Phys. Rev. A 69, 062311 (2004).
[Crossref]

R. Raussendorf, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68, 022312 (2003).
[Crossref]

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324 (2012).
[Crossref]

Phys. Rev. Lett. (12)

F. Wilczek, “Magnetic flux, angular momentum, and statistics,” Phys. Rev. Lett. 48, 1144–1146 (1982).
[Crossref]

S. D. Sarma, M. Freedman, and C. Nayak, “Topologically protected qubits from a possible non-abelian fractional quantum Hall state,” Phys. Rev. Lett. 94, 166802 (2005).
[Crossref]

R. Raussendorf and J. Harrington, “Fault-tolerant quantum computation with high threshold in two dimensions,” Phys. Rev. Lett. 98, 190504 (2007).
[Crossref]

C. Song, D. Xu, P. Zhang, J. Wang, Q. Guo, W. Liu, K. Xu, H. Deng, K. Huang, D. Zheng, and S. B. Zheng, “Demonstration of topological robustness of anyonic braiding statistics with a superconducting quantum circuit,” Phys. Rev. Lett. 121, 030502 (2018).
[Crossref]

M. Takita, A. D. Córcoles, E. Magesan, B. Abdo, M. Brink, A. Cross, J. M. Chow, and J. M. Gambetta, “Demonstration of weight-four parity measurements in the surface code architecture,” Phys. Rev. Lett. 117, 210505 (2016).
[Crossref]

F. Wilczek, “Quantum mechanics of fractional-spin particles,” Phys. Rev. Lett. 49, 957–959 (1982).
[Crossref]

E.-A. Kim, M. Lawler, S. Vishveshwara, and E. Fradkin, “Signatures of fractional statistics in noise experiments in quantum Hall fluids,” Phys. Rev. Lett. 95, 176402 (2005).
[Crossref]

C.-Y. Lu, W.-B. Gao, O. Gühne, X.-Q. Zhou, Z.-B. Chen, and J.-W. Pan, “Demonstrating anyonic fractional statistics with a six-qubit quantum simulator,” Phys. Rev. Lett. 102, 030502 (2009).
[Crossref]

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188–5191 (2001).
[Crossref]

Y.-J. Han, R. Raussendorf, and L.-M. Duan, “Scheme for demonstration of fractional statistics of anyons in an exactly solvable model,” Phys. Rev. Lett. 98, 150404 (2007).
[Crossref]

Y. Zhong, D. Xu, P. Wang, C. Song, Q. Guo, W. Liu, K. Xu, B. Xia, C.-Y. Lu, S. Han, and J. W. Pan, “Emulating anyonic fractional statistical behavior in a superconducting quantum circuit,” Phys. Rev. Lett. 117, 110501 (2016).
[Crossref]

X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, and H. Lu, “Experimental ten-photon entanglement,” Phys. Rev. Lett. 117, 210502 (2016).
[Crossref]

Quantum Sci. Technol. (1)

J. R. Wootton, “Demonstrating non-abelian braiding of surface code defects in a five qubit experiment,” Quantum Sci. Technol. 2, 015006 (2017).
[Crossref]

Rev. Mod. Phys. (1)

C. Nayak, “Non-abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
[Crossref]

Sci. Adv. (1)

N. M. Linke, M. Gutierrez, K. A. Landsman, C. Figgatt, S. Debnath, K. R. Brown, and C. Monroe, “Fault-tolerant quantum error detection,” Sci. Adv. 3, e1701074 (2017).
[Crossref]

Other (2)

A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault-tolerant quantum computing with color codes,” arXiv:1108.5738 (2011).

C. Vuillot, “Is error detection helpful on IBM 5q chips?” arXiv:1705.08957 (2017).

Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. (a) Qubit configuration for the toric code. The Hamiltonian (1) is a sum over all plaquette and star operators that involve four qubits each. An example of a star and plaquette is shown. Anyons are excited from the ground state by applying X and Z operators to the ground state (2). A pair of m anyons is excited by applying X operators to the ground state (2), with a corresponding string connecting them. Similarly, e anyons are excited by applying Z operators with their associated strings. (b) Our experimental configuration. An e ( m ) anyon pair is created by applying a Z ( X ) operator on qubit 3 (4). The m anyons are taken around a loop by applying the operators X 7 X 9 X 8 X 6 (loop 1), X 6 X 5 X 3 X 4 (loop 2), and X 7 X 9 X 8 X 5 X 3 X 4 (loop 3). The labeling conventions for the star A s and plaquette B p operators are shown.
Fig. 2.
Fig. 2. Experimental setup. Ultraviolet laser pulses with a central wavelength of 394 nm, pulse duration of 150 fs, and repetition rate of 80 MHz pass through three HWP-sandwiched BBO crystals and a single BBO to produce three entangled photon pairs | H V + | V H (in spatial modes 1–3, 5–6, and 7–9) and a pair of photons | H V (in spatial modes 4–8), respectively. All of these photons are put into a linear optical network to prepare the ground state | ϕ . The steps of ground state preparation, anyon creation, anyon braiding, anyon annihilation, and measurement are marked. C-BBO, sandwich-like BBO + HWP + BBO combination; S-BBO, single BBO; QWP, quarter-wave plate; HWP, half-wave plate; PBS, polarizing beam splitter; D-PBS, double PBS; D-BS, double beam splitter.
Fig. 3.
Fig. 3. Measured expectation values of the stabilizers A s and B p with respect to the ground state of the planar code (3) and the excited state (4). Error bars represent one standard deviation, deduced from propagated Poissonian counting statistics of the raw detection events. Star and plaquette operators are defined by A 1 = X 1 X 2 X 3 , A 2 = X 3 X 4 X 5 X 6 , A 3 = X 6 X 7 X 8 X 9 , B 1 = Z 1 Z 3 Z 4 , B 2 = Z 2 Z 3 Z 5 , B 3 = Z 4 Z 6 Z 7 , B 4 = Z 5 Z 6 Z 8 , B 5 = Z 7 Z 9 , and B 6 = Z 8 Z 9 .
Fig. 4.
Fig. 4. (a)–(d) Measured fringes for the state | ϕ + e i ϕ | e 1 , e 2 without braiding operation, and the states after loop 1, loop 2, and loop 3 braiding operations. Error bars represent one standard deviation, deduced from propagated Poissonian counting statistics of the raw detection events. (e)–(h) Measured expectation values of the stabilizers A s and B p with respect to the state after further applying a Z 3 operation on the state | ϕ + e i ϕ | e 1 , e 2 . Error bars represent one standard deviation.

Equations (10)

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H = s stars A s p plaquettes B p ,
| ϕ = s stars ( 1 + A s ) | 0 0 .
| ϕ = | 000000000 + | 111000000 + | 110111000 + | 001111000 + | 110110111 + | 001110111 + | 000001111 + | 111001111 .
| e 1 , e 2 , m 1 , m 3 = Z 3 X 4 | ϕ ,
| m 1 , m 3 = X 4 | ϕ .
| m 1 , m 3 + | e 1 , e 2 , m 1 , m 3 .
| m 1 , m 3 + e i ϕ | e 1 , e 2 , m 1 , m 3 ,
| ϕ + e i ϕ | e 1 , e 2 ,
P ( θ ) = | ++ θ 123 ++ θ | 123 ,
P ( θ ) = 1 8 ( 1 + cos ( θ ϕ ) ) .

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