Abstract

Having been introduced in the field of many bodies of statistical mechanics, the Yang–Baxter equation has become an important tool in a variety of fields of physics. In this work, we report the first direct experimental simulation of the Yang–Baxter equation using linear quantum optics. The equality between the two sides of the Yang–Baxter equation in two dimension has been demonstrated directly, and the spectral parameter transformation in the Yang–Baxter equation is explicitly confirmed.

© 2013 Optical Society of America

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    [CrossRef]
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  37. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2012

D. P. Hou and C. M. Bai, “J-dendriform algebras,” Front. Math. Chin. 7, 29–49 (2012).
[CrossRef]

X. C. Yao, T. X. Wang, H. Z. Chen, W. B. Gao, A. G. Fowler, R. Raussendorf, Z. B. Chen, N. L. Liu, C. Y. Lu, Y. J. Deng, Y. A. Chen, and J. W. Pan, “Experimental demonstration of topological error correction,” Nature 482, 489–494 (2012).
[CrossRef]

2010

E. C. Rowell, Y. Zhang, Y. S. Wu, and M. L. Ge, “Extraspecial two-groups, generalized Yang–Baxter equations and braiding quantum gates,” Quantum Inf. Comput. 10, 685–702 (2010).

Y. A. Liao, A. S. C. Rittner, T. Paprotta, W. Li, G. B. Partridge, R. G. Hulet, S. K. Baur, and E. J. Mueller, “Spin-imbalance in a one-dimensional Fermi gas,” Nature 467, 567–569 (2010).
[CrossRef]

2009

J. Dubail, J. L. Jacobsen, and H. Saleur, “Exact solution of the anisotropic special transition in the O(n) model in two dimensions,” Phys. Rev. Lett. 103, 145701 (2009).
[CrossRef]

2008

S. W. Hu, K. Xue, and M. L. Ge, “Optical simulation of the Yang-Baxter equation,” Phys. Rev. A 78, 022319 (2008).
[CrossRef]

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
[CrossRef]

K. Hikami, “Skein theory and topological quantum registers: braiding matrices and topological entanglement entropy of non-Abelian quantum Hall states,” Ann. Phys. 323, 1729–1769 (2008).
[CrossRef]

2007

E. Ardonne and K. Schoutens, “Wavefunctions for topological quantum registers,” Ann. Phys. 322, 201–235 (2007).
[CrossRef]

A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. H. Freedman, “Interacting anyons in topological quantum liquids: the golden chain,” Phys. Rev. Lett. 98, 160409 (2007).
[CrossRef]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Y. Zhang and M. L. Ge, “GHZ states, almost-complex structure and Yang–Baxter equation,” Quantum Inf. Process. 6, 363–379 (2007).
[CrossRef]

J. L. Chen, K. Xue, and M. L. Ge, “Braiding transformation, entanglement swapping, and Berry phase in entanglement space,” Phys. Rev. A 76, 042324 (2007).
[CrossRef]

2006

J. M. Franko, E. C. Rowell, and Z. Wang, “Extraspecial 2-groups and images of braid group representations,” J. Knot Theory Ramif. 15, 413–427 (2006).
[CrossRef]

2005

Y. Zhang, L. H. Kauffman, and M. L. Ge, “Universal quantum gate, Yang–Baxterization and Hamiltonian,” Int. J. Quantum. Inform. 3, 669–678 (2005).
[CrossRef]

2004

L. H. Kauffman and S. J. Lomonaco, “Braiding operators are universal quantum gates,” New J. Phys. 6, 134 (2004).
[CrossRef]

J. Lu, L. Zhou, and L. M. Kuang, “Linear optics implementation for quantum game with two players,” Phys. Lett. A 330, 48–53 (2004).
[CrossRef]

2003

J. L. O’ Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[CrossRef]

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

H. A. Dye, “Unitary solutions to the Yang–Baxter equation in dimension four,” Quant. Info. Proc. 2, 117–152 (2003).
[CrossRef]

2002

M. H. Freedman, A. Y. Kitaev, and Z. H. Wang, “Simulation of topological field theories by quantum computers,” Commun. Math. Phys. 227, 587–603 (2002).
[CrossRef]

2001

S. S. Li, G. L. Long, F. S. Bai, S. L. Feng, and H. Z. Zheng, “Quantum computing,” Proc. Natl. Acad. Sci. USA 98, 11847–11848 (2001).
[CrossRef]

1999

H. X. Chen, “Cpcycle deformations, braided monoidal categories and quasitriangularity,” Chin. Sci. Bull. 44, 510–513 (1999).
[CrossRef]

1997

S. C. Billey, “Kostant polynomials and the cohomology ring for G/B,” Proc. Natl. Acad. Sci. USA 94, 29–32 (1997).

P. Gui, “Another solution of Yang–Baxter equation on set and ‘metahomomorphisms on groups’,” Chin. Sci. Bull. 42, 1852–1855 (1997).
[CrossRef]

1995

D. A. Tennant, R. A. Cowley, S. E. Nagler, and A. M. Tsvelik, “Measurement of the spin-excitation continuum in one-dimensional KCuF3 using neutron scattering,” Phys. Rev. B 52, 13368–13380 (1995).
[CrossRef]

1993

D. A. Tennant, T. G. Perring, R. A. Cowley, and S. E. Nagler, “Unbound spinons in the S=1/2 antiferromagnetic chain KCuF3,” Phys. Rev. Lett. 70, 4003–4006 (1993).
[CrossRef]

1992

C. P. Sun, “The differential realization of new solutions for Yang–Baxter equation,” Chin. Sci. Bull. 37, 379 (1992).

1990

M. Gerstenhaber and S. D. Schack, “Bialgebra cohomology, deformations, and quantum groups,” Proc. Natl. Acad. Sci. USA 87, 478–481 (1990).
[CrossRef]

1988

I. B. Frenkel and N. Jing, “Vertex representations of quantum affine algebras,” Proc. Natl. Acad. Sci. USA 85, 9373–9377 (1988).
[CrossRef]

1972

R. J. Baxter, “Partition function of the Eight-Vertex lattice model,” Ann. Phys. 70, 193–228 (1972).
[CrossRef]

1971

H. N. V. Temperley and E. H. Lieb, “Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem,” Proc. R. Soc. London Ser. A 322, 251–280 (1971).
[CrossRef]

1968

C. N. Yang, “S matrix for the one-dimensional N-body problem with repulsive or attractive δ-function interaction,” Phys. Rev. 168, 1920–1923 (1968).
[CrossRef]

1967

C. N. Yang, “Some exact results for the many-body problem in one dimension with repulsive delta-function interaction,” Phys. Rev. Lett. 19, 1312–1315 (1967).
[CrossRef]

Ardonne, E.

E. Ardonne and K. Schoutens, “Wavefunctions for topological quantum registers,” Ann. Phys. 322, 201–235 (2007).
[CrossRef]

Bai, C. M.

D. P. Hou and C. M. Bai, “J-dendriform algebras,” Front. Math. Chin. 7, 29–49 (2012).
[CrossRef]

Bai, F. S.

S. S. Li, G. L. Long, F. S. Bai, S. L. Feng, and H. Z. Zheng, “Quantum computing,” Proc. Natl. Acad. Sci. USA 98, 11847–11848 (2001).
[CrossRef]

Baur, S. K.

Y. A. Liao, A. S. C. Rittner, T. Paprotta, W. Li, G. B. Partridge, R. G. Hulet, S. K. Baur, and E. J. Mueller, “Spin-imbalance in a one-dimensional Fermi gas,” Nature 467, 567–569 (2010).
[CrossRef]

Baxter, R. J.

R. J. Baxter, “Partition function of the Eight-Vertex lattice model,” Ann. Phys. 70, 193–228 (1972).
[CrossRef]

R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, 1982).

Billey, S. C.

S. C. Billey, “Kostant polynomials and the cohomology ring for G/B,” Proc. Natl. Acad. Sci. USA 94, 29–32 (1997).

Branning, D.

J. L. O’ Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[CrossRef]

Chen, H. X.

H. X. Chen, “Cpcycle deformations, braided monoidal categories and quasitriangularity,” Chin. Sci. Bull. 44, 510–513 (1999).
[CrossRef]

Chen, H. Z.

X. C. Yao, T. X. Wang, H. Z. Chen, W. B. Gao, A. G. Fowler, R. Raussendorf, Z. B. Chen, N. L. Liu, C. Y. Lu, Y. J. Deng, Y. A. Chen, and J. W. Pan, “Experimental demonstration of topological error correction,” Nature 482, 489–494 (2012).
[CrossRef]

Chen, J. L.

J. L. Chen, K. Xue, and M. L. Ge, “Braiding transformation, entanglement swapping, and Berry phase in entanglement space,” Phys. Rev. A 76, 042324 (2007).
[CrossRef]

Chen, Y. A.

X. C. Yao, T. X. Wang, H. Z. Chen, W. B. Gao, A. G. Fowler, R. Raussendorf, Z. B. Chen, N. L. Liu, C. Y. Lu, Y. J. Deng, Y. A. Chen, and J. W. Pan, “Experimental demonstration of topological error correction,” Nature 482, 489–494 (2012).
[CrossRef]

Chen, Z. B.

X. C. Yao, T. X. Wang, H. Z. Chen, W. B. Gao, A. G. Fowler, R. Raussendorf, Z. B. Chen, N. L. Liu, C. Y. Lu, Y. J. Deng, Y. A. Chen, and J. W. Pan, “Experimental demonstration of topological error correction,” Nature 482, 489–494 (2012).
[CrossRef]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000), pp. 1–607.

Cowley, R. A.

D. A. Tennant, R. A. Cowley, S. E. Nagler, and A. M. Tsvelik, “Measurement of the spin-excitation continuum in one-dimensional KCuF3 using neutron scattering,” Phys. Rev. B 52, 13368–13380 (1995).
[CrossRef]

D. A. Tennant, T. G. Perring, R. A. Cowley, and S. E. Nagler, “Unbound spinons in the S=1/2 antiferromagnetic chain KCuF3,” Phys. Rev. Lett. 70, 4003–4006 (1993).
[CrossRef]

Deng, Y. J.

X. C. Yao, T. X. Wang, H. Z. Chen, W. B. Gao, A. G. Fowler, R. Raussendorf, Z. B. Chen, N. L. Liu, C. Y. Lu, Y. J. Deng, Y. A. Chen, and J. W. Pan, “Experimental demonstration of topological error correction,” Nature 482, 489–494 (2012).
[CrossRef]

Dowling, J. P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Dubail, J.

J. Dubail, J. L. Jacobsen, and H. Saleur, “Exact solution of the anisotropic special transition in the O(n) model in two dimensions,” Phys. Rev. Lett. 103, 145701 (2009).
[CrossRef]

Dye, H. A.

H. A. Dye, “Unitary solutions to the Yang–Baxter equation in dimension four,” Quant. Info. Proc. 2, 117–152 (2003).
[CrossRef]

Feiguin, A.

A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. H. Freedman, “Interacting anyons in topological quantum liquids: the golden chain,” Phys. Rev. Lett. 98, 160409 (2007).
[CrossRef]

Feng, S. L.

S. S. Li, G. L. Long, F. S. Bai, S. L. Feng, and H. Z. Zheng, “Quantum computing,” Proc. Natl. Acad. Sci. USA 98, 11847–11848 (2001).
[CrossRef]

Fowler, A. G.

X. C. Yao, T. X. Wang, H. Z. Chen, W. B. Gao, A. G. Fowler, R. Raussendorf, Z. B. Chen, N. L. Liu, C. Y. Lu, Y. J. Deng, Y. A. Chen, and J. W. Pan, “Experimental demonstration of topological error correction,” Nature 482, 489–494 (2012).
[CrossRef]

Franko, J. M.

J. M. Franko, E. C. Rowell, and Z. Wang, “Extraspecial 2-groups and images of braid group representations,” J. Knot Theory Ramif. 15, 413–427 (2006).
[CrossRef]

Freedman, M.

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
[CrossRef]

Freedman, M. H.

A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. H. Freedman, “Interacting anyons in topological quantum liquids: the golden chain,” Phys. Rev. Lett. 98, 160409 (2007).
[CrossRef]

M. H. Freedman, A. Y. Kitaev, and Z. H. Wang, “Simulation of topological field theories by quantum computers,” Commun. Math. Phys. 227, 587–603 (2002).
[CrossRef]

Frenkel, I. B.

I. B. Frenkel and N. Jing, “Vertex representations of quantum affine algebras,” Proc. Natl. Acad. Sci. USA 85, 9373–9377 (1988).
[CrossRef]

Gao, W. B.

X. C. Yao, T. X. Wang, H. Z. Chen, W. B. Gao, A. G. Fowler, R. Raussendorf, Z. B. Chen, N. L. Liu, C. Y. Lu, Y. J. Deng, Y. A. Chen, and J. W. Pan, “Experimental demonstration of topological error correction,” Nature 482, 489–494 (2012).
[CrossRef]

Ge, M. L.

E. C. Rowell, Y. Zhang, Y. S. Wu, and M. L. Ge, “Extraspecial two-groups, generalized Yang–Baxter equations and braiding quantum gates,” Quantum Inf. Comput. 10, 685–702 (2010).

S. W. Hu, K. Xue, and M. L. Ge, “Optical simulation of the Yang-Baxter equation,” Phys. Rev. A 78, 022319 (2008).
[CrossRef]

Y. Zhang and M. L. Ge, “GHZ states, almost-complex structure and Yang–Baxter equation,” Quantum Inf. Process. 6, 363–379 (2007).
[CrossRef]

J. L. Chen, K. Xue, and M. L. Ge, “Braiding transformation, entanglement swapping, and Berry phase in entanglement space,” Phys. Rev. A 76, 042324 (2007).
[CrossRef]

Y. Zhang, L. H. Kauffman, and M. L. Ge, “Universal quantum gate, Yang–Baxterization and Hamiltonian,” Int. J. Quantum. Inform. 3, 669–678 (2005).
[CrossRef]

C. N. Yang and M. L. Ge, Braid Group, Knot Theory and Statistical Mechanics, 2nd ed. (World Scientific, 1994), pp. 1–176.

Gerstenhaber, M.

M. Gerstenhaber and S. D. Schack, “Bialgebra cohomology, deformations, and quantum groups,” Proc. Natl. Acad. Sci. USA 87, 478–481 (1990).
[CrossRef]

Gui, P.

P. Gui, “Another solution of Yang–Baxter equation on set and ‘metahomomorphisms on groups’,” Chin. Sci. Bull. 42, 1852–1855 (1997).
[CrossRef]

Hikami, K.

K. Hikami, “Skein theory and topological quantum registers: braiding matrices and topological entanglement entropy of non-Abelian quantum Hall states,” Ann. Phys. 323, 1729–1769 (2008).
[CrossRef]

Hou, D. P.

D. P. Hou and C. M. Bai, “J-dendriform algebras,” Front. Math. Chin. 7, 29–49 (2012).
[CrossRef]

Hu, S. W.

S. W. Hu, K. Xue, and M. L. Ge, “Optical simulation of the Yang-Baxter equation,” Phys. Rev. A 78, 022319 (2008).
[CrossRef]

Hulet, R. G.

Y. A. Liao, A. S. C. Rittner, T. Paprotta, W. Li, G. B. Partridge, R. G. Hulet, S. K. Baur, and E. J. Mueller, “Spin-imbalance in a one-dimensional Fermi gas,” Nature 467, 567–569 (2010).
[CrossRef]

Jacobsen, J. L.

J. Dubail, J. L. Jacobsen, and H. Saleur, “Exact solution of the anisotropic special transition in the O(n) model in two dimensions,” Phys. Rev. Lett. 103, 145701 (2009).
[CrossRef]

Jing, N.

I. B. Frenkel and N. Jing, “Vertex representations of quantum affine algebras,” Proc. Natl. Acad. Sci. USA 85, 9373–9377 (1988).
[CrossRef]

Kauffman, L. H.

Y. Zhang, L. H. Kauffman, and M. L. Ge, “Universal quantum gate, Yang–Baxterization and Hamiltonian,” Int. J. Quantum. Inform. 3, 669–678 (2005).
[CrossRef]

L. H. Kauffman and S. J. Lomonaco, “Braiding operators are universal quantum gates,” New J. Phys. 6, 134 (2004).
[CrossRef]

Kitaev, A.

A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. H. Freedman, “Interacting anyons in topological quantum liquids: the golden chain,” Phys. Rev. Lett. 98, 160409 (2007).
[CrossRef]

Kitaev, A. Y.

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

M. H. Freedman, A. Y. Kitaev, and Z. H. Wang, “Simulation of topological field theories by quantum computers,” Commun. Math. Phys. 227, 587–603 (2002).
[CrossRef]

Kok, P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Kuang, L. M.

J. Lu, L. Zhou, and L. M. Kuang, “Linear optics implementation for quantum game with two players,” Phys. Lett. A 330, 48–53 (2004).
[CrossRef]

Li, S. S.

S. S. Li, G. L. Long, F. S. Bai, S. L. Feng, and H. Z. Zheng, “Quantum computing,” Proc. Natl. Acad. Sci. USA 98, 11847–11848 (2001).
[CrossRef]

Li, W.

Y. A. Liao, A. S. C. Rittner, T. Paprotta, W. Li, G. B. Partridge, R. G. Hulet, S. K. Baur, and E. J. Mueller, “Spin-imbalance in a one-dimensional Fermi gas,” Nature 467, 567–569 (2010).
[CrossRef]

Liao, Y. A.

Y. A. Liao, A. S. C. Rittner, T. Paprotta, W. Li, G. B. Partridge, R. G. Hulet, S. K. Baur, and E. J. Mueller, “Spin-imbalance in a one-dimensional Fermi gas,” Nature 467, 567–569 (2010).
[CrossRef]

Lieb, E. H.

H. N. V. Temperley and E. H. Lieb, “Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem,” Proc. R. Soc. London Ser. A 322, 251–280 (1971).
[CrossRef]

Liu, N. L.

X. C. Yao, T. X. Wang, H. Z. Chen, W. B. Gao, A. G. Fowler, R. Raussendorf, Z. B. Chen, N. L. Liu, C. Y. Lu, Y. J. Deng, Y. A. Chen, and J. W. Pan, “Experimental demonstration of topological error correction,” Nature 482, 489–494 (2012).
[CrossRef]

Lomonaco, S. J.

L. H. Kauffman and S. J. Lomonaco, “Braiding operators are universal quantum gates,” New J. Phys. 6, 134 (2004).
[CrossRef]

Long, G. L.

S. S. Li, G. L. Long, F. S. Bai, S. L. Feng, and H. Z. Zheng, “Quantum computing,” Proc. Natl. Acad. Sci. USA 98, 11847–11848 (2001).
[CrossRef]

Lu, C. Y.

X. C. Yao, T. X. Wang, H. Z. Chen, W. B. Gao, A. G. Fowler, R. Raussendorf, Z. B. Chen, N. L. Liu, C. Y. Lu, Y. J. Deng, Y. A. Chen, and J. W. Pan, “Experimental demonstration of topological error correction,” Nature 482, 489–494 (2012).
[CrossRef]

Lu, J.

J. Lu, L. Zhou, and L. M. Kuang, “Linear optics implementation for quantum game with two players,” Phys. Lett. A 330, 48–53 (2004).
[CrossRef]

Ludwig, A. W. W.

A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. H. Freedman, “Interacting anyons in topological quantum liquids: the golden chain,” Phys. Rev. Lett. 98, 160409 (2007).
[CrossRef]

Mattis, D. C.

D. C. Mattis, The Many-Body Problem (World Scientific, 1993), pp. 419–472.

Milburn, G. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Mueller, E. J.

Y. A. Liao, A. S. C. Rittner, T. Paprotta, W. Li, G. B. Partridge, R. G. Hulet, S. K. Baur, and E. J. Mueller, “Spin-imbalance in a one-dimensional Fermi gas,” Nature 467, 567–569 (2010).
[CrossRef]

Munro, W. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Nagler, S. E.

D. A. Tennant, R. A. Cowley, S. E. Nagler, and A. M. Tsvelik, “Measurement of the spin-excitation continuum in one-dimensional KCuF3 using neutron scattering,” Phys. Rev. B 52, 13368–13380 (1995).
[CrossRef]

D. A. Tennant, T. G. Perring, R. A. Cowley, and S. E. Nagler, “Unbound spinons in the S=1/2 antiferromagnetic chain KCuF3,” Phys. Rev. Lett. 70, 4003–4006 (1993).
[CrossRef]

Nayak, C.

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
[CrossRef]

Nemoto, K.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000), pp. 1–607.

O’ Brien, J. L.

J. L. O’ Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[CrossRef]

Pan, J. W.

X. C. Yao, T. X. Wang, H. Z. Chen, W. B. Gao, A. G. Fowler, R. Raussendorf, Z. B. Chen, N. L. Liu, C. Y. Lu, Y. J. Deng, Y. A. Chen, and J. W. Pan, “Experimental demonstration of topological error correction,” Nature 482, 489–494 (2012).
[CrossRef]

Paprotta, T.

Y. A. Liao, A. S. C. Rittner, T. Paprotta, W. Li, G. B. Partridge, R. G. Hulet, S. K. Baur, and E. J. Mueller, “Spin-imbalance in a one-dimensional Fermi gas,” Nature 467, 567–569 (2010).
[CrossRef]

Partridge, G. B.

Y. A. Liao, A. S. C. Rittner, T. Paprotta, W. Li, G. B. Partridge, R. G. Hulet, S. K. Baur, and E. J. Mueller, “Spin-imbalance in a one-dimensional Fermi gas,” Nature 467, 567–569 (2010).
[CrossRef]

Perring, T. G.

D. A. Tennant, T. G. Perring, R. A. Cowley, and S. E. Nagler, “Unbound spinons in the S=1/2 antiferromagnetic chain KCuF3,” Phys. Rev. Lett. 70, 4003–4006 (1993).
[CrossRef]

Pryde, G. J.

J. L. O’ Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[CrossRef]

Ralph, T. C.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

J. L. O’ Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[CrossRef]

Raussendorf, R.

X. C. Yao, T. X. Wang, H. Z. Chen, W. B. Gao, A. G. Fowler, R. Raussendorf, Z. B. Chen, N. L. Liu, C. Y. Lu, Y. J. Deng, Y. A. Chen, and J. W. Pan, “Experimental demonstration of topological error correction,” Nature 482, 489–494 (2012).
[CrossRef]

Rittner, A. S. C.

Y. A. Liao, A. S. C. Rittner, T. Paprotta, W. Li, G. B. Partridge, R. G. Hulet, S. K. Baur, and E. J. Mueller, “Spin-imbalance in a one-dimensional Fermi gas,” Nature 467, 567–569 (2010).
[CrossRef]

Rowell, E. C.

E. C. Rowell, Y. Zhang, Y. S. Wu, and M. L. Ge, “Extraspecial two-groups, generalized Yang–Baxter equations and braiding quantum gates,” Quantum Inf. Comput. 10, 685–702 (2010).

J. M. Franko, E. C. Rowell, and Z. Wang, “Extraspecial 2-groups and images of braid group representations,” J. Knot Theory Ramif. 15, 413–427 (2006).
[CrossRef]

Saleur, H.

J. Dubail, J. L. Jacobsen, and H. Saleur, “Exact solution of the anisotropic special transition in the O(n) model in two dimensions,” Phys. Rev. Lett. 103, 145701 (2009).
[CrossRef]

Sarma, S. D.

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
[CrossRef]

Schack, S. D.

M. Gerstenhaber and S. D. Schack, “Bialgebra cohomology, deformations, and quantum groups,” Proc. Natl. Acad. Sci. USA 87, 478–481 (1990).
[CrossRef]

Schoutens, K.

E. Ardonne and K. Schoutens, “Wavefunctions for topological quantum registers,” Ann. Phys. 322, 201–235 (2007).
[CrossRef]

Simon, S. H.

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
[CrossRef]

Stern, A.

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
[CrossRef]

Sun, C. P.

C. P. Sun, “The differential realization of new solutions for Yang–Baxter equation,” Chin. Sci. Bull. 37, 379 (1992).

Temperley, H. N. V.

H. N. V. Temperley and E. H. Lieb, “Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem,” Proc. R. Soc. London Ser. A 322, 251–280 (1971).
[CrossRef]

Tennant, D. A.

D. A. Tennant, R. A. Cowley, S. E. Nagler, and A. M. Tsvelik, “Measurement of the spin-excitation continuum in one-dimensional KCuF3 using neutron scattering,” Phys. Rev. B 52, 13368–13380 (1995).
[CrossRef]

D. A. Tennant, T. G. Perring, R. A. Cowley, and S. E. Nagler, “Unbound spinons in the S=1/2 antiferromagnetic chain KCuF3,” Phys. Rev. Lett. 70, 4003–4006 (1993).
[CrossRef]

Trebst, S.

A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. H. Freedman, “Interacting anyons in topological quantum liquids: the golden chain,” Phys. Rev. Lett. 98, 160409 (2007).
[CrossRef]

Troyer, M.

A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. H. Freedman, “Interacting anyons in topological quantum liquids: the golden chain,” Phys. Rev. Lett. 98, 160409 (2007).
[CrossRef]

Tsvelik, A. M.

D. A. Tennant, R. A. Cowley, S. E. Nagler, and A. M. Tsvelik, “Measurement of the spin-excitation continuum in one-dimensional KCuF3 using neutron scattering,” Phys. Rev. B 52, 13368–13380 (1995).
[CrossRef]

Wang, T. X.

X. C. Yao, T. X. Wang, H. Z. Chen, W. B. Gao, A. G. Fowler, R. Raussendorf, Z. B. Chen, N. L. Liu, C. Y. Lu, Y. J. Deng, Y. A. Chen, and J. W. Pan, “Experimental demonstration of topological error correction,” Nature 482, 489–494 (2012).
[CrossRef]

Wang, Z.

A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. H. Freedman, “Interacting anyons in topological quantum liquids: the golden chain,” Phys. Rev. Lett. 98, 160409 (2007).
[CrossRef]

J. M. Franko, E. C. Rowell, and Z. Wang, “Extraspecial 2-groups and images of braid group representations,” J. Knot Theory Ramif. 15, 413–427 (2006).
[CrossRef]

Wang, Z. H.

M. H. Freedman, A. Y. Kitaev, and Z. H. Wang, “Simulation of topological field theories by quantum computers,” Commun. Math. Phys. 227, 587–603 (2002).
[CrossRef]

White, A. G.

J. L. O’ Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[CrossRef]

Wu, Y. S.

E. C. Rowell, Y. Zhang, Y. S. Wu, and M. L. Ge, “Extraspecial two-groups, generalized Yang–Baxter equations and braiding quantum gates,” Quantum Inf. Comput. 10, 685–702 (2010).

Xue, K.

S. W. Hu, K. Xue, and M. L. Ge, “Optical simulation of the Yang-Baxter equation,” Phys. Rev. A 78, 022319 (2008).
[CrossRef]

J. L. Chen, K. Xue, and M. L. Ge, “Braiding transformation, entanglement swapping, and Berry phase in entanglement space,” Phys. Rev. A 76, 042324 (2007).
[CrossRef]

Yang, C. N.

C. N. Yang, “S matrix for the one-dimensional N-body problem with repulsive or attractive δ-function interaction,” Phys. Rev. 168, 1920–1923 (1968).
[CrossRef]

C. N. Yang, “Some exact results for the many-body problem in one dimension with repulsive delta-function interaction,” Phys. Rev. Lett. 19, 1312–1315 (1967).
[CrossRef]

C. N. Yang and M. L. Ge, Braid Group, Knot Theory and Statistical Mechanics, 2nd ed. (World Scientific, 1994), pp. 1–176.

Yao, X. C.

X. C. Yao, T. X. Wang, H. Z. Chen, W. B. Gao, A. G. Fowler, R. Raussendorf, Z. B. Chen, N. L. Liu, C. Y. Lu, Y. J. Deng, Y. A. Chen, and J. W. Pan, “Experimental demonstration of topological error correction,” Nature 482, 489–494 (2012).
[CrossRef]

Zhang, Y.

E. C. Rowell, Y. Zhang, Y. S. Wu, and M. L. Ge, “Extraspecial two-groups, generalized Yang–Baxter equations and braiding quantum gates,” Quantum Inf. Comput. 10, 685–702 (2010).

Y. Zhang and M. L. Ge, “GHZ states, almost-complex structure and Yang–Baxter equation,” Quantum Inf. Process. 6, 363–379 (2007).
[CrossRef]

Y. Zhang, L. H. Kauffman, and M. L. Ge, “Universal quantum gate, Yang–Baxterization and Hamiltonian,” Int. J. Quantum. Inform. 3, 669–678 (2005).
[CrossRef]

Zheng, H. Z.

S. S. Li, G. L. Long, F. S. Bai, S. L. Feng, and H. Z. Zheng, “Quantum computing,” Proc. Natl. Acad. Sci. USA 98, 11847–11848 (2001).
[CrossRef]

Zhou, L.

J. Lu, L. Zhou, and L. M. Kuang, “Linear optics implementation for quantum game with two players,” Phys. Lett. A 330, 48–53 (2004).
[CrossRef]

Ann. Phys.

R. J. Baxter, “Partition function of the Eight-Vertex lattice model,” Ann. Phys. 70, 193–228 (1972).
[CrossRef]

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

E. Ardonne and K. Schoutens, “Wavefunctions for topological quantum registers,” Ann. Phys. 322, 201–235 (2007).
[CrossRef]

K. Hikami, “Skein theory and topological quantum registers: braiding matrices and topological entanglement entropy of non-Abelian quantum Hall states,” Ann. Phys. 323, 1729–1769 (2008).
[CrossRef]

Chin. Sci. Bull.

C. P. Sun, “The differential realization of new solutions for Yang–Baxter equation,” Chin. Sci. Bull. 37, 379 (1992).

P. Gui, “Another solution of Yang–Baxter equation on set and ‘metahomomorphisms on groups’,” Chin. Sci. Bull. 42, 1852–1855 (1997).
[CrossRef]

H. X. Chen, “Cpcycle deformations, braided monoidal categories and quasitriangularity,” Chin. Sci. Bull. 44, 510–513 (1999).
[CrossRef]

Commun. Math. Phys.

M. H. Freedman, A. Y. Kitaev, and Z. H. Wang, “Simulation of topological field theories by quantum computers,” Commun. Math. Phys. 227, 587–603 (2002).
[CrossRef]

Front. Math. Chin.

D. P. Hou and C. M. Bai, “J-dendriform algebras,” Front. Math. Chin. 7, 29–49 (2012).
[CrossRef]

Int. J. Quantum. Inform.

Y. Zhang, L. H. Kauffman, and M. L. Ge, “Universal quantum gate, Yang–Baxterization and Hamiltonian,” Int. J. Quantum. Inform. 3, 669–678 (2005).
[CrossRef]

J. Knot Theory Ramif.

J. M. Franko, E. C. Rowell, and Z. Wang, “Extraspecial 2-groups and images of braid group representations,” J. Knot Theory Ramif. 15, 413–427 (2006).
[CrossRef]

Nature

X. C. Yao, T. X. Wang, H. Z. Chen, W. B. Gao, A. G. Fowler, R. Raussendorf, Z. B. Chen, N. L. Liu, C. Y. Lu, Y. J. Deng, Y. A. Chen, and J. W. Pan, “Experimental demonstration of topological error correction,” Nature 482, 489–494 (2012).
[CrossRef]

Y. A. Liao, A. S. C. Rittner, T. Paprotta, W. Li, G. B. Partridge, R. G. Hulet, S. K. Baur, and E. J. Mueller, “Spin-imbalance in a one-dimensional Fermi gas,” Nature 467, 567–569 (2010).
[CrossRef]

J. L. O’ Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003).
[CrossRef]

New J. Phys.

L. H. Kauffman and S. J. Lomonaco, “Braiding operators are universal quantum gates,” New J. Phys. 6, 134 (2004).
[CrossRef]

Phys. Lett. A

J. Lu, L. Zhou, and L. M. Kuang, “Linear optics implementation for quantum game with two players,” Phys. Lett. A 330, 48–53 (2004).
[CrossRef]

Phys. Rev.

C. N. Yang, “S matrix for the one-dimensional N-body problem with repulsive or attractive δ-function interaction,” Phys. Rev. 168, 1920–1923 (1968).
[CrossRef]

Phys. Rev. A

J. L. Chen, K. Xue, and M. L. Ge, “Braiding transformation, entanglement swapping, and Berry phase in entanglement space,” Phys. Rev. A 76, 042324 (2007).
[CrossRef]

S. W. Hu, K. Xue, and M. L. Ge, “Optical simulation of the Yang-Baxter equation,” Phys. Rev. A 78, 022319 (2008).
[CrossRef]

Phys. Rev. B

D. A. Tennant, R. A. Cowley, S. E. Nagler, and A. M. Tsvelik, “Measurement of the spin-excitation continuum in one-dimensional KCuF3 using neutron scattering,” Phys. Rev. B 52, 13368–13380 (1995).
[CrossRef]

Phys. Rev. Lett.

D. A. Tennant, T. G. Perring, R. A. Cowley, and S. E. Nagler, “Unbound spinons in the S=1/2 antiferromagnetic chain KCuF3,” Phys. Rev. Lett. 70, 4003–4006 (1993).
[CrossRef]

A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. H. Freedman, “Interacting anyons in topological quantum liquids: the golden chain,” Phys. Rev. Lett. 98, 160409 (2007).
[CrossRef]

J. Dubail, J. L. Jacobsen, and H. Saleur, “Exact solution of the anisotropic special transition in the O(n) model in two dimensions,” Phys. Rev. Lett. 103, 145701 (2009).
[CrossRef]

C. N. Yang, “Some exact results for the many-body problem in one dimension with repulsive delta-function interaction,” Phys. Rev. Lett. 19, 1312–1315 (1967).
[CrossRef]

Proc. Natl. Acad. Sci. USA

S. C. Billey, “Kostant polynomials and the cohomology ring for G/B,” Proc. Natl. Acad. Sci. USA 94, 29–32 (1997).

I. B. Frenkel and N. Jing, “Vertex representations of quantum affine algebras,” Proc. Natl. Acad. Sci. USA 85, 9373–9377 (1988).
[CrossRef]

M. Gerstenhaber and S. D. Schack, “Bialgebra cohomology, deformations, and quantum groups,” Proc. Natl. Acad. Sci. USA 87, 478–481 (1990).
[CrossRef]

S. S. Li, G. L. Long, F. S. Bai, S. L. Feng, and H. Z. Zheng, “Quantum computing,” Proc. Natl. Acad. Sci. USA 98, 11847–11848 (2001).
[CrossRef]

Proc. R. Soc. London Ser. A

H. N. V. Temperley and E. H. Lieb, “Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem,” Proc. R. Soc. London Ser. A 322, 251–280 (1971).
[CrossRef]

Quant. Info. Proc.

H. A. Dye, “Unitary solutions to the Yang–Baxter equation in dimension four,” Quant. Info. Proc. 2, 117–152 (2003).
[CrossRef]

Quantum Inf. Comput.

E. C. Rowell, Y. Zhang, Y. S. Wu, and M. L. Ge, “Extraspecial two-groups, generalized Yang–Baxter equations and braiding quantum gates,” Quantum Inf. Comput. 10, 685–702 (2010).

Quantum Inf. Process.

Y. Zhang and M. L. Ge, “GHZ states, almost-complex structure and Yang–Baxter equation,” Quantum Inf. Process. 6, 363–379 (2007).
[CrossRef]

Rev. Mod. Phys.

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
[CrossRef]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[CrossRef]

Other

S. Bose and V. Korepin, “Quantum gates between flying qubits via spin-independent scattering,” http://arxiv.org/abs/1106.2329v1 .

D. C. Mattis, The Many-Body Problem (World Scientific, 1993), pp. 419–472.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000), pp. 1–607.

C. N. Yang and M. L. Ge, Braid Group, Knot Theory and Statistical Mechanics, 2nd ed. (World Scientific, 1994), pp. 1–176.

R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, 1982).

Yang-Baxter Equation in Integrable Systems, M. Jimbo, ed. (World Scientific, 1990).

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Figures (6)

Fig. 1.
Fig. 1.

Realization of operations (a) A(θ) and B(θ) by optical elements. UQ(θ) and UH(θ) are the matrices of QWP and HWP, respectively, and θ is the angle between the optical device axes and the vertical direction.

Fig. 2.
Fig. 2.

Optical realization of the (a) LHS and (b) RHS of YBE. The angles of these QWPs (filled) and HWPs (empty) must satisfy the Lorentz-like relation given in Eq. (12).

Fig. 3.
Fig. 3.

Experimental setup. The left part generates a horizontally polarized state, which is then transformed into a desired input state with arbitrary linear or elliptical polarization by using a HWP or QWP, respectively, following the PBS. The middle part is the left-hand right-hand side of the YBE, consisting of a series of wave plates. The right part, containing a Glan–Laser prism and a single photon detector, is used to detect the polarized state of the output state. A QWP may be inserted to determine the handedness of an elliptically polarized photon.

Fig. 4.
Fig. 4.

Curve of CYBE versus θ2, where θ1 and θ3 are kept fixed at 56° and 23°, respectively. The dots are the experimental data while the line is the theoretical curve. The input state is |, the vertical polarization state. When the input state is chosen as |, the horizontally polarized state, both experimental and theoretical results are identical to those for the vertical polarization input state.

Fig. 5.
Fig. 5.

Curve of CYBE versus θ2 where θ1 and θ3 are kept fixed at 56° and 23°, respectively. The dots are the experimental data while the line is the theoretical curve. The input state is an elliptically polarized state with the form (0.70710.5417i)|0.4545i|. CYBE reaches 0.9999±0.0356 when θ2=49.49°.

Fig. 6.
Fig. 6.

θ2 versus θ1 curve while fixing θ3. The blue dots, the green squares, and the red triangles are the experimental data for θ3=32°, 56°, and 146°, while the lines are the corresponding theoretical results.

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

12(u)23(u23)12(v)=23(v)12(u23)23(u),
u23=u+v1+βuv,
A(u)B(u+v1+β2uv)A(v)=B(v)A(u+v1+β2uv)B(u),
A(u)=ρ(u)(1+β2u2+2iϵβu1+β2u22iϵβu001),
B(u)=ρ(u)1+β2u22iϵβu(1+β2u22iϵβu2iϵβu1+β2u2),
1+β2u2+2iϵβu1+β2u22iϵβue2iθ,
ρ(u)eiθ.
A(θ)=(eiθ00eiθ),
B(θ)=(cosθisinθisinθcosθ).
A(θ1)B(θ2)A(θ3)=B(θ3)A(θ2)B(θ1).
(e2iθ2+1)[isec(θ1θ3)sin(θ1+θ3)]=2i.
θ2=arctan(sin(θ1+θ3)cos(θ1θ3)).
|ψ=α|+iβ|,
UQ(θ)=12(1icos(2θ)isin(2θ)isin(2θ)1+icos(2θ))
UH(θ)=UQ2(θ)=i(cos(2θ)sin(2θ)sin(2θ)cos(2θ)),
A(θ)=UQ(π4)UH(π4+θ2)UQ(π4)
B(θ)=UQ(π2)UH(θ2)UQ(π2),
|ψoutL=αL|L+iβL|L,
|ψoutR=αR|R+iβR|R.
CYBE=|ψout|ψoutRL|,

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