Abstract

A quantum walk is the quantum analog of the classical random walk, which provides a powerful tool for quantum computation and quantum simulation. However, the current experimental realization of the quantum walk based on photonic systems has been restricted to evolution in one dimension or small scale in two dimensions so far, and the growing demands cannot be fulfilled for further application of certain quantum computation tasks. Here, we theoretically propose a feasible scheme of a two-dimensional quantum walk with infinite steps by exploiting the orbital angular momentum of the photons and the phase information of the auxiliary coherent state induced by cross-Kerr nonlinearity. The numerical simulation shows that a proper parameter setting guarantees the walking for infinite steps in a two-dimensional grid with nearly $100\%$ success probability.

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

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  1. Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48(2), 1687–1690 (1993).
    [Crossref]
  2. A. Aspuruguzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8(4), 285–291 (2012).
    [Crossref]
  3. A. Ambainis, “Quantum walks and their algorithmic applications,” Int. J. Quantum Inform. 01(04), 507–518 (2003).
    [Crossref]
  4. A. M. Childs and J. Goldstone, “Spatial search by quantum walk,” Phys. Rev. A 70(2), 022314 (2004).
    [Crossref]
  5. N. Shenvi, J. Kempe, and K. B. Whaley, “Quantum random-walk search algorithm,” Phys. Rev. A 67(5), 052307 (2003).
    [Crossref]
  6. M. Schuld, I. Sinayskiy, and F. Petruccione, “The quest for a quantum neural network,” Quantum Inf. Process. 13(11), 2567–2586 (2014).
    [Crossref]
  7. G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, “Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems,” Nature 446(7137), 782–786 (2007).
    [Crossref]
  8. M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, “Environment-assisted quantum walks in photosynthetic energy transfer,” J. Chem. Phys. 129(17), 174106 (2008).
    [Crossref]
  9. B. L. Douglas and J. B. Wang, “A classical approach to the graph isomorphism problem using quantum walks,” J. Phys. A: Math. Theor. 41(7), 075303 (2008).
    [Crossref]
  10. H. Obuse and N. Kawakami, “Topological phases and delocalization of quantum walks in random environments,” Phys. Rev. B 84(19), 195139 (2011).
    [Crossref]
  11. T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82(3), 033429 (2010).
    [Crossref]
  12. N. P. Kumar, R. Balu, R. Laflamme, and C. M. Chandrashekar, “Bounds on the dynamics of periodic quantum walks and emergence of the gapless and gapped dirac equation,” Phys. Rev. A 97(1), 012116 (2018).
    [Crossref]
  13. F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7(1), 11439 (2016).
    [Crossref]
  14. A. M. Childs, “Universal computation by quantum walk,” Phys. Rev. Lett. 102(18), 180501 (2009).
    [Crossref]
  15. N. B. Lovett, S. Cooper, M. Everitt, M. Trevers, and V. Kendon, “Universal quantum computation using the discrete-time quantum walk,” Phys. Rev. A 81(4), 042330 (2010).
    [Crossref]
  16. C. A. Ryan, M. Laforest, J. C. Boileau, and R. Laflamme, “Experimental implementation of a discrete-time quantum random walk on an nmr quantum-information processor,” Phys. Rev. A 72(6), 062317 (2005).
    [Crossref]
  17. J. Du, H. Li, X. Xu, M. Shi, J. Wu, X. Zhou, and R. Han, “Experimental implementation of the quantum random-walk algorithm,” Phys. Rev. A 67(4), 042316 (2003).
    [Crossref]
  18. M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, “Quantum walk in position space with single optically trapped atoms,” Science 325(5937), 174–177 (2009).
    [Crossref]
  19. H. Schmitz, R. Matjeschk, C. Schneider, J. Glueckert, M. Enderlein, T. Huber, and T. Schaetz, “Quantum walk of a trapped ion in phase space,” Phys. Rev. Lett. 103(9), 090504 (2009).
    [Crossref]
  20. F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. F. Roos, “Realization of a quantum walk with one and two trapped ions,” Phys. Rev. Lett. 104(10), 100503 (2010).
    [Crossref]
  21. N. Satapathy, H. Hagman, M. Zelán, A. Kastberg, and H. Ramachandran, “Theoretical investigations of quantum walks by cold atoms in a double optical lattice,” Phys. Rev. A 80(1), 012302 (2009).
    [Crossref]
  22. S. Mugel, A. Celi, P. Massignan, J. K. Asbóth, M. Lewenstein, and C. Lobo, “Topological bound states of a quantum walk with cold atoms,” Phys. Rev. A 94(2), 023631 (2016).
    [Crossref]
  23. A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: A quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104(5), 050502 (2010).
    [Crossref]
  24. M. A. Broome, A. Fedrizzi, B. P. Lanyon, I. Kassal, A. Aspuru-Guzik, and A. G. White, “Discrete single-photon quantum walks with tunable decoherence,” Phys. Rev. Lett. 104(15), 153602 (2010).
    [Crossref]
  25. A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329(5998), 1500–1503 (2010).
    [Crossref]
  26. H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
    [Crossref]
  27. H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, L.-F. Qiao, A.-L. Yang, and X.-M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4(5), eaat3174 (2018).
    [Crossref]
  28. T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Commun. 3(1), 882 (2012).
    [Crossref]
  29. Z.-H. Bian, J. Li, X. Zhan, J. Twamley, and P. Xue, “Experimental implementation of a quantum walk on a circle with single photons,” Phys. Rev. A 95(5), 052338 (2017).
    [Crossref]
  30. L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13(11), 1117–1123 (2017).
    [Crossref]
  31. B. Wang, T. Chen, and X. Zhang, “Experimental observation of topologically protected bound states with vanishing chern numbers in a two-dimensional quantum walk,” Phys. Rev. Lett. 121(10), 100501 (2018).
    [Crossref]
  32. L. Xiao, X. Qiu, K. Wang, Z. Bian, X. Zhan, H. Obuse, B. C. Sanders, W. Yi, and P. Xue, “Higher winding number in a nonunitary photonic quantum walk,” Phys. Rev. A 98(6), 063847 (2018).
    [Crossref]
  33. K. Poulios, R. Keil, D. Fry, J. D. A. Meinecke, J. C. F. Matthews, A. Politi, M. Lobino, M. Gräfe, M. Heinrich, S. Nolte, A. Szameit, and J. L. O’Brien, “Quantum walks of correlated photon pairs in two-dimensional waveguide arrays,” Phys. Rev. Lett. 112(14), 143604 (2014).
    [Crossref]
  34. N. Wiebe, D. Braun, and S. Lloyd, “Quantum algorithm for data fitting,” Phys. Rev. Lett. 109(5), 050505 (2012).
    [Crossref]
  35. Q. Lin and J. Li, “Quantum control gates with weak cross-kerr nonlinearity,” Phys. Rev. A 79(2), 022301 (2009).
    [Crossref]
  36. D. J. Brod and J. Combes, “Passive cphase gate via cross-kerr nonlinearities,” Phys. Rev. Lett. 117(8), 080502 (2016).
    [Crossref]
  37. T. Wang, H. W. Lau, H. Kaviani, R. Ghobadi, and C. Simon, “Strong micro-macro entanglement from a weak cross-kerr nonlinearity,” Phys. Rev. A 92(1), 012316 (2015).
    [Crossref]
  38. F.-G. Deng, “Optimal nonlocal multipartite entanglement concentration based on projection measurements,” Phys. Rev. A 85(2), 022311 (2012).
    [Crossref]
  39. Y.-B. Sheng, F.-G. Deng, and H.-Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-kerr nonlinearity,” Phys. Rev. A 77(4), 042308 (2008).
    [Crossref]
  40. Y.-B. Sheng, F.-G. Deng, and G. L. Long, “Complete hyperentangled-bell-state analysis for quantum communication,” Phys. Rev. A 82(3), 032318 (2010).
    [Crossref]
  41. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref]
  42. K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
    [Crossref]
  43. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
    [Crossref]
  44. M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
    [Crossref]
  45. M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6(2), 118–121 (2010).
    [Crossref]
  46. S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14(9), 3792–3805 (2006).
    [Crossref]
  47. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
    [Crossref]
  48. J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008).
    [Crossref]
  49. E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through hong-mandel coalescence,” Nat. Photonics 3(12), 720–723 (2009).
    [Crossref]
  50. V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
    [Crossref]
  51. A. Aadhi, G. K. Samanta, S. C. Kumar, and M. Ebrahim-Zadeh, “Controlled switching of orbital angular momentum in an optical parametric oscillator,” Optica 4(3), 349–355 (2017).
    [Crossref]
  52. P. Zhang, B.-H. Liu, R.-F. Liu, H.-R. Li, F.-L. Li, and G.-C. Guo, “Implementation of one-dimensional quantum walks on spin-orbital angular momentum space of photons,” Phys. Rev. A 81(5), 052322 (2010).
    [Crossref]
  53. P. Zhang, X.-F. Ren, X.-B. Zou, B.-H. Liu, Y.-F. Huang, and G.-C. Guo, “Demonstration of one-dimensional quantum random walks using orbital angular momentum of photons,” Phys. Rev. A 75(5), 052310 (2007).
    [Crossref]
  54. S. K. Goyal, F. S. Roux, A. Forbes, and T. Konrad, “Implementing quantum walks using orbital angular momentum of classical light,” Phys. Rev. Lett. 110(26), 263602 (2013).
    [Crossref]
  55. W. J. Munro, K. Nemoto, and T. P. Spiller, “Weak nonlinearities: a new route to optical quantum computation,” New J. Phys. 7, 137 (2005).
    [Crossref]
  56. W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
    [Crossref]
  57. K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-not gate,” Phys. Rev. Lett. 93(25), 250502 (2004).
    [Crossref]
  58. L. S. Costanzo, A. S. Coelho, N. Biagi, J. Fiurášek, M. Bellini, and A. Zavatta, “Measurement-induced strong kerr nonlinearity for weak quantum states of light,” Phys. Rev. Lett. 119(1), 013601 (2017).
    [Crossref]
  59. P. Grangier, J. A. Levenson, and J.-P. Poizat, “Quantum non-demolition measurements in optics,” Nature 396(6711), 537–542 (1998).
    [Crossref]
  60. C. W. Gardiner and H. Haken, Quantum noise, vol. 2 (Springer, 1991).
  61. T. P. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock, and G. J. Milburn, “Quantum computation by communication,” New J. Phys. 8(2), 30 (2006).
    [Crossref]

2018 (4)

N. P. Kumar, R. Balu, R. Laflamme, and C. M. Chandrashekar, “Bounds on the dynamics of periodic quantum walks and emergence of the gapless and gapped dirac equation,” Phys. Rev. A 97(1), 012116 (2018).
[Crossref]

H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, L.-F. Qiao, A.-L. Yang, and X.-M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4(5), eaat3174 (2018).
[Crossref]

B. Wang, T. Chen, and X. Zhang, “Experimental observation of topologically protected bound states with vanishing chern numbers in a two-dimensional quantum walk,” Phys. Rev. Lett. 121(10), 100501 (2018).
[Crossref]

L. Xiao, X. Qiu, K. Wang, Z. Bian, X. Zhan, H. Obuse, B. C. Sanders, W. Yi, and P. Xue, “Higher winding number in a nonunitary photonic quantum walk,” Phys. Rev. A 98(6), 063847 (2018).
[Crossref]

2017 (4)

Z.-H. Bian, J. Li, X. Zhan, J. Twamley, and P. Xue, “Experimental implementation of a quantum walk on a circle with single photons,” Phys. Rev. A 95(5), 052338 (2017).
[Crossref]

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13(11), 1117–1123 (2017).
[Crossref]

A. Aadhi, G. K. Samanta, S. C. Kumar, and M. Ebrahim-Zadeh, “Controlled switching of orbital angular momentum in an optical parametric oscillator,” Optica 4(3), 349–355 (2017).
[Crossref]

L. S. Costanzo, A. S. Coelho, N. Biagi, J. Fiurášek, M. Bellini, and A. Zavatta, “Measurement-induced strong kerr nonlinearity for weak quantum states of light,” Phys. Rev. Lett. 119(1), 013601 (2017).
[Crossref]

2016 (3)

D. J. Brod and J. Combes, “Passive cphase gate via cross-kerr nonlinearities,” Phys. Rev. Lett. 117(8), 080502 (2016).
[Crossref]

S. Mugel, A. Celi, P. Massignan, J. K. Asbóth, M. Lewenstein, and C. Lobo, “Topological bound states of a quantum walk with cold atoms,” Phys. Rev. A 94(2), 023631 (2016).
[Crossref]

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7(1), 11439 (2016).
[Crossref]

2015 (1)

T. Wang, H. W. Lau, H. Kaviani, R. Ghobadi, and C. Simon, “Strong micro-macro entanglement from a weak cross-kerr nonlinearity,” Phys. Rev. A 92(1), 012316 (2015).
[Crossref]

2014 (2)

K. Poulios, R. Keil, D. Fry, J. D. A. Meinecke, J. C. F. Matthews, A. Politi, M. Lobino, M. Gräfe, M. Heinrich, S. Nolte, A. Szameit, and J. L. O’Brien, “Quantum walks of correlated photon pairs in two-dimensional waveguide arrays,” Phys. Rev. Lett. 112(14), 143604 (2014).
[Crossref]

M. Schuld, I. Sinayskiy, and F. Petruccione, “The quest for a quantum neural network,” Quantum Inf. Process. 13(11), 2567–2586 (2014).
[Crossref]

2013 (2)

S. K. Goyal, F. S. Roux, A. Forbes, and T. Konrad, “Implementing quantum walks using orbital angular momentum of classical light,” Phys. Rev. Lett. 110(26), 263602 (2013).
[Crossref]

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

2012 (5)

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

A. Aspuruguzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8(4), 285–291 (2012).
[Crossref]

N. Wiebe, D. Braun, and S. Lloyd, “Quantum algorithm for data fitting,” Phys. Rev. Lett. 109(5), 050505 (2012).
[Crossref]

F.-G. Deng, “Optimal nonlocal multipartite entanglement concentration based on projection measurements,” Phys. Rev. A 85(2), 022311 (2012).
[Crossref]

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Commun. 3(1), 882 (2012).
[Crossref]

2011 (2)

H. Obuse and N. Kawakami, “Topological phases and delocalization of quantum walks in random environments,” Phys. Rev. B 84(19), 195139 (2011).
[Crossref]

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[Crossref]

2010 (9)

Y.-B. Sheng, F.-G. Deng, and G. L. Long, “Complete hyperentangled-bell-state analysis for quantum communication,” Phys. Rev. A 82(3), 032318 (2010).
[Crossref]

M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6(2), 118–121 (2010).
[Crossref]

P. Zhang, B.-H. Liu, R.-F. Liu, H.-R. Li, F.-L. Li, and G.-C. Guo, “Implementation of one-dimensional quantum walks on spin-orbital angular momentum space of photons,” Phys. Rev. A 81(5), 052322 (2010).
[Crossref]

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82(3), 033429 (2010).
[Crossref]

N. B. Lovett, S. Cooper, M. Everitt, M. Trevers, and V. Kendon, “Universal quantum computation using the discrete-time quantum walk,” Phys. Rev. A 81(4), 042330 (2010).
[Crossref]

F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. F. Roos, “Realization of a quantum walk with one and two trapped ions,” Phys. Rev. Lett. 104(10), 100503 (2010).
[Crossref]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: A quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104(5), 050502 (2010).
[Crossref]

M. A. Broome, A. Fedrizzi, B. P. Lanyon, I. Kassal, A. Aspuru-Guzik, and A. G. White, “Discrete single-photon quantum walks with tunable decoherence,” Phys. Rev. Lett. 104(15), 153602 (2010).
[Crossref]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329(5998), 1500–1503 (2010).
[Crossref]

2009 (6)

N. Satapathy, H. Hagman, M. Zelán, A. Kastberg, and H. Ramachandran, “Theoretical investigations of quantum walks by cold atoms in a double optical lattice,” Phys. Rev. A 80(1), 012302 (2009).
[Crossref]

Q. Lin and J. Li, “Quantum control gates with weak cross-kerr nonlinearity,” Phys. Rev. A 79(2), 022301 (2009).
[Crossref]

M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, “Quantum walk in position space with single optically trapped atoms,” Science 325(5937), 174–177 (2009).
[Crossref]

H. Schmitz, R. Matjeschk, C. Schneider, J. Glueckert, M. Enderlein, T. Huber, and T. Schaetz, “Quantum walk of a trapped ion in phase space,” Phys. Rev. Lett. 103(9), 090504 (2009).
[Crossref]

A. M. Childs, “Universal computation by quantum walk,” Phys. Rev. Lett. 102(18), 180501 (2009).
[Crossref]

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through hong-mandel coalescence,” Nat. Photonics 3(12), 720–723 (2009).
[Crossref]

2008 (5)

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008).
[Crossref]

M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, “Environment-assisted quantum walks in photosynthetic energy transfer,” J. Chem. Phys. 129(17), 174106 (2008).
[Crossref]

B. L. Douglas and J. B. Wang, “A classical approach to the graph isomorphism problem using quantum walks,” J. Phys. A: Math. Theor. 41(7), 075303 (2008).
[Crossref]

Y.-B. Sheng, F.-G. Deng, and H.-Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-kerr nonlinearity,” Phys. Rev. A 77(4), 042308 (2008).
[Crossref]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
[Crossref]

2007 (2)

G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, “Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems,” Nature 446(7137), 782–786 (2007).
[Crossref]

P. Zhang, X.-F. Ren, X.-B. Zou, B.-H. Liu, Y.-F. Huang, and G.-C. Guo, “Demonstration of one-dimensional quantum random walks using orbital angular momentum of photons,” Phys. Rev. A 75(5), 052310 (2007).
[Crossref]

2006 (2)

S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14(9), 3792–3805 (2006).
[Crossref]

T. P. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock, and G. J. Milburn, “Quantum computation by communication,” New J. Phys. 8(2), 30 (2006).
[Crossref]

2005 (3)

W. J. Munro, K. Nemoto, and T. P. Spiller, “Weak nonlinearities: a new route to optical quantum computation,” New J. Phys. 7, 137 (2005).
[Crossref]

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
[Crossref]

C. A. Ryan, M. Laforest, J. C. Boileau, and R. Laflamme, “Experimental implementation of a discrete-time quantum random walk on an nmr quantum-information processor,” Phys. Rev. A 72(6), 062317 (2005).
[Crossref]

2004 (2)

A. M. Childs and J. Goldstone, “Spatial search by quantum walk,” Phys. Rev. A 70(2), 022314 (2004).
[Crossref]

K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-not gate,” Phys. Rev. Lett. 93(25), 250502 (2004).
[Crossref]

2003 (3)

N. Shenvi, J. Kempe, and K. B. Whaley, “Quantum random-walk search algorithm,” Phys. Rev. A 67(5), 052307 (2003).
[Crossref]

A. Ambainis, “Quantum walks and their algorithmic applications,” Int. J. Quantum Inform. 01(04), 507–518 (2003).
[Crossref]

J. Du, H. Li, X. Xu, M. Shi, J. Wu, X. Zhou, and R. Han, “Experimental implementation of the quantum random-walk algorithm,” Phys. Rev. A 67(4), 042316 (2003).
[Crossref]

2002 (1)

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[Crossref]

2001 (1)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref]

1998 (1)

P. Grangier, J. A. Levenson, and J.-P. Poizat, “Quantum non-demolition measurements in optics,” Nature 396(6711), 537–542 (1998).
[Crossref]

1993 (1)

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48(2), 1687–1690 (1993).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Aadhi, A.

Aharonov, Y.

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48(2), 1687–1690 (1993).
[Crossref]

Ahmed, N.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Ahn, T.-K.

G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, “Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems,” Nature 446(7137), 782–786 (2007).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Alt, W.

M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, “Quantum walk in position space with single optically trapped atoms,” Science 325(5937), 174–177 (2009).
[Crossref]

Ambainis, A.

A. Ambainis, “Quantum walks and their algorithmic applications,” Int. J. Quantum Inform. 01(04), 507–518 (2003).
[Crossref]

Andersson, E.

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: A quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104(5), 050502 (2010).
[Crossref]

Aolita, L.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

Arlt, J.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[Crossref]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref]

Asbóth, J. K.

S. Mugel, A. Celi, P. Massignan, J. K. Asbóth, M. Lewenstein, and C. Lobo, “Topological bound states of a quantum walk with cold atoms,” Phys. Rev. A 94(2), 023631 (2016).
[Crossref]

Aspuruguzik, A.

A. Aspuruguzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8(4), 285–291 (2012).
[Crossref]

Aspuru-Guzik, A.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Commun. 3(1), 882 (2012).
[Crossref]

M. A. Broome, A. Fedrizzi, B. P. Lanyon, I. Kassal, A. Aspuru-Guzik, and A. G. White, “Discrete single-photon quantum walks with tunable decoherence,” Phys. Rev. Lett. 104(15), 153602 (2010).
[Crossref]

M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, “Environment-assisted quantum walks in photosynthetic energy transfer,” J. Chem. Phys. 129(17), 174106 (2008).
[Crossref]

Balu, R.

N. P. Kumar, R. Balu, R. Laflamme, and C. M. Chandrashekar, “Bounds on the dynamics of periodic quantum walks and emergence of the gapless and gapped dirac equation,” Phys. Rev. A 97(1), 012116 (2018).
[Crossref]

Barreiro, J. T.

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008).
[Crossref]

Beausoleil, R. G.

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Bellini, M.

L. S. Costanzo, A. S. Coelho, N. Biagi, J. Fiurášek, M. Bellini, and A. Zavatta, “Measurement-induced strong kerr nonlinearity for weak quantum states of light,” Phys. Rev. Lett. 119(1), 013601 (2017).
[Crossref]

Berg, E.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Commun. 3(1), 882 (2012).
[Crossref]

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82(3), 033429 (2010).
[Crossref]

Bernet, S.

Biagi, N.

L. S. Costanzo, A. S. Coelho, N. Biagi, J. Fiurášek, M. Bellini, and A. Zavatta, “Measurement-induced strong kerr nonlinearity for weak quantum states of light,” Phys. Rev. Lett. 119(1), 013601 (2017).
[Crossref]

Bian, Z.

L. Xiao, X. Qiu, K. Wang, Z. Bian, X. Zhan, H. Obuse, B. C. Sanders, W. Yi, and P. Xue, “Higher winding number in a nonunitary photonic quantum walk,” Phys. Rev. A 98(6), 063847 (2018).
[Crossref]

Bian, Z. H.

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13(11), 1117–1123 (2017).
[Crossref]

Bian, Z.-H.

Z.-H. Bian, J. Li, X. Zhan, J. Twamley, and P. Xue, “Experimental implementation of a quantum walk on a circle with single photons,” Phys. Rev. A 95(5), 052338 (2017).
[Crossref]

Blankenship, R. E.

G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, “Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems,” Nature 446(7137), 782–786 (2007).
[Crossref]

Blatt, R.

F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. F. Roos, “Realization of a quantum walk with one and two trapped ions,” Phys. Rev. Lett. 104(10), 100503 (2010).
[Crossref]

Boileau, J. C.

C. A. Ryan, M. Laforest, J. C. Boileau, and R. Laflamme, “Experimental implementation of a discrete-time quantum random walk on an nmr quantum-information processor,” Phys. Rev. A 72(6), 062317 (2005).
[Crossref]

Braun, D.

N. Wiebe, D. Braun, and S. Lloyd, “Quantum algorithm for data fitting,” Phys. Rev. Lett. 109(5), 050505 (2012).
[Crossref]

Braunstein, S. L.

T. P. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock, and G. J. Milburn, “Quantum computation by communication,” New J. Phys. 8(2), 30 (2006).
[Crossref]

Brod, D. J.

D. J. Brod and J. Combes, “Passive cphase gate via cross-kerr nonlinearities,” Phys. Rev. Lett. 117(8), 080502 (2016).
[Crossref]

Bromberg, Y.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329(5998), 1500–1503 (2010).
[Crossref]

Broome, M. A.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Commun. 3(1), 882 (2012).
[Crossref]

M. A. Broome, A. Fedrizzi, B. P. Lanyon, I. Kassal, A. Aspuru-Guzik, and A. G. White, “Discrete single-photon quantum walks with tunable decoherence,” Phys. Rev. Lett. 104(15), 153602 (2010).
[Crossref]

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref]

Calhoun, T. R.

G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, “Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems,” Nature 446(7137), 782–786 (2007).
[Crossref]

Cardano, F.

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7(1), 11439 (2016).
[Crossref]

Cassemiro, K. N.

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: A quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104(5), 050502 (2010).
[Crossref]

Cataudella, V.

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7(1), 11439 (2016).
[Crossref]

Celi, A.

S. Mugel, A. Celi, P. Massignan, J. K. Asbóth, M. Lewenstein, and C. Lobo, “Topological bound states of a quantum walk with cold atoms,” Phys. Rev. A 94(2), 023631 (2016).
[Crossref]

Chandrashekar, C. M.

N. P. Kumar, R. Balu, R. Laflamme, and C. M. Chandrashekar, “Bounds on the dynamics of periodic quantum walks and emergence of the gapless and gapped dirac equation,” Phys. Rev. A 97(1), 012116 (2018).
[Crossref]

Chen, J.-Y.

H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, L.-F. Qiao, A.-L. Yang, and X.-M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4(5), eaat3174 (2018).
[Crossref]

Chen, T.

B. Wang, T. Chen, and X. Zhang, “Experimental observation of topologically protected bound states with vanishing chern numbers in a two-dimensional quantum walk,” Phys. Rev. Lett. 121(10), 100501 (2018).
[Crossref]

Cheng, Y.-C.

G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, “Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems,” Nature 446(7137), 782–786 (2007).
[Crossref]

Childs, A. M.

A. M. Childs, “Universal computation by quantum walk,” Phys. Rev. Lett. 102(18), 180501 (2009).
[Crossref]

A. M. Childs and J. Goldstone, “Spatial search by quantum walk,” Phys. Rev. A 70(2), 022314 (2004).
[Crossref]

Choi, J.-M.

M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, “Quantum walk in position space with single optically trapped atoms,” Science 325(5937), 174–177 (2009).
[Crossref]

Cizmar, T.

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[Crossref]

Coelho, A. S.

L. S. Costanzo, A. S. Coelho, N. Biagi, J. Fiurášek, M. Bellini, and A. Zavatta, “Measurement-induced strong kerr nonlinearity for weak quantum states of light,” Phys. Rev. Lett. 119(1), 013601 (2017).
[Crossref]

Combes, J.

D. J. Brod and J. Combes, “Passive cphase gate via cross-kerr nonlinearities,” Phys. Rev. Lett. 117(8), 080502 (2016).
[Crossref]

Cooper, S.

N. B. Lovett, S. Cooper, M. Everitt, M. Trevers, and V. Kendon, “Universal quantum computation using the discrete-time quantum walk,” Phys. Rev. A 81(4), 042330 (2010).
[Crossref]

Costanzo, L. S.

L. S. Costanzo, A. S. Coelho, N. Biagi, J. Fiurášek, M. Bellini, and A. Zavatta, “Measurement-induced strong kerr nonlinearity for weak quantum states of light,” Phys. Rev. Lett. 119(1), 013601 (2017).
[Crossref]

D’Ambrosio, V.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

Davidovich, L.

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48(2), 1687–1690 (1993).
[Crossref]

De Filippis, G.

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7(1), 11439 (2016).
[Crossref]

de Lisio, C.

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7(1), 11439 (2016).
[Crossref]

De Martini, F.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through hong-mandel coalescence,” Nat. Photonics 3(12), 720–723 (2009).
[Crossref]

Del Re, L.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

Demler, E.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Commun. 3(1), 882 (2012).
[Crossref]

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82(3), 033429 (2010).
[Crossref]

Deng, F.-G.

F.-G. Deng, “Optimal nonlocal multipartite entanglement concentration based on projection measurements,” Phys. Rev. A 85(2), 022311 (2012).
[Crossref]

Y.-B. Sheng, F.-G. Deng, and G. L. Long, “Complete hyperentangled-bell-state analysis for quantum communication,” Phys. Rev. A 82(3), 032318 (2010).
[Crossref]

Y.-B. Sheng, F.-G. Deng, and H.-Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-kerr nonlinearity,” Phys. Rev. A 77(4), 042308 (2008).
[Crossref]

Dennis, M. R.

M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6(2), 118–121 (2010).
[Crossref]

Dholakia, K.

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[Crossref]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[Crossref]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref]

Dolinar, S.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Douglas, B. L.

B. L. Douglas and J. B. Wang, “A classical approach to the graph isomorphism problem using quantum walks,” J. Phys. A: Math. Theor. 41(7), 075303 (2008).
[Crossref]

Du, J.

J. Du, H. Li, X. Xu, M. Shi, J. Wu, X. Zhou, and R. Han, “Experimental implementation of the quantum random-walk algorithm,” Phys. Rev. A 67(4), 042316 (2003).
[Crossref]

Ebrahim-Zadeh, M.

Enderlein, M.

H. Schmitz, R. Matjeschk, C. Schneider, J. Glueckert, M. Enderlein, T. Huber, and T. Schaetz, “Quantum walk of a trapped ion in phase space,” Phys. Rev. Lett. 103(9), 090504 (2009).
[Crossref]

Engel, G. S.

G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, “Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems,” Nature 446(7137), 782–786 (2007).
[Crossref]

Everitt, M.

N. B. Lovett, S. Cooper, M. Everitt, M. Trevers, and V. Kendon, “Universal quantum computation using the discrete-time quantum walk,” Phys. Rev. A 81(4), 042330 (2010).
[Crossref]

Fazal, I. M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Fedrizzi, A.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Commun. 3(1), 882 (2012).
[Crossref]

M. A. Broome, A. Fedrizzi, B. P. Lanyon, I. Kassal, A. Aspuru-Guzik, and A. G. White, “Discrete single-photon quantum walks with tunable decoherence,” Phys. Rev. Lett. 104(15), 153602 (2010).
[Crossref]

Feng, Z.

H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, L.-F. Qiao, A.-L. Yang, and X.-M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4(5), eaat3174 (2018).
[Crossref]

Fiurášek, J.

L. S. Costanzo, A. S. Coelho, N. Biagi, J. Fiurášek, M. Bellini, and A. Zavatta, “Measurement-induced strong kerr nonlinearity for weak quantum states of light,” Phys. Rev. Lett. 119(1), 013601 (2017).
[Crossref]

Fleming, G. R.

G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, “Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems,” Nature 446(7137), 782–786 (2007).
[Crossref]

Forbes, A.

S. K. Goyal, F. S. Roux, A. Forbes, and T. Konrad, “Implementing quantum walks using orbital angular momentum of classical light,” Phys. Rev. Lett. 110(26), 263602 (2013).
[Crossref]

Förster, L.

M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, “Quantum walk in position space with single optically trapped atoms,” Science 325(5937), 174–177 (2009).
[Crossref]

Fry, D.

K. Poulios, R. Keil, D. Fry, J. D. A. Meinecke, J. C. F. Matthews, A. Politi, M. Lobino, M. Gräfe, M. Heinrich, S. Nolte, A. Szameit, and J. L. O’Brien, “Quantum walks of correlated photon pairs in two-dimensional waveguide arrays,” Phys. Rev. Lett. 112(14), 143604 (2014).
[Crossref]

Fürhapter, S.

Gábris, A.

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: A quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104(5), 050502 (2010).
[Crossref]

Gao, J.

H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, L.-F. Qiao, A.-L. Yang, and X.-M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4(5), eaat3174 (2018).
[Crossref]

Gardiner, C. W.

C. W. Gardiner and H. Haken, Quantum noise, vol. 2 (Springer, 1991).

Gerritsma, R.

F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. F. Roos, “Realization of a quantum walk with one and two trapped ions,” Phys. Rev. Lett. 104(10), 100503 (2010).
[Crossref]

Ghobadi, R.

T. Wang, H. W. Lau, H. Kaviani, R. Ghobadi, and C. Simon, “Strong micro-macro entanglement from a weak cross-kerr nonlinearity,” Phys. Rev. A 92(1), 012316 (2015).
[Crossref]

Glueckert, J.

H. Schmitz, R. Matjeschk, C. Schneider, J. Glueckert, M. Enderlein, T. Huber, and T. Schaetz, “Quantum walk of a trapped ion in phase space,” Phys. Rev. Lett. 103(9), 090504 (2009).
[Crossref]

Goldstone, J.

A. M. Childs and J. Goldstone, “Spatial search by quantum walk,” Phys. Rev. A 70(2), 022314 (2004).
[Crossref]

Goyal, S. K.

S. K. Goyal, F. S. Roux, A. Forbes, and T. Konrad, “Implementing quantum walks using orbital angular momentum of classical light,” Phys. Rev. Lett. 110(26), 263602 (2013).
[Crossref]

Gräfe, M.

K. Poulios, R. Keil, D. Fry, J. D. A. Meinecke, J. C. F. Matthews, A. Politi, M. Lobino, M. Gräfe, M. Heinrich, S. Nolte, A. Szameit, and J. L. O’Brien, “Quantum walks of correlated photon pairs in two-dimensional waveguide arrays,” Phys. Rev. Lett. 112(14), 143604 (2014).
[Crossref]

Grangier, P.

P. Grangier, J. A. Levenson, and J.-P. Poizat, “Quantum non-demolition measurements in optics,” Nature 396(6711), 537–542 (1998).
[Crossref]

Guo, G.-C.

P. Zhang, B.-H. Liu, R.-F. Liu, H.-R. Li, F.-L. Li, and G.-C. Guo, “Implementation of one-dimensional quantum walks on spin-orbital angular momentum space of photons,” Phys. Rev. A 81(5), 052322 (2010).
[Crossref]

P. Zhang, X.-F. Ren, X.-B. Zou, B.-H. Liu, Y.-F. Huang, and G.-C. Guo, “Demonstration of one-dimensional quantum random walks using orbital angular momentum of photons,” Phys. Rev. A 75(5), 052310 (2007).
[Crossref]

Hagman, H.

N. Satapathy, H. Hagman, M. Zelán, A. Kastberg, and H. Ramachandran, “Theoretical investigations of quantum walks by cold atoms in a double optical lattice,” Phys. Rev. A 80(1), 012302 (2009).
[Crossref]

Haken, H.

C. W. Gardiner and H. Haken, Quantum noise, vol. 2 (Springer, 1991).

Han, R.

J. Du, H. Li, X. Xu, M. Shi, J. Wu, X. Zhou, and R. Han, “Experimental implementation of the quantum random-walk algorithm,” Phys. Rev. A 67(4), 042316 (2003).
[Crossref]

Heinrich, M.

K. Poulios, R. Keil, D. Fry, J. D. A. Meinecke, J. C. F. Matthews, A. Politi, M. Lobino, M. Gräfe, M. Heinrich, S. Nolte, A. Szameit, and J. L. O’Brien, “Quantum walks of correlated photon pairs in two-dimensional waveguide arrays,” Phys. Rev. Lett. 112(14), 143604 (2014).
[Crossref]

Huang, H.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Huang, Y.-F.

P. Zhang, X.-F. Ren, X.-B. Zou, B.-H. Liu, Y.-F. Huang, and G.-C. Guo, “Demonstration of one-dimensional quantum random walks using orbital angular momentum of photons,” Phys. Rev. A 75(5), 052310 (2007).
[Crossref]

Huber, T.

H. Schmitz, R. Matjeschk, C. Schneider, J. Glueckert, M. Enderlein, T. Huber, and T. Schaetz, “Quantum walk of a trapped ion in phase space,” Phys. Rev. Lett. 103(9), 090504 (2009).
[Crossref]

Ismail, N.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329(5998), 1500–1503 (2010).
[Crossref]

Jack, B.

M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6(2), 118–121 (2010).
[Crossref]

Jesacher, A.

Jex, I.

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: A quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104(5), 050502 (2010).
[Crossref]

Jin, X.-M.

H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, L.-F. Qiao, A.-L. Yang, and X.-M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4(5), eaat3174 (2018).
[Crossref]

Karimi, E.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through hong-mandel coalescence,” Nat. Photonics 3(12), 720–723 (2009).
[Crossref]

Karski, M.

M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, “Quantum walk in position space with single optically trapped atoms,” Science 325(5937), 174–177 (2009).
[Crossref]

Kassal, I.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Commun. 3(1), 882 (2012).
[Crossref]

M. A. Broome, A. Fedrizzi, B. P. Lanyon, I. Kassal, A. Aspuru-Guzik, and A. G. White, “Discrete single-photon quantum walks with tunable decoherence,” Phys. Rev. Lett. 104(15), 153602 (2010).
[Crossref]

Kastberg, A.

N. Satapathy, H. Hagman, M. Zelán, A. Kastberg, and H. Ramachandran, “Theoretical investigations of quantum walks by cold atoms in a double optical lattice,” Phys. Rev. A 80(1), 012302 (2009).
[Crossref]

Kaviani, H.

T. Wang, H. W. Lau, H. Kaviani, R. Ghobadi, and C. Simon, “Strong micro-macro entanglement from a weak cross-kerr nonlinearity,” Phys. Rev. A 92(1), 012316 (2015).
[Crossref]

Kawakami, N.

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13(11), 1117–1123 (2017).
[Crossref]

H. Obuse and N. Kawakami, “Topological phases and delocalization of quantum walks in random environments,” Phys. Rev. B 84(19), 195139 (2011).
[Crossref]

Keil, R.

K. Poulios, R. Keil, D. Fry, J. D. A. Meinecke, J. C. F. Matthews, A. Politi, M. Lobino, M. Gräfe, M. Heinrich, S. Nolte, A. Szameit, and J. L. O’Brien, “Quantum walks of correlated photon pairs in two-dimensional waveguide arrays,” Phys. Rev. Lett. 112(14), 143604 (2014).
[Crossref]

Kempe, J.

N. Shenvi, J. Kempe, and K. B. Whaley, “Quantum random-walk search algorithm,” Phys. Rev. A 67(5), 052307 (2003).
[Crossref]

Kendon, V.

N. B. Lovett, S. Cooper, M. Everitt, M. Trevers, and V. Kendon, “Universal quantum computation using the discrete-time quantum walk,” Phys. Rev. A 81(4), 042330 (2010).
[Crossref]

Kim, D.

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13(11), 1117–1123 (2017).
[Crossref]

King, R. P.

M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6(2), 118–121 (2010).
[Crossref]

Kirchmair, G.

F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. F. Roos, “Realization of a quantum walk with one and two trapped ions,” Phys. Rev. Lett. 104(10), 100503 (2010).
[Crossref]

Kitagawa, T.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Commun. 3(1), 882 (2012).
[Crossref]

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82(3), 033429 (2010).
[Crossref]

Konrad, T.

S. K. Goyal, F. S. Roux, A. Forbes, and T. Konrad, “Implementing quantum walks using orbital angular momentum of classical light,” Phys. Rev. Lett. 110(26), 263602 (2013).
[Crossref]

Kumar, N. P.

N. P. Kumar, R. Balu, R. Laflamme, and C. M. Chandrashekar, “Bounds on the dynamics of periodic quantum walks and emergence of the gapless and gapped dirac equation,” Phys. Rev. A 97(1), 012116 (2018).
[Crossref]

Kumar, S. C.

Kwek, L. C.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

Kwiat, P. G.

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008).
[Crossref]

Laflamme, R.

N. P. Kumar, R. Balu, R. Laflamme, and C. M. Chandrashekar, “Bounds on the dynamics of periodic quantum walks and emergence of the gapless and gapped dirac equation,” Phys. Rev. A 97(1), 012116 (2018).
[Crossref]

C. A. Ryan, M. Laforest, J. C. Boileau, and R. Laflamme, “Experimental implementation of a discrete-time quantum random walk on an nmr quantum-information processor,” Phys. Rev. A 72(6), 062317 (2005).
[Crossref]

Laforest, M.

C. A. Ryan, M. Laforest, J. C. Boileau, and R. Laflamme, “Experimental implementation of a discrete-time quantum random walk on an nmr quantum-information processor,” Phys. Rev. A 72(6), 062317 (2005).
[Crossref]

Lahini, Y.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329(5998), 1500–1503 (2010).
[Crossref]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
[Crossref]

Lai, P.-C.

H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, L.-F. Qiao, A.-L. Yang, and X.-M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4(5), eaat3174 (2018).
[Crossref]

Lanyon, B. P.

M. A. Broome, A. Fedrizzi, B. P. Lanyon, I. Kassal, A. Aspuru-Guzik, and A. G. White, “Discrete single-photon quantum walks with tunable decoherence,” Phys. Rev. Lett. 104(15), 153602 (2010).
[Crossref]

Lau, H. W.

T. Wang, H. W. Lau, H. Kaviani, R. Ghobadi, and C. Simon, “Strong micro-macro entanglement from a weak cross-kerr nonlinearity,” Phys. Rev. A 92(1), 012316 (2015).
[Crossref]

Levenson, J. A.

P. Grangier, J. A. Levenson, and J.-P. Poizat, “Quantum non-demolition measurements in optics,” Nature 396(6711), 537–542 (1998).
[Crossref]

Lewenstein, M.

S. Mugel, A. Celi, P. Massignan, J. K. Asbóth, M. Lewenstein, and C. Lobo, “Topological bound states of a quantum walk with cold atoms,” Phys. Rev. A 94(2), 023631 (2016).
[Crossref]

Li, F.-L.

P. Zhang, B.-H. Liu, R.-F. Liu, H.-R. Li, F.-L. Li, and G.-C. Guo, “Implementation of one-dimensional quantum walks on spin-orbital angular momentum space of photons,” Phys. Rev. A 81(5), 052322 (2010).
[Crossref]

Li, H.

J. Du, H. Li, X. Xu, M. Shi, J. Wu, X. Zhou, and R. Han, “Experimental implementation of the quantum random-walk algorithm,” Phys. Rev. A 67(4), 042316 (2003).
[Crossref]

Li, H.-R.

P. Zhang, B.-H. Liu, R.-F. Liu, H.-R. Li, F.-L. Li, and G.-C. Guo, “Implementation of one-dimensional quantum walks on spin-orbital angular momentum space of photons,” Phys. Rev. A 81(5), 052322 (2010).
[Crossref]

Li, J.

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13(11), 1117–1123 (2017).
[Crossref]

Z.-H. Bian, J. Li, X. Zhan, J. Twamley, and P. Xue, “Experimental implementation of a quantum walk on a circle with single photons,” Phys. Rev. A 95(5), 052338 (2017).
[Crossref]

Q. Lin and J. Li, “Quantum control gates with weak cross-kerr nonlinearity,” Phys. Rev. A 79(2), 022301 (2009).
[Crossref]

Li, Y.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

Lin, Q.

Q. Lin and J. Li, “Quantum control gates with weak cross-kerr nonlinearity,” Phys. Rev. A 79(2), 022301 (2009).
[Crossref]

Lin, X.-F.

H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, L.-F. Qiao, A.-L. Yang, and X.-M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4(5), eaat3174 (2018).
[Crossref]

Liu, B.-H.

P. Zhang, B.-H. Liu, R.-F. Liu, H.-R. Li, F.-L. Li, and G.-C. Guo, “Implementation of one-dimensional quantum walks on spin-orbital angular momentum space of photons,” Phys. Rev. A 81(5), 052322 (2010).
[Crossref]

P. Zhang, X.-F. Ren, X.-B. Zou, B.-H. Liu, Y.-F. Huang, and G.-C. Guo, “Demonstration of one-dimensional quantum random walks using orbital angular momentum of photons,” Phys. Rev. A 75(5), 052310 (2007).
[Crossref]

Liu, R.-F.

P. Zhang, B.-H. Liu, R.-F. Liu, H.-R. Li, F.-L. Li, and G.-C. Guo, “Implementation of one-dimensional quantum walks on spin-orbital angular momentum space of photons,” Phys. Rev. A 81(5), 052322 (2010).
[Crossref]

Lloyd, S.

N. Wiebe, D. Braun, and S. Lloyd, “Quantum algorithm for data fitting,” Phys. Rev. Lett. 109(5), 050505 (2012).
[Crossref]

M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, “Environment-assisted quantum walks in photosynthetic energy transfer,” J. Chem. Phys. 129(17), 174106 (2008).
[Crossref]

Lobino, M.

K. Poulios, R. Keil, D. Fry, J. D. A. Meinecke, J. C. F. Matthews, A. Politi, M. Lobino, M. Gräfe, M. Heinrich, S. Nolte, A. Szameit, and J. L. O’Brien, “Quantum walks of correlated photon pairs in two-dimensional waveguide arrays,” Phys. Rev. Lett. 112(14), 143604 (2014).
[Crossref]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329(5998), 1500–1503 (2010).
[Crossref]

Lobo, C.

S. Mugel, A. Celi, P. Massignan, J. K. Asbóth, M. Lewenstein, and C. Lobo, “Topological bound states of a quantum walk with cold atoms,” Phys. Rev. A 94(2), 023631 (2016).
[Crossref]

Long, G. L.

Y.-B. Sheng, F.-G. Deng, and G. L. Long, “Complete hyperentangled-bell-state analysis for quantum communication,” Phys. Rev. A 82(3), 032318 (2010).
[Crossref]

Lovett, N. B.

N. B. Lovett, S. Cooper, M. Everitt, M. Trevers, and V. Kendon, “Universal quantum computation using the discrete-time quantum walk,” Phys. Rev. A 81(4), 042330 (2010).
[Crossref]

MacDonald, M. P.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[Crossref]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref]

Maffei, M.

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7(1), 11439 (2016).
[Crossref]

Mancal, T.

G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, “Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems,” Nature 446(7137), 782–786 (2007).
[Crossref]

Marrucci, L.

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7(1), 11439 (2016).
[Crossref]

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through hong-mandel coalescence,” Nat. Photonics 3(12), 720–723 (2009).
[Crossref]

Massa, F.

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7(1), 11439 (2016).
[Crossref]

Massignan, P.

S. Mugel, A. Celi, P. Massignan, J. K. Asbóth, M. Lewenstein, and C. Lobo, “Topological bound states of a quantum walk with cold atoms,” Phys. Rev. A 94(2), 023631 (2016).
[Crossref]

Matjeschk, R.

H. Schmitz, R. Matjeschk, C. Schneider, J. Glueckert, M. Enderlein, T. Huber, and T. Schaetz, “Quantum walk of a trapped ion in phase space,” Phys. Rev. Lett. 103(9), 090504 (2009).
[Crossref]

Matsuda, N.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329(5998), 1500–1503 (2010).
[Crossref]

Matthews, J. C. F.

K. Poulios, R. Keil, D. Fry, J. D. A. Meinecke, J. C. F. Matthews, A. Politi, M. Lobino, M. Gräfe, M. Heinrich, S. Nolte, A. Szameit, and J. L. O’Brien, “Quantum walks of correlated photon pairs in two-dimensional waveguide arrays,” Phys. Rev. Lett. 112(14), 143604 (2014).
[Crossref]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329(5998), 1500–1503 (2010).
[Crossref]

Maurer, C.

Meinecke, J. D. A.

K. Poulios, R. Keil, D. Fry, J. D. A. Meinecke, J. C. F. Matthews, A. Politi, M. Lobino, M. Gräfe, M. Heinrich, S. Nolte, A. Szameit, and J. L. O’Brien, “Quantum walks of correlated photon pairs in two-dimensional waveguide arrays,” Phys. Rev. Lett. 112(14), 143604 (2014).
[Crossref]

Meschede, D.

M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, “Quantum walk in position space with single optically trapped atoms,” Science 325(5937), 174–177 (2009).
[Crossref]

Milburn, G. J.

T. P. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock, and G. J. Milburn, “Quantum computation by communication,” New J. Phys. 8(2), 30 (2006).
[Crossref]

Mochizuki, K.

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13(11), 1117–1123 (2017).
[Crossref]

Mohseni, M.

M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, “Environment-assisted quantum walks in photosynthetic energy transfer,” J. Chem. Phys. 129(17), 174106 (2008).
[Crossref]

Morandotti, R.

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
[Crossref]

Mosley, P. J.

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: A quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104(5), 050502 (2010).
[Crossref]

Mugel, S.

S. Mugel, A. Celi, P. Massignan, J. K. Asbóth, M. Lewenstein, and C. Lobo, “Topological bound states of a quantum walk with cold atoms,” Phys. Rev. A 94(2), 023631 (2016).
[Crossref]

Munro, W. J.

T. P. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock, and G. J. Milburn, “Quantum computation by communication,” New J. Phys. 8(2), 30 (2006).
[Crossref]

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
[Crossref]

W. J. Munro, K. Nemoto, and T. P. Spiller, “Weak nonlinearities: a new route to optical quantum computation,” New J. Phys. 7, 137 (2005).
[Crossref]

K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-not gate,” Phys. Rev. Lett. 93(25), 250502 (2004).
[Crossref]

Nagali, E.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through hong-mandel coalescence,” Nat. Photonics 3(12), 720–723 (2009).
[Crossref]

Nemoto, K.

T. P. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock, and G. J. Milburn, “Quantum computation by communication,” New J. Phys. 8(2), 30 (2006).
[Crossref]

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
[Crossref]

W. J. Munro, K. Nemoto, and T. P. Spiller, “Weak nonlinearities: a new route to optical quantum computation,” New J. Phys. 7, 137 (2005).
[Crossref]

K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-not gate,” Phys. Rev. Lett. 93(25), 250502 (2004).
[Crossref]

Nolte, S.

K. Poulios, R. Keil, D. Fry, J. D. A. Meinecke, J. C. F. Matthews, A. Politi, M. Lobino, M. Gräfe, M. Heinrich, S. Nolte, A. Szameit, and J. L. O’Brien, “Quantum walks of correlated photon pairs in two-dimensional waveguide arrays,” Phys. Rev. Lett. 112(14), 143604 (2014).
[Crossref]

O’Brien, J. L.

K. Poulios, R. Keil, D. Fry, J. D. A. Meinecke, J. C. F. Matthews, A. Politi, M. Lobino, M. Gräfe, M. Heinrich, S. Nolte, A. Szameit, and J. L. O’Brien, “Quantum walks of correlated photon pairs in two-dimensional waveguide arrays,” Phys. Rev. Lett. 112(14), 143604 (2014).
[Crossref]

O’Holleran, K.

M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6(2), 118–121 (2010).
[Crossref]

OBrien, J. L.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329(5998), 1500–1503 (2010).
[Crossref]

Obuse, H.

L. Xiao, X. Qiu, K. Wang, Z. Bian, X. Zhan, H. Obuse, B. C. Sanders, W. Yi, and P. Xue, “Higher winding number in a nonunitary photonic quantum walk,” Phys. Rev. A 98(6), 063847 (2018).
[Crossref]

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13(11), 1117–1123 (2017).
[Crossref]

H. Obuse and N. Kawakami, “Topological phases and delocalization of quantum walks in random environments,” Phys. Rev. B 84(19), 195139 (2011).
[Crossref]

Padgett, M. J.

M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6(2), 118–121 (2010).
[Crossref]

Paterson, L.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[Crossref]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref]

Perets, H. B.

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
[Crossref]

Peruzzo, A.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329(5998), 1500–1503 (2010).
[Crossref]

Petruccione, F.

M. Schuld, I. Sinayskiy, and F. Petruccione, “The quest for a quantum neural network,” Quantum Inf. Process. 13(11), 2567–2586 (2014).
[Crossref]

Piccirillo, B.

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7(1), 11439 (2016).
[Crossref]

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through hong-mandel coalescence,” Nat. Photonics 3(12), 720–723 (2009).
[Crossref]

Poizat, J.-P.

P. Grangier, J. A. Levenson, and J.-P. Poizat, “Quantum non-demolition measurements in optics,” Nature 396(6711), 537–542 (1998).
[Crossref]

Politi, A.

K. Poulios, R. Keil, D. Fry, J. D. A. Meinecke, J. C. F. Matthews, A. Politi, M. Lobino, M. Gräfe, M. Heinrich, S. Nolte, A. Szameit, and J. L. O’Brien, “Quantum walks of correlated photon pairs in two-dimensional waveguide arrays,” Phys. Rev. Lett. 112(14), 143604 (2014).
[Crossref]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329(5998), 1500–1503 (2010).
[Crossref]

Potocek, V.

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: A quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104(5), 050502 (2010).
[Crossref]

Poulios, K.

K. Poulios, R. Keil, D. Fry, J. D. A. Meinecke, J. C. F. Matthews, A. Politi, M. Lobino, M. Gräfe, M. Heinrich, S. Nolte, A. Szameit, and J. L. O’Brien, “Quantum walks of correlated photon pairs in two-dimensional waveguide arrays,” Phys. Rev. Lett. 112(14), 143604 (2014).
[Crossref]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329(5998), 1500–1503 (2010).
[Crossref]

Pozzi, F.

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
[Crossref]

Qiao, L.-F.

H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, L.-F. Qiao, A.-L. Yang, and X.-M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4(5), eaat3174 (2018).
[Crossref]

Qiu, X.

L. Xiao, X. Qiu, K. Wang, Z. Bian, X. Zhan, H. Obuse, B. C. Sanders, W. Yi, and P. Xue, “Higher winding number in a nonunitary photonic quantum walk,” Phys. Rev. A 98(6), 063847 (2018).
[Crossref]

Ramachandran, H.

N. Satapathy, H. Hagman, M. Zelán, A. Kastberg, and H. Ramachandran, “Theoretical investigations of quantum walks by cold atoms in a double optical lattice,” Phys. Rev. A 80(1), 012302 (2009).
[Crossref]

Read, E. L.

G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, “Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems,” Nature 446(7137), 782–786 (2007).
[Crossref]

Rebentrost, P.

M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, “Environment-assisted quantum walks in photosynthetic energy transfer,” J. Chem. Phys. 129(17), 174106 (2008).
[Crossref]

Ren, X.-F.

P. Zhang, X.-F. Ren, X.-B. Zou, B.-H. Liu, Y.-F. Huang, and G.-C. Guo, “Demonstration of one-dimensional quantum random walks using orbital angular momentum of photons,” Phys. Rev. A 75(5), 052310 (2007).
[Crossref]

Ren, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Ritsch-Marte, M.

Roos, C. F.

F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. F. Roos, “Realization of a quantum walk with one and two trapped ions,” Phys. Rev. Lett. 104(10), 100503 (2010).
[Crossref]

Roux, F. S.

S. K. Goyal, F. S. Roux, A. Forbes, and T. Konrad, “Implementing quantum walks using orbital angular momentum of classical light,” Phys. Rev. Lett. 110(26), 263602 (2013).
[Crossref]

Rudner, M. S.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Commun. 3(1), 882 (2012).
[Crossref]

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82(3), 033429 (2010).
[Crossref]

Ryan, C. A.

C. A. Ryan, M. Laforest, J. C. Boileau, and R. Laflamme, “Experimental implementation of a discrete-time quantum random walk on an nmr quantum-information processor,” Phys. Rev. A 72(6), 062317 (2005).
[Crossref]

Samanta, G. K.

Sanders, B. C.

L. Xiao, X. Qiu, K. Wang, Z. Bian, X. Zhan, H. Obuse, B. C. Sanders, W. Yi, and P. Xue, “Higher winding number in a nonunitary photonic quantum walk,” Phys. Rev. A 98(6), 063847 (2018).
[Crossref]

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13(11), 1117–1123 (2017).
[Crossref]

Sansoni, L.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through hong-mandel coalescence,” Nat. Photonics 3(12), 720–723 (2009).
[Crossref]

Santamato, E.

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7(1), 11439 (2016).
[Crossref]

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through hong-mandel coalescence,” Nat. Photonics 3(12), 720–723 (2009).
[Crossref]

Satapathy, N.

N. Satapathy, H. Hagman, M. Zelán, A. Kastberg, and H. Ramachandran, “Theoretical investigations of quantum walks by cold atoms in a double optical lattice,” Phys. Rev. A 80(1), 012302 (2009).
[Crossref]

Schaetz, T.

H. Schmitz, R. Matjeschk, C. Schneider, J. Glueckert, M. Enderlein, T. Huber, and T. Schaetz, “Quantum walk of a trapped ion in phase space,” Phys. Rev. Lett. 103(9), 090504 (2009).
[Crossref]

Schmitz, H.

H. Schmitz, R. Matjeschk, C. Schneider, J. Glueckert, M. Enderlein, T. Huber, and T. Schaetz, “Quantum walk of a trapped ion in phase space,” Phys. Rev. Lett. 103(9), 090504 (2009).
[Crossref]

Schneider, C.

H. Schmitz, R. Matjeschk, C. Schneider, J. Glueckert, M. Enderlein, T. Huber, and T. Schaetz, “Quantum walk of a trapped ion in phase space,” Phys. Rev. Lett. 103(9), 090504 (2009).
[Crossref]

Schreiber, A.

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: A quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104(5), 050502 (2010).
[Crossref]

Schuld, M.

M. Schuld, I. Sinayskiy, and F. Petruccione, “The quest for a quantum neural network,” Quantum Inf. Process. 13(11), 2567–2586 (2014).
[Crossref]

Sciarrino, F.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through hong-mandel coalescence,” Nat. Photonics 3(12), 720–723 (2009).
[Crossref]

Sheng, Y.-B.

Y.-B. Sheng, F.-G. Deng, and G. L. Long, “Complete hyperentangled-bell-state analysis for quantum communication,” Phys. Rev. A 82(3), 032318 (2010).
[Crossref]

Y.-B. Sheng, F.-G. Deng, and H.-Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-kerr nonlinearity,” Phys. Rev. A 77(4), 042308 (2008).
[Crossref]

Shenvi, N.

N. Shenvi, J. Kempe, and K. B. Whaley, “Quantum random-walk search algorithm,” Phys. Rev. A 67(5), 052307 (2003).
[Crossref]

Shi, M.

J. Du, H. Li, X. Xu, M. Shi, J. Wu, X. Zhou, and R. Han, “Experimental implementation of the quantum random-walk algorithm,” Phys. Rev. A 67(4), 042316 (2003).
[Crossref]

Sibbett, W.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[Crossref]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref]

Silberberg, Y.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329(5998), 1500–1503 (2010).
[Crossref]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
[Crossref]

Silberhorn, C.

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: A quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104(5), 050502 (2010).
[Crossref]

Simon, C.

T. Wang, H. W. Lau, H. Kaviani, R. Ghobadi, and C. Simon, “Strong micro-macro entanglement from a weak cross-kerr nonlinearity,” Phys. Rev. A 92(1), 012316 (2015).
[Crossref]

Sinayskiy, I.

M. Schuld, I. Sinayskiy, and F. Petruccione, “The quest for a quantum neural network,” Quantum Inf. Process. 13(11), 2567–2586 (2014).
[Crossref]

Slussarenko, S.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

Solano, E.

F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. F. Roos, “Realization of a quantum walk with one and two trapped ions,” Phys. Rev. Lett. 104(10), 100503 (2010).
[Crossref]

Sorel, M.

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
[Crossref]

Spagnolo, N.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

Spiller, T. P.

T. P. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock, and G. J. Milburn, “Quantum computation by communication,” New J. Phys. 8(2), 30 (2006).
[Crossref]

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
[Crossref]

W. J. Munro, K. Nemoto, and T. P. Spiller, “Weak nonlinearities: a new route to optical quantum computation,” New J. Phys. 7, 137 (2005).
[Crossref]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Steffen, A.

M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, “Quantum walk in position space with single optically trapped atoms,” Science 325(5937), 174–177 (2009).
[Crossref]

Sun, K.

H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, L.-F. Qiao, A.-L. Yang, and X.-M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4(5), eaat3174 (2018).
[Crossref]

Szameit, A.

K. Poulios, R. Keil, D. Fry, J. D. A. Meinecke, J. C. F. Matthews, A. Politi, M. Lobino, M. Gräfe, M. Heinrich, S. Nolte, A. Szameit, and J. L. O’Brien, “Quantum walks of correlated photon pairs in two-dimensional waveguide arrays,” Phys. Rev. Lett. 112(14), 143604 (2014).
[Crossref]

Tang, H.

H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, L.-F. Qiao, A.-L. Yang, and X.-M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4(5), eaat3174 (2018).
[Crossref]

Thompson, M. G.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329(5998), 1500–1503 (2010).
[Crossref]

Trevers, M.

N. B. Lovett, S. Cooper, M. Everitt, M. Trevers, and V. Kendon, “Universal quantum computation using the discrete-time quantum walk,” Phys. Rev. A 81(4), 042330 (2010).
[Crossref]

Tur, M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Twamley, J.

Z.-H. Bian, J. Li, X. Zhan, J. Twamley, and P. Xue, “Experimental implementation of a quantum walk on a circle with single photons,” Phys. Rev. A 95(5), 052338 (2017).
[Crossref]

van Loock, P.

T. P. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock, and G. J. Milburn, “Quantum computation by communication,” New J. Phys. 8(2), 30 (2006).
[Crossref]

Volke-Sepulveda, K.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[Crossref]

Walborn, S. P.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

Walther, P.

A. Aspuruguzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8(4), 285–291 (2012).
[Crossref]

Wang, B.

B. Wang, T. Chen, and X. Zhang, “Experimental observation of topologically protected bound states with vanishing chern numbers in a two-dimensional quantum walk,” Phys. Rev. Lett. 121(10), 100501 (2018).
[Crossref]

Wang, C.-Y.

H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, L.-F. Qiao, A.-L. Yang, and X.-M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4(5), eaat3174 (2018).
[Crossref]

Wang, J.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Wang, J. B.

B. L. Douglas and J. B. Wang, “A classical approach to the graph isomorphism problem using quantum walks,” J. Phys. A: Math. Theor. 41(7), 075303 (2008).
[Crossref]

Wang, K.

L. Xiao, X. Qiu, K. Wang, Z. Bian, X. Zhan, H. Obuse, B. C. Sanders, W. Yi, and P. Xue, “Higher winding number in a nonunitary photonic quantum walk,” Phys. Rev. A 98(6), 063847 (2018).
[Crossref]

Wang, K. K.

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13(11), 1117–1123 (2017).
[Crossref]

Wang, T.

T. Wang, H. W. Lau, H. Kaviani, R. Ghobadi, and C. Simon, “Strong micro-macro entanglement from a weak cross-kerr nonlinearity,” Phys. Rev. A 92(1), 012316 (2015).
[Crossref]

Wang, X. P.

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13(11), 1117–1123 (2017).
[Crossref]

Wang, Y.

H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, L.-F. Qiao, A.-L. Yang, and X.-M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4(5), eaat3174 (2018).
[Crossref]

Wei, T.-C.

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008).
[Crossref]

Whaley, K. B.

N. Shenvi, J. Kempe, and K. B. Whaley, “Quantum random-walk search algorithm,” Phys. Rev. A 67(5), 052307 (2003).
[Crossref]

White, A. G.

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Commun. 3(1), 882 (2012).
[Crossref]

M. A. Broome, A. Fedrizzi, B. P. Lanyon, I. Kassal, A. Aspuru-Guzik, and A. G. White, “Discrete single-photon quantum walks with tunable decoherence,” Phys. Rev. Lett. 104(15), 153602 (2010).
[Crossref]

Widera, A.

M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, “Quantum walk in position space with single optically trapped atoms,” Science 325(5937), 174–177 (2009).
[Crossref]

Wiebe, N.

N. Wiebe, D. Braun, and S. Lloyd, “Quantum algorithm for data fitting,” Phys. Rev. Lett. 109(5), 050505 (2012).
[Crossref]

Willner, A. E.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Wörhoff, K.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329(5998), 1500–1503 (2010).
[Crossref]

Wu, J.

J. Du, H. Li, X. Xu, M. Shi, J. Wu, X. Zhou, and R. Han, “Experimental implementation of the quantum random-walk algorithm,” Phys. Rev. A 67(4), 042316 (2003).
[Crossref]

Xiao, L.

L. Xiao, X. Qiu, K. Wang, Z. Bian, X. Zhan, H. Obuse, B. C. Sanders, W. Yi, and P. Xue, “Higher winding number in a nonunitary photonic quantum walk,” Phys. Rev. A 98(6), 063847 (2018).
[Crossref]

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13(11), 1117–1123 (2017).
[Crossref]

Xu, X.

J. Du, H. Li, X. Xu, M. Shi, J. Wu, X. Zhou, and R. Han, “Experimental implementation of the quantum random-walk algorithm,” Phys. Rev. A 67(4), 042316 (2003).
[Crossref]

Xu, X.-Y.

H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, L.-F. Qiao, A.-L. Yang, and X.-M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4(5), eaat3174 (2018).
[Crossref]

Xue, P.

L. Xiao, X. Qiu, K. Wang, Z. Bian, X. Zhan, H. Obuse, B. C. Sanders, W. Yi, and P. Xue, “Higher winding number in a nonunitary photonic quantum walk,” Phys. Rev. A 98(6), 063847 (2018).
[Crossref]

Z.-H. Bian, J. Li, X. Zhan, J. Twamley, and P. Xue, “Experimental implementation of a quantum walk on a circle with single photons,” Phys. Rev. A 95(5), 052338 (2017).
[Crossref]

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13(11), 1117–1123 (2017).
[Crossref]

Yan, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yang, A.-L.

H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, L.-F. Qiao, A.-L. Yang, and X.-M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4(5), eaat3174 (2018).
[Crossref]

Yang, J.-Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yi, W.

L. Xiao, X. Qiu, K. Wang, Z. Bian, X. Zhan, H. Obuse, B. C. Sanders, W. Yi, and P. Xue, “Higher winding number in a nonunitary photonic quantum walk,” Phys. Rev. A 98(6), 063847 (2018).
[Crossref]

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13(11), 1117–1123 (2017).
[Crossref]

Yue, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Zagury, N.

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48(2), 1687–1690 (1993).
[Crossref]

Zähringer, F.

F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. F. Roos, “Realization of a quantum walk with one and two trapped ions,” Phys. Rev. Lett. 104(10), 100503 (2010).
[Crossref]

Zavatta, A.

L. S. Costanzo, A. S. Coelho, N. Biagi, J. Fiurášek, M. Bellini, and A. Zavatta, “Measurement-induced strong kerr nonlinearity for weak quantum states of light,” Phys. Rev. Lett. 119(1), 013601 (2017).
[Crossref]

Zelán, M.

N. Satapathy, H. Hagman, M. Zelán, A. Kastberg, and H. Ramachandran, “Theoretical investigations of quantum walks by cold atoms in a double optical lattice,” Phys. Rev. A 80(1), 012302 (2009).
[Crossref]

Zhan, X.

L. Xiao, X. Qiu, K. Wang, Z. Bian, X. Zhan, H. Obuse, B. C. Sanders, W. Yi, and P. Xue, “Higher winding number in a nonunitary photonic quantum walk,” Phys. Rev. A 98(6), 063847 (2018).
[Crossref]

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13(11), 1117–1123 (2017).
[Crossref]

Z.-H. Bian, J. Li, X. Zhan, J. Twamley, and P. Xue, “Experimental implementation of a quantum walk on a circle with single photons,” Phys. Rev. A 95(5), 052338 (2017).
[Crossref]

Zhang, P.

P. Zhang, B.-H. Liu, R.-F. Liu, H.-R. Li, F.-L. Li, and G.-C. Guo, “Implementation of one-dimensional quantum walks on spin-orbital angular momentum space of photons,” Phys. Rev. A 81(5), 052322 (2010).
[Crossref]

P. Zhang, X.-F. Ren, X.-B. Zou, B.-H. Liu, Y.-F. Huang, and G.-C. Guo, “Demonstration of one-dimensional quantum random walks using orbital angular momentum of photons,” Phys. Rev. A 75(5), 052310 (2007).
[Crossref]

Zhang, X.

B. Wang, T. Chen, and X. Zhang, “Experimental observation of topologically protected bound states with vanishing chern numbers in a two-dimensional quantum walk,” Phys. Rev. Lett. 121(10), 100501 (2018).
[Crossref]

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13(11), 1117–1123 (2017).
[Crossref]

Zhou, H.-Y.

Y.-B. Sheng, F.-G. Deng, and H.-Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-kerr nonlinearity,” Phys. Rev. A 77(4), 042308 (2008).
[Crossref]

Zhou, X.

J. Du, H. Li, X. Xu, M. Shi, J. Wu, X. Zhou, and R. Han, “Experimental implementation of the quantum random-walk algorithm,” Phys. Rev. A 67(4), 042316 (2003).
[Crossref]

Zhou, X.-Q.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329(5998), 1500–1503 (2010).
[Crossref]

Zou, X.-B.

P. Zhang, X.-F. Ren, X.-B. Zou, B.-H. Liu, Y.-F. Huang, and G.-C. Guo, “Demonstration of one-dimensional quantum random walks using orbital angular momentum of photons,” Phys. Rev. A 75(5), 052310 (2007).
[Crossref]

Int. J. Quantum Inform. (1)

A. Ambainis, “Quantum walks and their algorithmic applications,” Int. J. Quantum Inform. 01(04), 507–518 (2003).
[Crossref]

J. Chem. Phys. (1)

M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, “Environment-assisted quantum walks in photosynthetic energy transfer,” J. Chem. Phys. 129(17), 174106 (2008).
[Crossref]

J. Phys. A: Math. Theor. (1)

B. L. Douglas and J. B. Wang, “A classical approach to the graph isomorphism problem using quantum walks,” J. Phys. A: Math. Theor. 41(7), 075303 (2008).
[Crossref]

Nat. Commun. (3)

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7(1), 11439 (2016).
[Crossref]

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Commun. 3(1), 882 (2012).
[Crossref]

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

Nat. Photonics (3)

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through hong-mandel coalescence,” Nat. Photonics 3(12), 720–723 (2009).
[Crossref]

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[Crossref]

Nat. Phys. (4)

M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6(2), 118–121 (2010).
[Crossref]

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008).
[Crossref]

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, K. Mochizuki, D. Kim, N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, “Observation of topological edge states in parity-time-symmetric quantum walks,” Nat. Phys. 13(11), 1117–1123 (2017).
[Crossref]

A. Aspuruguzik and P. Walther, “Photonic quantum simulators,” Nat. Phys. 8(4), 285–291 (2012).
[Crossref]

Nature (2)

G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, “Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems,” Nature 446(7137), 782–786 (2007).
[Crossref]

P. Grangier, J. A. Levenson, and J.-P. Poizat, “Quantum non-demolition measurements in optics,” Nature 396(6711), 537–542 (1998).
[Crossref]

New J. Phys. (2)

T. P. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock, and G. J. Milburn, “Quantum computation by communication,” New J. Phys. 8(2), 30 (2006).
[Crossref]

W. J. Munro, K. Nemoto, and T. P. Spiller, “Weak nonlinearities: a new route to optical quantum computation,” New J. Phys. 7, 137 (2005).
[Crossref]

Opt. Express (1)

Optica (1)

Phys. Rev. A (21)

P. Zhang, B.-H. Liu, R.-F. Liu, H.-R. Li, F.-L. Li, and G.-C. Guo, “Implementation of one-dimensional quantum walks on spin-orbital angular momentum space of photons,” Phys. Rev. A 81(5), 052322 (2010).
[Crossref]

P. Zhang, X.-F. Ren, X.-B. Zou, B.-H. Liu, Y.-F. Huang, and G.-C. Guo, “Demonstration of one-dimensional quantum random walks using orbital angular momentum of photons,” Phys. Rev. A 75(5), 052310 (2007).
[Crossref]

T. Wang, H. W. Lau, H. Kaviani, R. Ghobadi, and C. Simon, “Strong micro-macro entanglement from a weak cross-kerr nonlinearity,” Phys. Rev. A 92(1), 012316 (2015).
[Crossref]

F.-G. Deng, “Optimal nonlocal multipartite entanglement concentration based on projection measurements,” Phys. Rev. A 85(2), 022311 (2012).
[Crossref]

Y.-B. Sheng, F.-G. Deng, and H.-Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-kerr nonlinearity,” Phys. Rev. A 77(4), 042308 (2008).
[Crossref]

Y.-B. Sheng, F.-G. Deng, and G. L. Long, “Complete hyperentangled-bell-state analysis for quantum communication,” Phys. Rev. A 82(3), 032318 (2010).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
[Crossref]

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48(2), 1687–1690 (1993).
[Crossref]

A. M. Childs and J. Goldstone, “Spatial search by quantum walk,” Phys. Rev. A 70(2), 022314 (2004).
[Crossref]

N. Shenvi, J. Kempe, and K. B. Whaley, “Quantum random-walk search algorithm,” Phys. Rev. A 67(5), 052307 (2003).
[Crossref]

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82(3), 033429 (2010).
[Crossref]

N. P. Kumar, R. Balu, R. Laflamme, and C. M. Chandrashekar, “Bounds on the dynamics of periodic quantum walks and emergence of the gapless and gapped dirac equation,” Phys. Rev. A 97(1), 012116 (2018).
[Crossref]

N. B. Lovett, S. Cooper, M. Everitt, M. Trevers, and V. Kendon, “Universal quantum computation using the discrete-time quantum walk,” Phys. Rev. A 81(4), 042330 (2010).
[Crossref]

C. A. Ryan, M. Laforest, J. C. Boileau, and R. Laflamme, “Experimental implementation of a discrete-time quantum random walk on an nmr quantum-information processor,” Phys. Rev. A 72(6), 062317 (2005).
[Crossref]

J. Du, H. Li, X. Xu, M. Shi, J. Wu, X. Zhou, and R. Han, “Experimental implementation of the quantum random-walk algorithm,” Phys. Rev. A 67(4), 042316 (2003).
[Crossref]

Z.-H. Bian, J. Li, X. Zhan, J. Twamley, and P. Xue, “Experimental implementation of a quantum walk on a circle with single photons,” Phys. Rev. A 95(5), 052338 (2017).
[Crossref]

L. Xiao, X. Qiu, K. Wang, Z. Bian, X. Zhan, H. Obuse, B. C. Sanders, W. Yi, and P. Xue, “Higher winding number in a nonunitary photonic quantum walk,” Phys. Rev. A 98(6), 063847 (2018).
[Crossref]

Q. Lin and J. Li, “Quantum control gates with weak cross-kerr nonlinearity,” Phys. Rev. A 79(2), 022301 (2009).
[Crossref]

N. Satapathy, H. Hagman, M. Zelán, A. Kastberg, and H. Ramachandran, “Theoretical investigations of quantum walks by cold atoms in a double optical lattice,” Phys. Rev. A 80(1), 012302 (2009).
[Crossref]

S. Mugel, A. Celi, P. Massignan, J. K. Asbóth, M. Lewenstein, and C. Lobo, “Topological bound states of a quantum walk with cold atoms,” Phys. Rev. A 94(2), 023631 (2016).
[Crossref]

Phys. Rev. B (1)

H. Obuse and N. Kawakami, “Topological phases and delocalization of quantum walks in random environments,” Phys. Rev. B 84(19), 195139 (2011).
[Crossref]

Phys. Rev. Lett. (13)

A. M. Childs, “Universal computation by quantum walk,” Phys. Rev. Lett. 102(18), 180501 (2009).
[Crossref]

A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: A quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104(5), 050502 (2010).
[Crossref]

M. A. Broome, A. Fedrizzi, B. P. Lanyon, I. Kassal, A. Aspuru-Guzik, and A. G. White, “Discrete single-photon quantum walks with tunable decoherence,” Phys. Rev. Lett. 104(15), 153602 (2010).
[Crossref]

H. Schmitz, R. Matjeschk, C. Schneider, J. Glueckert, M. Enderlein, T. Huber, and T. Schaetz, “Quantum walk of a trapped ion in phase space,” Phys. Rev. Lett. 103(9), 090504 (2009).
[Crossref]

F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. F. Roos, “Realization of a quantum walk with one and two trapped ions,” Phys. Rev. Lett. 104(10), 100503 (2010).
[Crossref]

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100(17), 170506 (2008).
[Crossref]

D. J. Brod and J. Combes, “Passive cphase gate via cross-kerr nonlinearities,” Phys. Rev. Lett. 117(8), 080502 (2016).
[Crossref]

K. Poulios, R. Keil, D. Fry, J. D. A. Meinecke, J. C. F. Matthews, A. Politi, M. Lobino, M. Gräfe, M. Heinrich, S. Nolte, A. Szameit, and J. L. O’Brien, “Quantum walks of correlated photon pairs in two-dimensional waveguide arrays,” Phys. Rev. Lett. 112(14), 143604 (2014).
[Crossref]

N. Wiebe, D. Braun, and S. Lloyd, “Quantum algorithm for data fitting,” Phys. Rev. Lett. 109(5), 050505 (2012).
[Crossref]

B. Wang, T. Chen, and X. Zhang, “Experimental observation of topologically protected bound states with vanishing chern numbers in a two-dimensional quantum walk,” Phys. Rev. Lett. 121(10), 100501 (2018).
[Crossref]

K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-not gate,” Phys. Rev. Lett. 93(25), 250502 (2004).
[Crossref]

L. S. Costanzo, A. S. Coelho, N. Biagi, J. Fiurášek, M. Bellini, and A. Zavatta, “Measurement-induced strong kerr nonlinearity for weak quantum states of light,” Phys. Rev. Lett. 119(1), 013601 (2017).
[Crossref]

S. K. Goyal, F. S. Roux, A. Forbes, and T. Konrad, “Implementing quantum walks using orbital angular momentum of classical light,” Phys. Rev. Lett. 110(26), 263602 (2013).
[Crossref]

Quantum Inf. Process. (1)

M. Schuld, I. Sinayskiy, and F. Petruccione, “The quest for a quantum neural network,” Quantum Inf. Process. 13(11), 2567–2586 (2014).
[Crossref]

Sci. Adv. (1)

H. Tang, X.-F. Lin, Z. Feng, J.-Y. Chen, J. Gao, K. Sun, C.-Y. Wang, P.-C. Lai, X.-Y. Xu, Y. Wang, L.-F. Qiao, A.-L. Yang, and X.-M. Jin, “Experimental two-dimensional quantum walk on a photonic chip,” Sci. Adv. 4(5), eaat3174 (2018).
[Crossref]

Science (4)

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum walks of correlated photons,” Science 329(5998), 1500–1503 (2010).
[Crossref]

M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, “Quantum walk in position space with single optically trapped atoms,” Science 325(5937), 174–177 (2009).
[Crossref]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[Crossref]

Other (1)

C. W. Gardiner and H. Haken, Quantum noise, vol. 2 (Springer, 1991).

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Figures (3)

Fig. 1.
Fig. 1. Schematic diagram showing the principle of implementing 2D QW. The coin state of 2D QW is encoded by the path modes $|U\rangle$ and $|D\rangle$, and the 2D lattice is mapped to a grid with the help of the phase information of the coherent state $|\alpha \rangle$ indeced by cross-Kerr media and the OAM states of the photon number state resorting to q-plate. SLM (spatial light modulator) and $\textbf {X}$ (homodyne-heterodyne measurement) are employed to read the site $x$ and site $y$, respectively.
Fig. 2.
Fig. 2. Probability distribution after 20 steps of the 2D QW with auxiliary coherent state and corresponding deterministic measurement success probability as a function of the evolution scale. In (a), (b), and (c) we have simulated the evolution pattern of 2D QW based on $\textbf {X}$ homodyne analysis and the walker starts from the lattice site (0, 0). The initial input state are (a) $|U\rangle$, (b) $|D\rangle$, (c) $1/\sqrt {2}(|U\rangle +|D\rangle )$. All phases caused by cross-Kerr media are set to $\theta ' = 1^\circ$. (d) When the step length is set to $\theta ' = 1^\circ$ and $\theta ' = 0.5^\circ$, the success probability of deterministic measurements as the number $\textbf {n}$ of walking steps increases for different $\alpha$ is shown.
Fig. 3.
Fig. 3. The success probability of 2D QW using $\textbf {X}$ Homodyne analysis. (a) Deterministic success probability of resolution measurement against the site y (the value of $\theta$) and amplitude $\alpha$ for 2D QW. (b) Three different $\theta '$s were chosen as the walking step length to simulate the success rate against the amplitude $\alpha$, inset, small $\theta '$ for big growth $\alpha$.

Equations (13)

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UcrossKerr|ϕ=eiχnanc(μ|0+ν|1)|α=μ|0|α+ν|1|αeiθ,
U=SyC2SxC1,
H[|U|D]=12[|U+|D|U|D]:=C1[|U|D],
l|l+1l||UU|+|l1l||DD|,
x|x+1x||UU|+|x1x||DD|:=Sx.
C2[|U|D]=12[|U+|D|U|D].
θ|αeiθα||UU|+|αeiθα||DD|,
y|y+1y||UU|+|y1y||DD|:=Sy.
UQW=SyC2SxC1.
UQW|ψ012|U|x+1|y+1+12|D|x+1|y1+12|U|x1|y+112|D|x1|y1.
X|αe±iθ=Σx,y=nnPnu(x,y)f(x,αcosyθ)|x|U+Σx,y=nnPnu(x,y)f(x,αcosyθ)|x|D
f(x,αcosyθ)=αsinyθ(x2αcosyθ)
Psuccess=112erfc[α2n(1cosθ)].

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