Abstract

We introduce an efficient method for fully characterizing multimode linear-optical networks. Our approach requires only a standard laser source and intensity measurements to directly and uniquely determine all moduli and non-trivial phases of the matrix describing a network. We experimentally demonstrate the characterization of a 6×6 fiber-optic network and independently verify the results via nonclassical two-photon interference.

© 2013 OSA

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  1. M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett.73, 58–61 (1994).
    [CrossRef] [PubMed]
  2. J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett.93, 080502 (2004).
    [CrossRef]
  3. A. M. Childs, I. L. Chuang, and D. W. Leung, “Realization of quantum process tomography in NMR,” Phys. Rev. A64, 012314 (2001).
    [CrossRef]
  4. M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M. Steinberg, “Diagnosis, prescription, and prognosis of a Bell-state filter by quantum process tomography,” Phys. Rev. Lett.91, 120402 (2003).
    [CrossRef] [PubMed]
  5. M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, and A. I. Lvovsky, “Complete characterization of quantum-optical processes,” Science322, 563–566 (2008).
    [CrossRef] [PubMed]
  6. S. Rahimi-Keshari, A. Scherer, A. Mann, A. T. Rezakhani, A. I. Lvovsky, and B. C. Sanders, “Quantum process tomography with coherent states,” New J. Phys.13, 013006 (2011).
    [CrossRef]
  7. A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett.106, 100401 (2011).
    [CrossRef] [PubMed]
  8. G. VanWiggeren and D. Baney, “Swept-wavelength interferometric analysis of multiport components,” IEEE Photon. Technol. Lett.15, 1267–1269 (2003).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  23. J. O. Owens, M. A. Broome, D. N. Biggerstaff, M. E. Goggin, A. Fedrizzi, T. Linjordet, M. Ams, G. D. Marshall, J. Twamley, M. J. Withford, and A. G. White, “Two-photon quantum walks in an elliptical direct-write waveguide array,” New J. Phys.13, 075003 (2011).
    [CrossRef]
  24. S. Aaronson and A. Arkhipov, “The computational complexity of linear optics,” Proc. ACM Symposium on Theory of Computing, San Jose, CA pp. 333–342 (2011).
  25. 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,” Science339, 794–798 (2013).
    [CrossRef]
  26. J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-Min 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 in a photonic chip,” Science339, 798–801 (2013).
    [CrossRef]
  27. B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nature Communications4, 1356 (2013).
    [CrossRef]
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2013

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,” Science339, 794–798 (2013).
[CrossRef]

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-Min 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 in a photonic chip,” Science339, 798–801 (2013).
[CrossRef]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nature Communications4, 1356 (2013).
[CrossRef]

2012

A. Schreiber, A. Gábris, P. P. Rohde, K. Laiho, M. Štefaňák, V. Potoček, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science336, 55–58 (2012).
[CrossRef] [PubMed]

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Two-particle bosonicfermionic quantum walk via integrated photonics,” Phys. Rev. Lett.108, 010502 (2012).
[CrossRef] [PubMed]

A. Laing and J. L. O’Brien, “Super-stable tomography of any linear optical device,” arXiv:1208.2868 (2012).

2011

A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun.2, 224 (2011).
[CrossRef] [PubMed]

S. Rahimi-Keshari, A. Scherer, A. Mann, A. T. Rezakhani, A. I. Lvovsky, and B. C. Sanders, “Quantum process tomography with coherent states,” New J. Phys.13, 013006 (2011).
[CrossRef]

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett.106, 100401 (2011).
[CrossRef] [PubMed]

J. O. Owens, M. A. Broome, D. N. Biggerstaff, M. E. Goggin, A. Fedrizzi, T. Linjordet, M. Ams, G. D. Marshall, J. Twamley, M. J. Withford, and A. G. White, “Two-photon quantum walks in an elliptical direct-write waveguide array,” New J. Phys.13, 075003 (2011).
[CrossRef]

2010

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science329, 1500–1503 (2010).
[CrossRef] [PubMed]

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nature Chemistry2, 106–111 (2010).
[CrossRef] [PubMed]

R. Keil, A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, and A. Tünnermann, “Photon correlations in two-dimensional waveguide arrays and their classical estimate,” Phys. Rev. A81, 023834 (2010).
[CrossRef]

2009

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

2008

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, and A. I. Lvovsky, “Complete characterization of quantum-optical processes,” Science322, 563–566 (2008).
[CrossRef] [PubMed]

2007

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. V. James, A. Gilchrist, and A. G. White, “Experimental demonstration of a compiled version of Shor’s algorithm with quantum entanglement,” Phys. Rev. Lett.99, 250505 (2007).
[CrossRef]

2004

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett.93, 080502 (2004).
[CrossRef]

2003

M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M. Steinberg, “Diagnosis, prescription, and prognosis of a Bell-state filter by quantum process tomography,” Phys. Rev. Lett.91, 120402 (2003).
[CrossRef] [PubMed]

G. VanWiggeren and D. Baney, “Swept-wavelength interferometric analysis of multiport components,” IEEE Photon. Technol. Lett.15, 1267–1269 (2003).
[CrossRef]

2002

2001

A. M. Childs, I. L. Chuang, and D. W. Leung, “Realization of quantum process tomography in NMR,” Phys. Rev. A64, 012314 (2001).
[CrossRef]

1995

C. Mattle, M. Michler, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Non-classical statistics at multiport beam splitters,” Appl. Phys. B60, S111–S117 (1995).

1994

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett.73, 58–61 (1994).
[CrossRef] [PubMed]

1989

A. Peres, “Construction of unitary matrices from observable transition probabilities,” Nucl. Phys. B6, 243245 (1989).

1987

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett.59, 20442046 (1987).
[CrossRef]

1974

H. J. Bernstein, “Must quantum theory assume unrestricted superposition?,” J. Math. Phys.15, 1677 (1974).
[CrossRef]

1955

K. Fan and A. J. Hoffman, “Some metric inequalities in the space of matrices,” Proc. Amer. Math. Soc.6, 111–116 (1955).
[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,” Science339, 794–798 (2013).
[CrossRef]

S. Aaronson and A. Arkhipov, “The computational complexity of linear optics,” Proc. ACM Symposium on Theory of Computing, San Jose, CA pp. 333–342 (2011).

Almeida, M. P.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett.106, 100401 (2011).
[CrossRef] [PubMed]

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nature Chemistry2, 106–111 (2010).
[CrossRef] [PubMed]

Ams, M.

J. O. Owens, M. A. Broome, D. N. Biggerstaff, M. E. Goggin, A. Fedrizzi, T. Linjordet, M. Ams, G. D. Marshall, J. Twamley, M. J. Withford, and A. G. White, “Two-photon quantum walks in an elliptical direct-write waveguide array,” New J. Phys.13, 075003 (2011).
[CrossRef]

Arkhipov, A.

S. Aaronson and A. Arkhipov, “The computational complexity of linear optics,” Proc. ACM Symposium on Theory of Computing, San Jose, CA pp. 333–342 (2011).

Aspuru-Guzik, A.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nature Chemistry2, 106–111 (2010).
[CrossRef] [PubMed]

Baney, D.

G. VanWiggeren and D. Baney, “Swept-wavelength interferometric analysis of multiport components,” IEEE Photon. Technol. Lett.15, 1267–1269 (2003).
[CrossRef]

Barbieri, M.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-Min 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 in a photonic chip,” Science339, 798–801 (2013).
[CrossRef]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nature Communications4, 1356 (2013).
[CrossRef]

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nature Chemistry2, 106–111 (2010).
[CrossRef] [PubMed]

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. V. James, A. Gilchrist, and A. G. White, “Experimental demonstration of a compiled version of Shor’s algorithm with quantum entanglement,” Phys. Rev. Lett.99, 250505 (2007).
[CrossRef]

Bernstein, H. J.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett.73, 58–61 (1994).
[CrossRef] [PubMed]

H. J. Bernstein, “Must quantum theory assume unrestricted superposition?,” J. Math. Phys.15, 1677 (1974).
[CrossRef]

Bertani, P.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett.73, 58–61 (1994).
[CrossRef] [PubMed]

Biamonte, J. D.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nature Chemistry2, 106–111 (2010).
[CrossRef] [PubMed]

Biggerstaff, D. N.

J. O. Owens, M. A. Broome, D. N. Biggerstaff, M. E. Goggin, A. Fedrizzi, T. Linjordet, M. Ams, G. D. Marshall, J. Twamley, M. J. Withford, and A. G. White, “Two-photon quantum walks in an elliptical direct-write waveguide array,” New J. Phys.13, 075003 (2011).
[CrossRef]

Bromberg, Y.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science329, 1500–1503 (2010).
[CrossRef] [PubMed]

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

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,” Science339, 794–798 (2013).
[CrossRef]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nature Communications4, 1356 (2013).
[CrossRef]

J. O. Owens, M. A. Broome, D. N. Biggerstaff, M. E. Goggin, A. Fedrizzi, T. Linjordet, M. Ams, G. D. Marshall, J. Twamley, M. J. Withford, and A. G. White, “Two-photon quantum walks in an elliptical direct-write waveguide array,” New J. Phys.13, 075003 (2011).
[CrossRef]

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett.106, 100401 (2011).
[CrossRef] [PubMed]

Childs, A. M.

A. M. Childs, I. L. Chuang, and D. W. Leung, “Realization of quantum process tomography in NMR,” Phys. Rev. A64, 012314 (2001).
[CrossRef]

Chuang, I. L.

A. M. Childs, I. L. Chuang, and D. W. Leung, “Realization of quantum process tomography in NMR,” Phys. Rev. A64, 012314 (2001).
[CrossRef]

Crespi, A.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Two-particle bosonicfermionic quantum walk via integrated photonics,” Phys. Rev. Lett.108, 010502 (2012).
[CrossRef] [PubMed]

Datta, A.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-Min 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 in a photonic chip,” Science339, 798–801 (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,” Science339, 794–798 (2013).
[CrossRef]

Dreisow, F.

R. Keil, A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, and A. Tünnermann, “Photon correlations in two-dimensional waveguide arrays and their classical estimate,” Phys. Rev. A81, 023834 (2010).
[CrossRef]

Ellenor, C. W.

M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M. Steinberg, “Diagnosis, prescription, and prognosis of a Bell-state filter by quantum process tomography,” Phys. Rev. Lett.91, 120402 (2003).
[CrossRef] [PubMed]

Fan, K.

K. Fan and A. J. Hoffman, “Some metric inequalities in the space of matrices,” Proc. Amer. Math. Soc.6, 111–116 (1955).
[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,” Science339, 794–798 (2013).
[CrossRef]

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett.106, 100401 (2011).
[CrossRef] [PubMed]

J. O. Owens, M. A. Broome, D. N. Biggerstaff, M. E. Goggin, A. Fedrizzi, T. Linjordet, M. Ams, G. D. Marshall, J. Twamley, M. J. Withford, and A. G. White, “Two-photon quantum walks in an elliptical direct-write waveguide array,” New J. Phys.13, 075003 (2011).
[CrossRef]

Figueroa, E.

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, and A. I. Lvovsky, “Complete characterization of quantum-optical processes,” Science322, 563–566 (2008).
[CrossRef] [PubMed]

Gábris, A.

A. Schreiber, A. Gábris, P. P. Rohde, K. Laiho, M. Štefaňák, V. Potoček, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science336, 55–58 (2012).
[CrossRef] [PubMed]

Gates, J. C.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-Min 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 in a photonic chip,” Science339, 798–801 (2013).
[CrossRef]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nature Communications4, 1356 (2013).
[CrossRef]

Gilchrist, A.

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. V. James, A. Gilchrist, and A. G. White, “Experimental demonstration of a compiled version of Shor’s algorithm with quantum entanglement,” Phys. Rev. Lett.99, 250505 (2007).
[CrossRef]

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett.93, 080502 (2004).
[CrossRef]

Gillett, G. G.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nature Chemistry2, 106–111 (2010).
[CrossRef] [PubMed]

Goggin, M. E.

J. O. Owens, M. A. Broome, D. N. Biggerstaff, M. E. Goggin, A. Fedrizzi, T. Linjordet, M. Ams, G. D. Marshall, J. Twamley, M. J. Withford, and A. G. White, “Two-photon quantum walks in an elliptical direct-write waveguide array,” New J. Phys.13, 075003 (2011).
[CrossRef]

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nature Chemistry2, 106–111 (2010).
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A. Schreiber, A. Gábris, P. P. Rohde, K. Laiho, M. Štefaňák, V. Potoček, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science336, 55–58 (2012).
[CrossRef] [PubMed]

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R. Keil, A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, and A. Tünnermann, “Photon correlations in two-dimensional waveguide arrays and their classical estimate,” Phys. Rev. A81, 023834 (2010).
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K. Fan and A. J. Hoffman, “Some metric inequalities in the space of matrices,” Proc. Amer. Math. Soc.6, 111–116 (1955).
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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, 20442046 (1987).
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B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nature Communications4, 1356 (2013).
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J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-Min 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 in a photonic chip,” Science339, 798–801 (2013).
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A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science329, 1500–1503 (2010).
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B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. V. James, A. Gilchrist, and A. G. White, “Experimental demonstration of a compiled version of Shor’s algorithm with quantum entanglement,” Phys. Rev. Lett.99, 250505 (2007).
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J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett.93, 080502 (2004).
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A. Schreiber, A. Gábris, P. P. Rohde, K. Laiho, M. Štefaňák, V. Potoček, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science336, 55–58 (2012).
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B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nature Communications4, 1356 (2013).
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J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-Min 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 in a photonic chip,” Science339, 798–801 (2013).
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B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nature Chemistry2, 106–111 (2010).
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R. Keil, A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, and A. Tünnermann, “Photon correlations in two-dimensional waveguide arrays and their classical estimate,” Phys. Rev. A81, 023834 (2010).
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J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-Min 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 in a photonic chip,” Science339, 798–801 (2013).
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B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nature Communications4, 1356 (2013).
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M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, and A. I. Lvovsky, “Complete characterization of quantum-optical processes,” Science322, 563–566 (2008).
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A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett.106, 100401 (2011).
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J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-Min 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 in a photonic chip,” Science339, 798–801 (2013).
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B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nature Communications4, 1356 (2013).
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M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, and A. I. Lvovsky, “Complete characterization of quantum-optical processes,” Science322, 563–566 (2008).
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A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science329, 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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A. Schreiber, A. Gábris, P. P. Rohde, K. Laiho, M. Štefaňák, V. Potoček, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science336, 55–58 (2012).
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A. Laing and J. L. O’Brien, “Super-stable tomography of any linear optical device,” arXiv:1208.2868 (2012).

A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun.2, 224 (2011).
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J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-Min 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 in a photonic chip,” Science339, 798–801 (2013).
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B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nature Communications4, 1356 (2013).
[CrossRef]

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. V. James, A. Gilchrist, and A. G. White, “Experimental demonstration of a compiled version of Shor’s algorithm with quantum entanglement,” Phys. Rev. Lett.99, 250505 (2007).
[CrossRef]

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett.93, 080502 (2004).
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B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nature Chemistry2, 106–111 (2010).
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B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. V. James, A. Gilchrist, and A. G. White, “Experimental demonstration of a compiled version of Shor’s algorithm with quantum entanglement,” Phys. Rev. Lett.99, 250505 (2007).
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J. O. Owens, M. A. Broome, D. N. Biggerstaff, M. E. Goggin, A. Fedrizzi, T. Linjordet, M. Ams, G. D. Marshall, J. Twamley, M. J. Withford, and A. G. White, “Two-photon quantum walks in an elliptical direct-write waveguide array,” New J. Phys.13, 075003 (2011).
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A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science329, 1500–1503 (2010).
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M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, and A. I. Lvovsky, “Complete characterization of quantum-optical processes,” Science322, 563–566 (2008).
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Lvovsky, A. I.

S. Rahimi-Keshari, A. Scherer, A. Mann, A. T. Rezakhani, A. I. Lvovsky, and B. C. Sanders, “Quantum process tomography with coherent states,” New J. Phys.13, 013006 (2011).
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M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, and A. I. Lvovsky, “Complete characterization of quantum-optical processes,” Science322, 563–566 (2008).
[CrossRef] [PubMed]

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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, 20442046 (1987).
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Mann, A.

S. Rahimi-Keshari, A. Scherer, A. Mann, A. T. Rezakhani, A. I. Lvovsky, and B. C. Sanders, “Quantum process tomography with coherent states,” New J. Phys.13, 013006 (2011).
[CrossRef]

Marshall, G. D.

J. O. Owens, M. A. Broome, D. N. Biggerstaff, M. E. Goggin, A. Fedrizzi, T. Linjordet, M. Ams, G. D. Marshall, J. Twamley, M. J. Withford, and A. G. White, “Two-photon quantum walks in an elliptical direct-write waveguide array,” New J. Phys.13, 075003 (2011).
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L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Two-particle bosonicfermionic quantum walk via integrated photonics,” Phys. Rev. Lett.108, 010502 (2012).
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A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science329, 1500–1503 (2010).
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A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science329, 1500–1503 (2010).
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B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nature Communications4, 1356 (2013).
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J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-Min 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 in a photonic chip,” Science339, 798–801 (2013).
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M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M. Steinberg, “Diagnosis, prescription, and prognosis of a Bell-state filter by quantum process tomography,” Phys. Rev. Lett.91, 120402 (2003).
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A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett.106, 100401 (2011).
[CrossRef] [PubMed]

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nature Chemistry2, 106–111 (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).
[CrossRef] [PubMed]

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R. Keil, A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, and A. Tünnermann, “Photon correlations in two-dimensional waveguide arrays and their classical estimate,” Phys. Rev. A81, 023834 (2010).
[CrossRef]

O’Brien, J. L.

A. Laing and J. L. O’Brien, “Super-stable tomography of any linear optical device,” arXiv:1208.2868 (2012).

A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun.2, 224 (2011).
[CrossRef] [PubMed]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science329, 1500–1503 (2010).
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J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett.93, 080502 (2004).
[CrossRef]

Osellame, R.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Two-particle bosonicfermionic quantum walk via integrated photonics,” Phys. Rev. Lett.108, 010502 (2012).
[CrossRef] [PubMed]

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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, 20442046 (1987).
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Owens, J. O.

J. O. Owens, M. A. Broome, D. N. Biggerstaff, M. E. Goggin, A. Fedrizzi, T. Linjordet, M. Ams, G. D. Marshall, J. Twamley, M. J. Withford, and A. G. White, “Two-photon quantum walks in an elliptical direct-write waveguide array,” New J. Phys.13, 075003 (2011).
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A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun.2, 224 (2011).
[CrossRef] [PubMed]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science329, 1500–1503 (2010).
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Politi, A.

A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun.2, 224 (2011).
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A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science329, 1500–1503 (2010).
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Potocek, V.

A. Schreiber, A. Gábris, P. P. Rohde, K. Laiho, M. Štefaňák, V. Potoček, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science336, 55–58 (2012).
[CrossRef] [PubMed]

Poulios, K.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science329, 1500–1503 (2010).
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B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nature Chemistry2, 106–111 (2010).
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Prather, D.

Pryde, G. J.

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett.93, 080502 (2004).
[CrossRef]

Rabitz, H.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett.106, 100401 (2011).
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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,” Science339, 794–798 (2013).
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S. Rahimi-Keshari, A. Scherer, A. Mann, A. T. Rezakhani, A. I. Lvovsky, and B. C. Sanders, “Quantum process tomography with coherent states,” New J. Phys.13, 013006 (2011).
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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,” Science339, 794–798 (2013).
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J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett.93, 080502 (2004).
[CrossRef]

Ramponi, R.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Two-particle bosonicfermionic quantum walk via integrated photonics,” Phys. Rev. Lett.108, 010502 (2012).
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M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett.73, 58–61 (1994).
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S. Rahimi-Keshari, A. Scherer, A. Mann, A. T. Rezakhani, A. I. Lvovsky, and B. C. Sanders, “Quantum process tomography with coherent states,” New J. Phys.13, 013006 (2011).
[CrossRef]

Rohde, P. P.

A. Schreiber, A. Gábris, P. P. Rohde, K. Laiho, M. Štefaňák, V. Potoček, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science336, 55–58 (2012).
[CrossRef] [PubMed]

Rudolph, T.

A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun.2, 224 (2011).
[CrossRef] [PubMed]

Sanders, B. C.

S. Rahimi-Keshari, A. Scherer, A. Mann, A. T. Rezakhani, A. I. Lvovsky, and B. C. Sanders, “Quantum process tomography with coherent states,” New J. Phys.13, 013006 (2011).
[CrossRef]

M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders, and A. I. Lvovsky, “Complete characterization of quantum-optical processes,” Science322, 563–566 (2008).
[CrossRef] [PubMed]

Sansoni, L.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Two-particle bosonicfermionic quantum walk via integrated photonics,” Phys. Rev. Lett.108, 010502 (2012).
[CrossRef] [PubMed]

Scherer, A.

S. Rahimi-Keshari, A. Scherer, A. Mann, A. T. Rezakhani, A. I. Lvovsky, and B. C. Sanders, “Quantum process tomography with coherent states,” New J. Phys.13, 013006 (2011).
[CrossRef]

Schneider, S.

M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M. Steinberg, “Diagnosis, prescription, and prognosis of a Bell-state filter by quantum process tomography,” Phys. Rev. Lett.91, 120402 (2003).
[CrossRef] [PubMed]

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A. Schreiber, A. Gábris, P. P. Rohde, K. Laiho, M. Štefaňák, V. Potoček, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science336, 55–58 (2012).
[CrossRef] [PubMed]

Sciarrino, F.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Two-particle bosonicfermionic quantum walk via integrated photonics,” Phys. Rev. Lett.108, 010502 (2012).
[CrossRef] [PubMed]

Shabani, A.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett.106, 100401 (2011).
[CrossRef] [PubMed]

Sharkawy, A.

Shi, S.

Silberberg, Y.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science329, 1500–1503 (2010).
[CrossRef] [PubMed]

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

Silberhorn, C.

A. Schreiber, A. Gábris, P. P. Rohde, K. Laiho, M. Štefaňák, V. Potoček, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science336, 55–58 (2012).
[CrossRef] [PubMed]

Smith, B. J.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-Min 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 in a photonic chip,” Science339, 798–801 (2013).
[CrossRef]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nature Communications4, 1356 (2013).
[CrossRef]

Smith, P. G. R.

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nature Communications4, 1356 (2013).
[CrossRef]

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-Min 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 in a photonic chip,” Science339, 798–801 (2013).
[CrossRef]

Soref, R.

Spring, J. B.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-Min 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 in a photonic chip,” Science339, 798–801 (2013).
[CrossRef]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nature Communications4, 1356 (2013).
[CrossRef]

Štefanák, M.

A. Schreiber, A. Gábris, P. P. Rohde, K. Laiho, M. Štefaňák, V. Potoček, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science336, 55–58 (2012).
[CrossRef] [PubMed]

Steinberg, A. M.

M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M. Steinberg, “Diagnosis, prescription, and prognosis of a Bell-state filter by quantum process tomography,” Phys. Rev. Lett.91, 120402 (2003).
[CrossRef] [PubMed]

Szameit, A.

R. Keil, A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, and A. Tünnermann, “Photon correlations in two-dimensional waveguide arrays and their classical estimate,” Phys. Rev. A81, 023834 (2010).
[CrossRef]

Thomas-Peter, N.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-Min 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 in a photonic chip,” Science339, 798–801 (2013).
[CrossRef]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nature Communications4, 1356 (2013).
[CrossRef]

Thompson, M. G.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science329, 1500–1503 (2010).
[CrossRef] [PubMed]

Tünnermann, A.

R. Keil, A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, and A. Tünnermann, “Photon correlations in two-dimensional waveguide arrays and their classical estimate,” Phys. Rev. A81, 023834 (2010).
[CrossRef]

Twamley, J.

J. O. Owens, M. A. Broome, D. N. Biggerstaff, M. E. Goggin, A. Fedrizzi, T. Linjordet, M. Ams, G. D. Marshall, J. Twamley, M. J. Withford, and A. G. White, “Two-photon quantum walks in an elliptical direct-write waveguide array,” New J. Phys.13, 075003 (2011).
[CrossRef]

Vallone, G.

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Two-particle bosonicfermionic quantum walk via integrated photonics,” Phys. Rev. Lett.108, 010502 (2012).
[CrossRef] [PubMed]

VanWiggeren, G.

G. VanWiggeren and D. Baney, “Swept-wavelength interferometric analysis of multiport components,” IEEE Photon. Technol. Lett.15, 1267–1269 (2003).
[CrossRef]

Walmsley, I. A.

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-Min 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 in a photonic chip,” Science339, 798–801 (2013).
[CrossRef]

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nature Communications4, 1356 (2013).
[CrossRef]

Weinfurter, H.

C. Mattle, M. Michler, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Non-classical statistics at multiport beam splitters,” Appl. Phys. B60, S111–S117 (1995).

Weinhold, T. J.

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. V. James, A. Gilchrist, and A. G. White, “Experimental demonstration of a compiled version of Shor’s algorithm with quantum entanglement,” Phys. Rev. Lett.99, 250505 (2007).
[CrossRef]

White, A. G.

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,” Science339, 794–798 (2013).
[CrossRef]

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett.106, 100401 (2011).
[CrossRef] [PubMed]

J. O. Owens, M. A. Broome, D. N. Biggerstaff, M. E. Goggin, A. Fedrizzi, T. Linjordet, M. Ams, G. D. Marshall, J. Twamley, M. J. Withford, and A. G. White, “Two-photon quantum walks in an elliptical direct-write waveguide array,” New J. Phys.13, 075003 (2011).
[CrossRef]

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nature Chemistry2, 106–111 (2010).
[CrossRef] [PubMed]

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. V. James, A. Gilchrist, and A. G. White, “Experimental demonstration of a compiled version of Shor’s algorithm with quantum entanglement,” Phys. Rev. Lett.99, 250505 (2007).
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J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett.93, 080502 (2004).
[CrossRef]

Whitfield, J. D.

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nature Chemistry2, 106–111 (2010).
[CrossRef] [PubMed]

Withford, M. J.

J. O. Owens, M. A. Broome, D. N. Biggerstaff, M. E. Goggin, A. Fedrizzi, T. Linjordet, M. Ams, G. D. Marshall, J. Twamley, M. J. Withford, and A. G. White, “Two-photon quantum walks in an elliptical direct-write waveguide array,” New J. Phys.13, 075003 (2011).
[CrossRef]

Wörhoff, K.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science329, 1500–1503 (2010).
[CrossRef] [PubMed]

Zeilinger, A.

C. Mattle, M. Michler, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Non-classical statistics at multiport beam splitters,” Appl. Phys. B60, S111–S117 (1995).

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett.73, 58–61 (1994).
[CrossRef] [PubMed]

Zhou, X.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science329, 1500–1503 (2010).
[CrossRef] [PubMed]

Zukowski, M.

C. Mattle, M. Michler, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Non-classical statistics at multiport beam splitters,” Appl. Phys. B60, S111–S117 (1995).

Appl. Phys. B

C. Mattle, M. Michler, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Non-classical statistics at multiport beam splitters,” Appl. Phys. B60, S111–S117 (1995).

IEEE Photon. Technol. Lett.

G. VanWiggeren and D. Baney, “Swept-wavelength interferometric analysis of multiport components,” IEEE Photon. Technol. Lett.15, 1267–1269 (2003).
[CrossRef]

J. Math. Phys.

H. J. Bernstein, “Must quantum theory assume unrestricted superposition?,” J. Math. Phys.15, 1677 (1974).
[CrossRef]

Nat. Commun.

A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun.2, 224 (2011).
[CrossRef] [PubMed]

Nature Chemistry

B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nature Chemistry2, 106–111 (2010).
[CrossRef] [PubMed]

Nature Communications

B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nature Communications4, 1356 (2013).
[CrossRef]

New J. Phys.

J. O. Owens, M. A. Broome, D. N. Biggerstaff, M. E. Goggin, A. Fedrizzi, T. Linjordet, M. Ams, G. D. Marshall, J. Twamley, M. J. Withford, and A. G. White, “Two-photon quantum walks in an elliptical direct-write waveguide array,” New J. Phys.13, 075003 (2011).
[CrossRef]

S. Rahimi-Keshari, A. Scherer, A. Mann, A. T. Rezakhani, A. I. Lvovsky, and B. C. Sanders, “Quantum process tomography with coherent states,” New J. Phys.13, 013006 (2011).
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Nucl. Phys. B

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R. Keil, A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, and A. Tünnermann, “Photon correlations in two-dimensional waveguide arrays and their classical estimate,” Phys. Rev. A81, 023834 (2010).
[CrossRef]

A. M. Childs, I. L. Chuang, and D. W. Leung, “Realization of quantum process tomography in NMR,” Phys. Rev. A64, 012314 (2001).
[CrossRef]

Phys. Rev. Lett.

M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M. Steinberg, “Diagnosis, prescription, and prognosis of a Bell-state filter by quantum process tomography,” Phys. Rev. Lett.91, 120402 (2003).
[CrossRef] [PubMed]

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett.73, 58–61 (1994).
[CrossRef] [PubMed]

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-NOT gate,” Phys. Rev. Lett.93, 080502 (2004).
[CrossRef]

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett.106, 100401 (2011).
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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).
[CrossRef] [PubMed]

B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. V. James, A. Gilchrist, and A. G. White, “Experimental demonstration of a compiled version of Shor’s algorithm with quantum entanglement,” Phys. Rev. Lett.99, 250505 (2007).
[CrossRef]

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Two-particle bosonicfermionic quantum walk via integrated photonics,” Phys. Rev. Lett.108, 010502 (2012).
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A. Schreiber, A. Gábris, P. P. Rohde, K. Laiho, M. Štefaňák, V. Potoček, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science336, 55–58 (2012).
[CrossRef] [PubMed]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science329, 1500–1503 (2010).
[CrossRef] [PubMed]

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,” Science339, 794–798 (2013).
[CrossRef]

J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-Min 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 in a photonic chip,” Science339, 798–801 (2013).
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A. Laing and J. L. O’Brien, “Super-stable tomography of any linear optical device,” arXiv:1208.2868 (2012).

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Figures (4)

Fig. 1
Fig. 1

Scheme for characterizing a linear-optical network M. (a) Using a 50:50 beam splitter (BS) and phase shifter (ϕ) a dual-mode coherent state, |α〉, is prepared and sent through M, where |α2〉= |eα1〉. By sequentially inputting |α2〉 into modes 2,3,...,N, and varying the phase over at least 2π, all phases of matrix M can be directly determined. (b) Experimental realization. The device-under-test is a 3×3-mode fused-fiber beam splitter (FBS), which constitutes a 6×6 optical network including polarization. Orthogonal polarization modes are resolved using fibre polarization beam splitters (FPBS) at its outputs. Interferometric probe states between pair-wise input combinations are prepared with two polarization beam displacers (BD), and half-wave plates (HWP). The outputs are monitored with fast photo-diodes (PD) connected to an oscilloscope (OSC) while the phase ϕ is scanned.

Fig. 2
Fig. 2

Experimental characterization of a linear-optical device. (a) Moduli rjk of the experimentally measured M. The x and y axes correspond to the input and output modes, j and k respectively. (b) Representative experimental data for obtaining θjk when injecting the variable-phase dual-mode coherent state into input modes 1 and 3. The amplitudes (voltage at output photo-diodes) of the six output modes (1–6 from top-to-bottom) oscillate as the phase ϕ is swept in time. Red and blue lines are measured data and theoretical fits to Acos(ϕθjk) respectively. (c) Phases, θjk, of the measured matrix M. The entire characterization method was performed 10 times to obtain experimental uncertainties; error bars are not visible on the scales shown.

Fig. 3
Fig. 3

Independent verification of the measured matrix M. (a) Experimental schematic. A pair of 820 nm photons is generated via type-I spontaneous parametric downconversion (SPDC) in a nonlinear β-barium-borate (BBO) crystal pumped with a mode-locked pulsed laser at 410 nm. After being spectrally filtered (FWHM 2 nm) individual downconverted photons are steered into the optical modes of the linear-optical network by a series of beam displacers (BD) and half-wave plates (HWP). The temporal overlap, Δt, between the input photons is controlled via a micro-translation stage at one of the inputs. Output photons are detected using avalanche photo-diodes (APD) whose coincident signals—monitored using a commercially available counting logic—are used to post-select two single photon events. (b) Measured nonclassical visibilities vs. predicted visibilities for photons input into modes {1, 3} and {1, 6}. Red bars show the directly measured nonclassical visibilities; blue bars show the predictions from the measured matrix M; errors are given at the top of each data point. Numbers on the x-axis show the corresponding output modes.

Fig. 4
Fig. 4

(a) Scheme for characterizing path-independent loss in the linear optical network. Virtual beam splitters are placed at each input mode of the optical network represented by a lossless matrix B. The resulting matrix V is a 2N×2N matrix and accounts for path-independent loss. (b) The matrix VV. The diagonal hatched squares are equal to unity by construction and the off-diagonal elements are coloured according to their value given by the color bar.

Equations (14)

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

a j = k = 1 N M j k b k .
β k = j = 1 N M j k α j ,
r j k = I k I , k = 1 , 2 , , N .
I k = I | M 1 k + M j k e i ϕ | 2 .
I 1 = I [ r 11 2 + r j 1 2 + 2 r 11 r j 1 cos ( ϕ ) ] ,
I k = I [ r 1 k 2 + r j k 2 + 2 r 1 k r j k cos ( ϕ + θ j k ) ] .
θ j k = 2 π ϕ .
Q i j k l = 1 1 + δ i j | M i k M j l + M i l M j k | 2 ,
C i j k l = | M i k M j l | 2 + | M i l M j k | 2 .
S j = ( η j 1 η j 2 1 η j 2 η j ) ,
S tot = ( η η ˜ η ˜ η ) ,
η = diag ( η 1 , η 2 , , η N ) , η ˜ = diag ( 1 η 1 2 , 1 η 2 2 , , 1 η N 2 ) .
V = S tot × ( B O O ) = ( η B η ˜ η ˜ B η ) ,
η j = 1 I k = 1 N I k .

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