Abstract

We show a simulation of quantum random walks (QRWs) with multiple photons using a staggered array of 50/50 beam splitters with a bank of detectors at any desired level. We discuss the multiphoton interference effects that are inherent to this setup, and introduce one, two, and threefold coincidence detection schemes. Feynman diagrams are used to intuitively explain the unique multiphoton interference effects of these QRWs.

© 2013 Optical Society of America

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  1. Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48, 1687–1690 (1993).
    [CrossRef]
  2. J. Kempe, “Quantum random walks: an introductory overview,” Contemp. Phys. 44, 307–327 (2003).
    [CrossRef]
  3. N. Shenvi, J. Kempe, and K. B. Whaley, “Quantum random-walk search algorithm,” Phys. Rev. A 67, 052307 (2003).
    [CrossRef]
  4. 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, 042330 (2010).
    [CrossRef]
  5. E. Farhi and S. Gutmann, “Quantum computation and decision trees,” Phys. Rev. A 58, 915–928 (1998).
    [CrossRef]
  6. 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, 170506 (2008).
    [CrossRef]
  7. V. Kendon, “Decoherence in quantum walks—a review,” Math. Struct. Comp. Sci. 17, 1169–1220 (2007).
    [CrossRef]
  8. B. C. Travaglione and G. J. Milburn, “Implementing the quantum random walk,” Phys. Rev. A 65, 032310 (2002).
    [CrossRef]
  9. 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, 042316 (2003).
    [CrossRef]
  10. J. K. Gamble, M. Friesen, D. Zhou, R. Joynt, and S. N. Coppersmith, “Two-particle quantum walks applied to the graph isomorphism problem,” Phys. Rev. A 81, 052313 (2010).
    [CrossRef]
  11. B. C. Sanders, S. D. Bartlett, B. Tregenna, and P. L. Knight, “Quantum quincunx in cavity quantum electrodynamics,” Phys. Rev. A 67, 042305 (2003).
    [CrossRef]
  12. W. Dür, R. Raussendorf, V. M. Kendon, and H. J. Briegel, “Quantum walks in optical lattices,” Phys. Rev. A 66, 052319 (2002).
    [CrossRef]
  13. P. L. Knight, E. Roldán, and J. E. Sipe, “Quantum walk on the line as an interference phenomenon,” Phys. Rev. A 68, 020301 (2003).
    [CrossRef]
  14. A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvao, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Experimental boson sampling in arbitrary integrated photonic circuits,” arXiv:1212.2783 [quant-ph].
  15. A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photonics 7, 322–328 (2013).
    [CrossRef]
  16. J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
    [CrossRef]
  17. M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” arXiv:1212.2240 [quant-ph] (2012).
  18. 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, 100503 (2010).
    [CrossRef]
  19. A. Peruzzo, M. Lobino, J. C. 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, 1500–1503 (2010).
    [CrossRef]
  20. Y. Omar, N. Paunković, L. Sheridan, and S. Bose, “Quantum walk on a line with two entangled particles,” Phys. Rev. A 74, 042304 (2006).
    [CrossRef]
  21. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
    [CrossRef]
  22. T. D. Mackay, S. D. Bartlett, L. T. Stephenson, and B. C. Sanders, “Quantum walks in higher dimensions,” J. Phys. A 35, 2745 (2002).
    [CrossRef]
  23. H. Jeong, M. Paternostro, and M. S. Kim, “Simulation of quantum random walks using the interference of a classical field,” Phys. Rev. A 69, 012310 (2004).
    [CrossRef]
  24. C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University, 2004), pp. 122–139.
  25. P. P. Rohde, A. Schreiber, M. Štefaňák, I. Jex, and C. Silberhorn, “Multi-walker discrete time quantum walks on arbitrary graphs, their properties and their photonic implementation,” New J. Phys. 13, 013001 (2011).
    [CrossRef]
  26. Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
    [CrossRef]
  27. M. Karski, L. Förster, J. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, “Quantum walk in position space with single optically trapped atoms,” Science 325, 174 (2009).
    [CrossRef]
  28. S. Aaronson and A. Arkhipov, “The computational complexity of linear optics,” in Proceedings of 43rd Annual ACM Symposium on Theory of Computing, (ACM, 2011), pp. 333–342.
  29. M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794–798 (2013).
    [CrossRef]
  30. P. P. Rohde, “Are quantum walks the saviour of optical quantum computing?” arXiv:1010.4608 [quant-ph] (2010).
  31. S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
    [CrossRef]
  32. M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: single-photon sources and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
    [CrossRef]

2013 (3)

A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photonics 7, 322–328 (2013).
[CrossRef]

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
[CrossRef]

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794–798 (2013).
[CrossRef]

2011 (2)

P. P. Rohde, A. Schreiber, M. Štefaňák, I. Jex, and C. Silberhorn, “Multi-walker discrete time quantum walks on arbitrary graphs, their properties and their photonic implementation,” New J. Phys. 13, 013001 (2011).
[CrossRef]

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: single-photon sources and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
[CrossRef]

2010 (4)

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, 100503 (2010).
[CrossRef]

A. Peruzzo, M. Lobino, J. C. 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, 1500–1503 (2010).
[CrossRef]

J. K. Gamble, M. Friesen, D. Zhou, R. Joynt, and S. N. Coppersmith, “Two-particle quantum walks applied to the graph isomorphism problem,” Phys. Rev. A 81, 052313 (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, 042330 (2010).
[CrossRef]

2009 (2)

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
[CrossRef]

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

2008 (2)

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (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, 170506 (2008).
[CrossRef]

2007 (1)

V. Kendon, “Decoherence in quantum walks—a review,” Math. Struct. Comp. Sci. 17, 1169–1220 (2007).
[CrossRef]

2006 (1)

Y. Omar, N. Paunković, L. Sheridan, and S. Bose, “Quantum walk on a line with two entangled particles,” Phys. Rev. A 74, 042304 (2006).
[CrossRef]

2004 (1)

H. Jeong, M. Paternostro, and M. S. Kim, “Simulation of quantum random walks using the interference of a classical field,” Phys. Rev. A 69, 012310 (2004).
[CrossRef]

2003 (5)

B. C. Sanders, S. D. Bartlett, B. Tregenna, and P. L. Knight, “Quantum quincunx in cavity quantum electrodynamics,” Phys. Rev. A 67, 042305 (2003).
[CrossRef]

P. L. Knight, E. Roldán, and J. E. Sipe, “Quantum walk on the line as an interference phenomenon,” Phys. Rev. A 68, 020301 (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, 042316 (2003).
[CrossRef]

J. Kempe, “Quantum random walks: an introductory overview,” Contemp. Phys. 44, 307–327 (2003).
[CrossRef]

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

2002 (3)

B. C. Travaglione and G. J. Milburn, “Implementing the quantum random walk,” Phys. Rev. A 65, 032310 (2002).
[CrossRef]

T. D. Mackay, S. D. Bartlett, L. T. Stephenson, and B. C. Sanders, “Quantum walks in higher dimensions,” J. Phys. A 35, 2745 (2002).
[CrossRef]

W. Dür, R. Raussendorf, V. M. Kendon, and H. J. Briegel, “Quantum walks in optical lattices,” Phys. Rev. A 66, 052319 (2002).
[CrossRef]

1998 (1)

E. Farhi and S. Gutmann, “Quantum computation and decision trees,” Phys. Rev. A 58, 915–928 (1998).
[CrossRef]

1993 (1)

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

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[CrossRef]

Aaronson, S.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794–798 (2013).
[CrossRef]

S. Aaronson and A. Arkhipov, “The computational complexity of linear optics,” in Proceedings of 43rd Annual ACM Symposium on Theory of Computing, (ACM, 2011), pp. 333–342.

Aharonov, Y.

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

Alt, W.

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

Arkhipov, A.

S. Aaronson and A. Arkhipov, “The computational complexity of linear optics,” in Proceedings of 43rd Annual ACM Symposium on Theory of Computing, (ACM, 2011), pp. 333–342.

Barbieri, M.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
[CrossRef]

Bartlett, S. D.

B. C. Sanders, S. D. Bartlett, B. Tregenna, and P. L. Knight, “Quantum quincunx in cavity quantum electrodynamics,” Phys. Rev. A 67, 042305 (2003).
[CrossRef]

T. D. Mackay, S. D. Bartlett, L. T. Stephenson, and B. C. Sanders, “Quantum walks in higher dimensions,” J. Phys. A 35, 2745 (2002).
[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, 100503 (2010).
[CrossRef]

Bose, S.

Y. Omar, N. Paunković, L. Sheridan, and S. Bose, “Quantum walk on a line with two entangled particles,” Phys. Rev. A 74, 042304 (2006).
[CrossRef]

Briegel, H. J.

W. Dür, R. Raussendorf, V. M. Kendon, and H. J. Briegel, “Quantum walks in optical lattices,” Phys. Rev. A 66, 052319 (2002).
[CrossRef]

Brod, D. J.

A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvao, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Experimental boson sampling in arbitrary integrated photonic circuits,” arXiv:1212.2783 [quant-ph].

Bromberg, Y.

A. Peruzzo, M. Lobino, J. C. 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, 1500–1503 (2010).
[CrossRef]

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
[CrossRef]

Broome, M. A.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794–798 (2013).
[CrossRef]

Choi, J.

M. Karski, L. Förster, J. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, “Quantum walk in position space with single optically trapped atoms,” Science 325, 174 (2009).
[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, 042330 (2010).
[CrossRef]

Coppersmith, S. N.

J. K. Gamble, M. Friesen, D. Zhou, R. Joynt, and S. N. Coppersmith, “Two-particle quantum walks applied to the graph isomorphism problem,” Phys. Rev. A 81, 052313 (2010).
[CrossRef]

Crespi, A.

A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photonics 7, 322–328 (2013).
[CrossRef]

A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvao, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Experimental boson sampling in arbitrary integrated photonic circuits,” arXiv:1212.2783 [quant-ph].

Dakic, B.

M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” arXiv:1212.2240 [quant-ph] (2012).

Datta, A.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
[CrossRef]

Davidovich, L.

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

De Nicola, F.

A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photonics 7, 322–328 (2013).
[CrossRef]

Dove, J.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794–798 (2013).
[CrossRef]

Dowling, J. P.

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (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, 042316 (2003).
[CrossRef]

Dür, W.

W. Dür, R. Raussendorf, V. M. Kendon, and H. J. Briegel, “Quantum walks in optical lattices,” Phys. Rev. A 66, 052319 (2002).
[CrossRef]

Eisaman, M. D.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: single-photon sources and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
[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, 042330 (2010).
[CrossRef]

Fan, J.

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: single-photon sources and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
[CrossRef]

Farhi, E.

E. Farhi and S. Gutmann, “Quantum computation and decision trees,” Phys. Rev. A 58, 915–928 (1998).
[CrossRef]

Fazio, R.

A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photonics 7, 322–328 (2013).
[CrossRef]

Fedrizzi, A.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794–798 (2013).
[CrossRef]

Förster, L.

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

Friesen, M.

J. K. Gamble, M. Friesen, D. Zhou, R. Joynt, and S. N. Coppersmith, “Two-particle quantum walks applied to the graph isomorphism problem,” Phys. Rev. A 81, 052313 (2010).
[CrossRef]

Galvao, E. F.

A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvao, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Experimental boson sampling in arbitrary integrated photonic circuits,” arXiv:1212.2783 [quant-ph].

Gamble, J. K.

J. K. Gamble, M. Friesen, D. Zhou, R. Joynt, and S. N. Coppersmith, “Two-particle quantum walks applied to the graph isomorphism problem,” Phys. Rev. A 81, 052313 (2010).
[CrossRef]

Gates, J. C.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
[CrossRef]

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, 100503 (2010).
[CrossRef]

Gerry, C.

C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University, 2004), pp. 122–139.

Giovannetti, V.

A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photonics 7, 322–328 (2013).
[CrossRef]

Gutmann, S.

E. Farhi and S. Gutmann, “Quantum computation and decision trees,” Phys. Rev. A 58, 915–928 (1998).
[CrossRef]

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, 042316 (2003).
[CrossRef]

Heilmann, R.

M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” arXiv:1212.2240 [quant-ph] (2012).

Hong, C. K.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[CrossRef]

Humphreys, P. C.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
[CrossRef]

Huver, S. D.

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
[CrossRef]

Ismail, N.

A. Peruzzo, M. Lobino, J. C. 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, 1500–1503 (2010).
[CrossRef]

Jeong, H.

H. Jeong, M. Paternostro, and M. S. Kim, “Simulation of quantum random walks using the interference of a classical field,” Phys. Rev. A 69, 012310 (2004).
[CrossRef]

Jex, I.

P. P. Rohde, A. Schreiber, M. Štefaňák, I. Jex, and C. Silberhorn, “Multi-walker discrete time quantum walks on arbitrary graphs, their properties and their photonic implementation,” New J. Phys. 13, 013001 (2011).
[CrossRef]

Jin, X.-M.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
[CrossRef]

Joynt, R.

J. K. Gamble, M. Friesen, D. Zhou, R. Joynt, and S. N. Coppersmith, “Two-particle quantum walks applied to the graph isomorphism problem,” Phys. Rev. A 81, 052313 (2010).
[CrossRef]

Karski, M.

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W. Dür, R. Raussendorf, V. M. Kendon, and H. J. Briegel, “Quantum walks in optical lattices,” Phys. Rev. A 66, 052319 (2002).
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H. Jeong, M. Paternostro, and M. S. Kim, “Simulation of quantum random walks using the interference of a classical field,” Phys. Rev. A 69, 012310 (2004).
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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, 100503 (2010).
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P. L. Knight, E. Roldán, and J. E. Sipe, “Quantum walk on the line as an interference phenomenon,” Phys. Rev. A 68, 020301 (2003).
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B. C. Sanders, S. D. Bartlett, B. Tregenna, and P. L. Knight, “Quantum quincunx in cavity quantum electrodynamics,” Phys. Rev. A 67, 042305 (2003).
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J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
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J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
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A. Peruzzo, M. Lobino, J. C. 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, 1500–1503 (2010).
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Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
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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, 170506 (2008).
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J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
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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, 042316 (2003).
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A. Peruzzo, M. Lobino, J. C. 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, 1500–1503 (2010).
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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, 042330 (2010).
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T. D. Mackay, S. D. Bartlett, L. T. Stephenson, and B. C. Sanders, “Quantum walks in higher dimensions,” J. Phys. A 35, 2745 (2002).
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A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvao, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Experimental boson sampling in arbitrary integrated photonic circuits,” arXiv:1212.2783 [quant-ph].

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C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
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Mataloni, P.

A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photonics 7, 322–328 (2013).
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A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvao, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Experimental boson sampling in arbitrary integrated photonic circuits,” arXiv:1212.2783 [quant-ph].

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A. Peruzzo, M. Lobino, J. C. 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, 1500–1503 (2010).
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A. Peruzzo, M. Lobino, J. C. 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, 1500–1503 (2010).
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M. Karski, L. Förster, J. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, “Quantum walk in position space with single optically trapped atoms,” Science 325, 174 (2009).
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J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
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B. C. Travaglione and G. J. Milburn, “Implementing the quantum random walk,” Phys. Rev. A 65, 032310 (2002).
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Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
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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, 170506 (2008).
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M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” arXiv:1212.2240 [quant-ph] (2012).

Obrien, J. L.

A. Peruzzo, M. Lobino, J. C. 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, 1500–1503 (2010).
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Y. Omar, N. Paunković, L. Sheridan, and S. Bose, “Quantum walk on a line with two entangled particles,” Phys. Rev. A 74, 042304 (2006).
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A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photonics 7, 322–328 (2013).
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A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvao, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Experimental boson sampling in arbitrary integrated photonic circuits,” arXiv:1212.2783 [quant-ph].

Ou, Z. Y.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
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Paternostro, M.

H. Jeong, M. Paternostro, and M. S. Kim, “Simulation of quantum random walks using the interference of a classical field,” Phys. Rev. A 69, 012310 (2004).
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Paunkovic, N.

Y. Omar, N. Paunković, L. Sheridan, and S. Bose, “Quantum walk on a line with two entangled particles,” Phys. Rev. A 74, 042304 (2006).
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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, 170506 (2008).
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Peruzzo, A.

A. Peruzzo, M. Lobino, J. C. 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, 1500–1503 (2010).
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Politi, A.

A. Peruzzo, M. Lobino, J. C. 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, 1500–1503 (2010).
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M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited review article: single-photon sources and detectors,” Rev. Sci. Instrum. 82, 071101 (2011).
[CrossRef]

Poulios, K.

A. Peruzzo, M. Lobino, J. C. 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, 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, 170506 (2008).
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Rahimi-Keshari, S.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794–798 (2013).
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Ralph, T. C.

M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794–798 (2013).
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Ramponi, R.

A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photonics 7, 322–328 (2013).
[CrossRef]

A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvao, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Experimental boson sampling in arbitrary integrated photonic circuits,” arXiv:1212.2783 [quant-ph].

Raussendorf, R.

W. Dür, R. Raussendorf, V. M. Kendon, and H. J. Briegel, “Quantum walks in optical lattices,” Phys. Rev. A 66, 052319 (2002).
[CrossRef]

Rohde, P. P.

P. P. Rohde, A. Schreiber, M. Štefaňák, I. Jex, and C. Silberhorn, “Multi-walker discrete time quantum walks on arbitrary graphs, their properties and their photonic implementation,” New J. Phys. 13, 013001 (2011).
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P. P. Rohde, “Are quantum walks the saviour of optical quantum computing?” arXiv:1010.4608 [quant-ph] (2010).

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P. L. Knight, E. Roldán, and J. E. Sipe, “Quantum walk on the line as an interference phenomenon,” Phys. Rev. A 68, 020301 (2003).
[CrossRef]

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, 100503 (2010).
[CrossRef]

Sanders, B. C.

B. C. Sanders, S. D. Bartlett, B. Tregenna, and P. L. Knight, “Quantum quincunx in cavity quantum electrodynamics,” Phys. Rev. A 67, 042305 (2003).
[CrossRef]

T. D. Mackay, S. D. Bartlett, L. T. Stephenson, and B. C. Sanders, “Quantum walks in higher dimensions,” J. Phys. A 35, 2745 (2002).
[CrossRef]

Sansoni, L.

A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photonics 7, 322–328 (2013).
[CrossRef]

Schreiber, A.

P. P. Rohde, A. Schreiber, M. Štefaňák, I. Jex, and C. Silberhorn, “Multi-walker discrete time quantum walks on arbitrary graphs, their properties and their photonic implementation,” New J. Phys. 13, 013001 (2011).
[CrossRef]

Sciarrino, F.

A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photonics 7, 322–328 (2013).
[CrossRef]

A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvao, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Experimental boson sampling in arbitrary integrated photonic circuits,” arXiv:1212.2783 [quant-ph].

Shenvi, N.

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

Sheridan, L.

Y. Omar, N. Paunković, L. Sheridan, and S. Bose, “Quantum walk on a line with two entangled particles,” Phys. Rev. A 74, 042304 (2006).
[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, 042316 (2003).
[CrossRef]

Silberberg, Y.

A. Peruzzo, M. Lobino, J. C. 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, 1500–1503 (2010).
[CrossRef]

Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
[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, 170506 (2008).
[CrossRef]

Silberhorn, C.

P. P. Rohde, A. Schreiber, M. Štefaňák, I. Jex, and C. Silberhorn, “Multi-walker discrete time quantum walks on arbitrary graphs, their properties and their photonic implementation,” New J. Phys. 13, 013001 (2011).
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Sipe, J. E.

P. L. Knight, E. Roldán, and J. E. Sipe, “Quantum walk on the line as an interference phenomenon,” Phys. Rev. A 68, 020301 (2003).
[CrossRef]

Smith, B. J.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
[CrossRef]

Smith, P. G. R.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (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, 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, 170506 (2008).
[CrossRef]

Spagnolo, N.

A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvao, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Experimental boson sampling in arbitrary integrated photonic circuits,” arXiv:1212.2783 [quant-ph].

Spring, J. B.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
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P. P. Rohde, A. Schreiber, M. Štefaňák, I. Jex, and C. Silberhorn, “Multi-walker discrete time quantum walks on arbitrary graphs, their properties and their photonic implementation,” New J. Phys. 13, 013001 (2011).
[CrossRef]

Steffen, A.

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

Stephenson, L. T.

T. D. Mackay, S. D. Bartlett, L. T. Stephenson, and B. C. Sanders, “Quantum walks in higher dimensions,” J. Phys. A 35, 2745 (2002).
[CrossRef]

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M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” arXiv:1212.2240 [quant-ph] (2012).

Thomas-Peter, N.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
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A. Peruzzo, M. Lobino, J. C. 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, 1500–1503 (2010).
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M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” arXiv:1212.2240 [quant-ph] (2012).

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B. C. Travaglione and G. J. Milburn, “Implementing the quantum random walk,” Phys. Rev. A 65, 032310 (2002).
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B. C. Sanders, S. D. Bartlett, B. Tregenna, and P. L. Knight, “Quantum quincunx in cavity quantum electrodynamics,” Phys. Rev. A 67, 042305 (2003).
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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, 042330 (2010).
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A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvao, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Experimental boson sampling in arbitrary integrated photonic circuits,” arXiv:1212.2783 [quant-ph].

Walmsley, I. A.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
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M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” arXiv:1212.2240 [quant-ph] (2012).

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N. Shenvi, J. Kempe, and K. B. Whaley, “Quantum random-walk search algorithm,” Phys. Rev. A 67, 052307 (2003).
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M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, “Photonic boson sampling in a tunable circuit,” Science 339, 794–798 (2013).
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M. Karski, L. Förster, J. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, “Quantum walk in position space with single optically trapped atoms,” Science 325, 174 (2009).
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S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
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A. Peruzzo, M. Lobino, J. C. 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, 1500–1503 (2010).
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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, 042316 (2003).
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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, 042316 (2003).
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Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48, 1687–1690 (1993).
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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, 100503 (2010).
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J. K. Gamble, M. Friesen, D. Zhou, R. Joynt, and S. N. Coppersmith, “Two-particle quantum walks applied to the graph isomorphism problem,” Phys. Rev. A 81, 052313 (2010).
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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, 042316 (2003).
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Figures (7)

Fig. 1.
Fig. 1.

(a) Classical random walk using a peg board and ping-pong balls. (b) Result of a classical random walk is a binomial distribution. The maximum of the binomial distribution corresponds to the location directly below the starting position.

Fig. 2.
Fig. 2.

(a) Classical description of a beam splitter. (b) Quantum-mechanical description of a beam splitter with the input and output field operators shown.

Fig. 3.
Fig. 3.

Pyramid structure of QRW model using beam splitters.

Fig. 4.
Fig. 4.

(a) Plot of onefold probabilities showing single-photon interference of |1,0 input state propagated through three levels of beam splitters. Complete destructive interference is shown at detector D3, while constructive interference occurs at detector D4. (b), (c) Two Feynman diagrams are required to explain the complete destructive interference at detector D3. (d), (e) Feynman diagrams explaining the constructive interference at detector D4. Only one path is shown per diagram since there is only one photon in the system.

Fig. 5.
Fig. 5.

(a) Plot showing twofold probabilities using a |1,1 input state propagated through three levels. (b) Feynman diagram showing one set of unique paths that reach detectors D1 and D5. (c) Feynman diagram showing the only other unique set of paths that the photons can take to reach detectors D1 and D5.

Fig. 6.
Fig. 6.

Plot of threefold probabilities using input state of |2,1 propagating through three levels of beam splitters. Each row, column, and pillar represents an individual detector, while the color of each cube represents the probability. Blue represents minimum probability, while red represents maximum, and white is exactly zero. (a), (b) Plots of the threefold probabilities using input state of |2,1 propagating through a three-level beam-splitter detection scheme looking into the space diagonals. These views display the symmetry along the diagonals. (c) “Exploded” view of the same parameters. This view allows you to see inside the cube.

Fig. 7.
Fig. 7.

Feynman diagrams explaining the threefold probability at detectors D1, D4, and D6 using |1,0 and |1,0 as input states along with the discussion in Section 2.A to represent a |2,1 input state. These diagrams, along with the discussion in Section 2.A, show complete destructive interference with D1, D4, and D6 in coincidence. Note that each input state has two paths to arrive at detector D4 and thus has two amplitudes at D4.

Equations (9)

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a^in=12(ia^out+b^out),b^in=12(a^out+ib^out).
|N,M=a^0Nb^0MN!M!|0,0Eq.1|N,M=1N!M!(ia^1+b^12)N(a^1+ib^12)M|0a,0b,
t=1/2(Photon is transmitted),r=i/2(Photon is reflected).
Figure5(b):{PathA=(rt2)=i/23/2PathB=(rtr)=1/23/2.
Figure5(c):{PathC=(t2r)=i/23/2PathD=(t3)=1/23/2.
Figure7:{PathA=(rt2)=i/23/2PathB=(r2t)=1/23/2PathC=(tr2)=1/23/2PathD=(t3)=1/23/2,
|2,11,4,6=12!1!((i23/2)1+(223/2)4+(123/2)6)2((123/2)1+(0)4+(i23/2)6),
|2,11,4,6=((123)1,4,6+(0)1,4,6(123)1,4,6)=0.
(x+y)(l1)=k=0l1(l1k)x(l1k)yk,

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