Abstract

We numerically investigate the implementation of Haar-random unitarity transformations and Fourier transformations in photonic devices consisting of beam splitters and phase shifters, which are used for integrated photonics implementations of boson sampling. The distribution of reflectivities required to implement an arbitrary unitary transformation is skewed towards low values, and this skew becomes stronger the larger the number of modes. A realistic implementation using Mach-Zehnder interferometers is incapable of achieving the low values required and thus has limited fidelity. We show that numerical optimisation and adding extra beam splitters to the network can help to restore fidelity.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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  19. All code used in this work is available at https://github.com/jrenema/RandomUnitaries
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    [Crossref]
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2017 (1)

N. J. Russell, L. Chakhmakhchyan, J.L. O’Brien, and A. Laing, “Direct dialling of haar random unitary matrices,” New J. Phys. 19, 033007 (2017).
[Crossref]

2016 (4)

2015 (5)

D. A. B. Miller, “Perfect optics with imperfect components,” Optica 2(8), 747–750 (2015).
[Crossref]

J. Mower, N. C. Harris, G. R. Steinbrecher, Y. Lahini, and D. Englund, “High-fidelity quantum state evolution in imperfect photonic integrated circuits,” Phys. Rev. A 92(3),1–7 (2015).
[Crossref]

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1(3), e1400255 (2015).
[Crossref] [PubMed]

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015).
[Crossref] [PubMed]

A. Arkhipov, “Boson Sampling is Robust to Small Errors in the Network Matrix,” Phys. Rev. A 92, 062326 (2015).
[Crossref]

2014 (2)

B.J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8(10), 770–774 (2014).
[Crossref]

J. C. Mikkelsen, W. D. Sacher, and J. K. S. Poon, “Dimensional variation tolerant silicon-on-insulator directional couplers,” Opt. Express 22(3), 3145–3150 (2014).
[Crossref] [PubMed]

2013 (8)

D. A. B. Miller, “Reconfigurable add-drop multiplexer for spatial modes,” Opt. Express 21(17), 20220–20229 (2013).
[Crossref] [PubMed]

D. A. B. Miller, “Self-aligning universal beam coupler,” Opt. Express 21(5), 6360–6370 (2013).
[Crossref] [PubMed]

D. A. B. Miller, “Self-configuring universal linear optical component,” Photonics Res. 1(1), 1 (2013).
[Crossref]

S. Aaronson and A. Arkhipov, “The Computational Complexity of Linear Optics,” Theory Comput. 9(4), 143–252 (2013).
[Crossref]

A. Crespi, R. Osellame, R. Ramponi, D. J. Bord, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, ‘multimode interferometers with arbitrary designs for photonic boson sampling,” Nat. Photonics 7, 545–549 (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(6121), 798–801 (2013).
[Crossref]

M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nat. Photonics 7(7), 540–544 (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]

2009 (1)

D. O. Kundys, H. C. Gates, S. Dasgupta, C. B. E. Gawith, and P. G. R. Smith, “Use of cross-couplers to decrease size of UV written photonic circuits,” IEEE Photonics Technol. Lett. 21(13), 947–949 (2009).
[Crossref]

2004 (1)

1999 (1)

C. A. Fuchs and J. van de Graaf, “Cryptographic Distinguishability Measures for Quantum Mechanical States,” IEEE Trans. Inf. Theory 45, 1216 (1999).
[Crossref]

1997 (1)

P. Shor, “Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer,” SIAM J. Comput. 26(5), 1484–1509 (1997).
[Crossref]

1994 (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73(1), 58–61 (1994).
[Crossref] [PubMed]

Aaronson, S.

S. Aaronson and A. Arkhipov, “The Computational Complexity of Linear Optics,” Theory Comput. 9(4), 143–252 (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]

Alippi, A.

Annoni, A.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D.A. B. Miller, A. Melloni, and F. Morichetti, “All-optical mode unscrambling on a silicon photonic chip,” arXiv:1512.06762

Arkhipov, A.

A. Arkhipov, “Boson Sampling is Robust to Small Errors in the Network Matrix,” Phys. Rev. A 92, 062326 (2015).
[Crossref]

S. Aaronson and A. Arkhipov, “The Computational Complexity of Linear Optics,” Theory Comput. 9(4), 143–252 (2013).
[Crossref]

Barbieri, M.

B.J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8(10), 770–774 (2014).
[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(6121), 798–801 (2013).
[Crossref]

Barkhofen, S.

C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, “Gaussian Boson Sampling,” arXiv:1612.01199.

Bentivegna, M.

A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N Viggianiello, L Innocenti, P Mataloni, and F Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
[Crossref] [PubMed]

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1(3), e1400255 (2015).
[Crossref] [PubMed]

N. Spagnolo, E. Maiorino, C. Vitelli, M. Bentivegna, A. Crespi, R. Ramponi, P. Mataloni, R. Osellame, and F. Sciarrino, “Learning an unknown transformation via a genetic approach,” arXiv:1610.03291.

Bernstein, H. J.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73(1), 58–61 (1994).
[Crossref] [PubMed]

Bertani, P.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73(1), 58–61 (1994).
[Crossref] [PubMed]

Bord, D. J.

A. Crespi, R. Osellame, R. Ramponi, D. J. Bord, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, ‘multimode interferometers with arbitrary designs for photonic boson sampling,” Nat. Photonics 7, 545–549 (2013).
[Crossref]

Brod, D. J.

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1(3), e1400255 (2015).
[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,” Science 339, 794–798 (2013).
[Crossref]

Carminati, M.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D.A. B. Miller, A. Melloni, and F. Morichetti, “All-optical mode unscrambling on a silicon photonic chip,” arXiv:1512.06762

Carolan, J.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015).
[Crossref] [PubMed]

Chakhmakhchyan, L.

N. J. Russell, L. Chakhmakhchyan, J.L. O’Brien, and A. Laing, “Direct dialling of haar random unitary matrices,” New J. Phys. 19, 033007 (2017).
[Crossref]

Chen, M.

H. Wang, Y. He, Y. Li, Z. Su, B. Li, H. Huang, X. Ding, M. Chen, C. Liu, J. Qin, and J. Li, “Multi-photon boson-sampling machines beating early classical computers,” arXiv:1612.06956.

Clements, W. R.

Crespi, A.

A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N Viggianiello, L Innocenti, P Mataloni, and F Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
[Crossref] [PubMed]

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1(3), e1400255 (2015).
[Crossref] [PubMed]

A. Crespi, R. Osellame, R. Ramponi, D. J. Bord, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, ‘multimode interferometers with arbitrary designs for photonic boson sampling,” Nat. Photonics 7, 545–549 (2013).
[Crossref]

N. Spagnolo, E. Maiorino, C. Vitelli, M. Bentivegna, A. Crespi, R. Ramponi, P. Mataloni, R. Osellame, and F. Sciarrino, “Learning an unknown transformation via a genetic approach,” arXiv:1610.03291.

Dakic, B.

M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nat. Photonics 7(7), 540–544 (2013).
[Crossref]

Dasgupta, S.

D. O. Kundys, H. C. Gates, S. Dasgupta, C. B. E. Gawith, and P. G. R. Smith, “Use of cross-couplers to decrease size of UV written photonic circuits,” IEEE Photonics Technol. Lett. 21(13), 947–949 (2009).
[Crossref]

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(6121), 798–801 (2013).
[Crossref]

Ding, X.

H. Wang, Y. He, Y. Li, Z. Su, B. Li, H. Huang, X. Ding, M. Chen, C. Liu, J. Qin, and J. Li, “Multi-photon boson-sampling machines beating early classical computers,” arXiv:1612.06956.

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]

Englund, D.

J. Mower, N. C. Harris, G. R. Steinbrecher, Y. Lahini, and D. Englund, “High-fidelity quantum state evolution in imperfect photonic integrated circuits,” Phys. Rev. A 92(3),1–7 (2015).
[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]

Ferrari, G.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D.A. B. Miller, A. Melloni, and F. Morichetti, “All-optical mode unscrambling on a silicon photonic chip,” arXiv:1512.06762

Flamini, F.

A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N Viggianiello, L Innocenti, P Mataloni, and F Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
[Crossref] [PubMed]

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1(3), e1400255 (2015).
[Crossref] [PubMed]

Fletcher, R.

R. Fletcher, Practical Methods of Optimization, 2nd ed. (John Wiley & Sons, 1987).

Fuchs, C. A.

C. A. Fuchs and J. van de Graaf, “Cryptographic Distinguishability Measures for Quantum Mechanical States,” IEEE Trans. Inf. Theory 45, 1216 (1999).
[Crossref]

Galvão, E. F.

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1(3), e1400255 (2015).
[Crossref] [PubMed]

A. Crespi, R. Osellame, R. Ramponi, D. J. Bord, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, ‘multimode interferometers with arbitrary designs for photonic boson sampling,” Nat. Photonics 7, 545–549 (2013).
[Crossref]

Gates, H. C.

D. O. Kundys, H. C. Gates, S. Dasgupta, C. B. E. Gawith, and P. G. R. Smith, “Use of cross-couplers to decrease size of UV written photonic circuits,” IEEE Photonics Technol. Lett. 21(13), 947–949 (2009).
[Crossref]

Gates, J. C.

B.J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8(10), 770–774 (2014).
[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(6121), 798–801 (2013).
[Crossref]

Gawith, C. B. E.

D. O. Kundys, H. C. Gates, S. Dasgupta, C. B. E. Gawith, and P. G. R. Smith, “Use of cross-couplers to decrease size of UV written photonic circuits,” IEEE Photonics Technol. Lett. 21(13), 947–949 (2009).
[Crossref]

Guglielmi, E.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D.A. B. Miller, A. Melloni, and F. Morichetti, “All-optical mode unscrambling on a silicon photonic chip,” arXiv:1512.06762

Hamilton, C. S.

C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, “Gaussian Boson Sampling,” arXiv:1612.01199.

Harris, N. C.

J. Mower, N. C. Harris, G. R. Steinbrecher, Y. Lahini, and D. Englund, “High-fidelity quantum state evolution in imperfect photonic integrated circuits,” Phys. Rev. A 92(3),1–7 (2015).
[Crossref]

Harrold, C.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015).
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Hashimoto, T.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015).
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H. Wang, Y. He, Y. Li, Z. Su, B. Li, H. Huang, X. Ding, M. Chen, C. Liu, J. Qin, and J. Li, “Multi-photon boson-sampling machines beating early classical computers,” arXiv:1612.06956.

Heilmann, R.

M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nat. Photonics 7(7), 540–544 (2013).
[Crossref]

Honjo, T.

Huang, H.

H. Wang, Y. He, Y. Li, Z. Su, B. Li, H. Huang, X. Ding, M. Chen, C. Liu, J. Qin, and J. Li, “Multi-photon boson-sampling machines beating early classical computers,” arXiv:1612.06956.

Humphreys, P. C.

W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, “An Optimal Design for Universal Multiport Interferometers,” Optica 2(8), 1–8 (2016).

B.J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8(10), 770–774 (2014).
[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(6121), 798–801 (2013).
[Crossref]

Innocenti, L

A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N Viggianiello, L Innocenti, P Mataloni, and F Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
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Inoue, K.

Itoh, M.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015).
[Crossref] [PubMed]

Jex, I.

C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, “Gaussian Boson Sampling,” arXiv:1612.01199.

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(6121), 798–801 (2013).
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Jin, X-M.

B.J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8(10), 770–774 (2014).
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Kolthammer, W. S.

W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, “An Optimal Design for Universal Multiport Interferometers,” Optica 2(8), 1–8 (2016).

B.J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8(10), 770–774 (2014).
[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(6121), 798–801 (2013).
[Crossref]

Kruse, R.

C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, “Gaussian Boson Sampling,” arXiv:1612.01199.

Kundys, D.

B.J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8(10), 770–774 (2014).
[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(6121), 798–801 (2013).
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Kundys, D. O.

D. O. Kundys, H. C. Gates, S. Dasgupta, C. B. E. Gawith, and P. G. R. Smith, “Use of cross-couplers to decrease size of UV written photonic circuits,” IEEE Photonics Technol. Lett. 21(13), 947–949 (2009).
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Lahini, Y.

J. Mower, N. C. Harris, G. R. Steinbrecher, Y. Lahini, and D. Englund, “High-fidelity quantum state evolution in imperfect photonic integrated circuits,” Phys. Rev. A 92(3),1–7 (2015).
[Crossref]

Laing, A.

N. J. Russell, L. Chakhmakhchyan, J.L. O’Brien, and A. Laing, “Direct dialling of haar random unitary matrices,” New J. Phys. 19, 033007 (2017).
[Crossref]

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015).
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N. J. Russell, J. L. O’Brien, and A. Laing, “Direct dialling of Haar random unitary matrices,” arXiv:1506.06220v1.

Langford, N. K.

B.J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8(10), 770–774 (2014).
[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(6121), 798–801 (2013).
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M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1(3), e1400255 (2015).
[Crossref] [PubMed]

Li, B.

H. Wang, Y. He, Y. Li, Z. Su, B. Li, H. Huang, X. Ding, M. Chen, C. Liu, J. Qin, and J. Li, “Multi-photon boson-sampling machines beating early classical computers,” arXiv:1612.06956.

Li, J.

H. Wang, Y. He, Y. Li, Z. Su, B. Li, H. Huang, X. Ding, M. Chen, C. Liu, J. Qin, and J. Li, “Multi-photon boson-sampling machines beating early classical computers,” arXiv:1612.06956.

Li, Y.

H. Wang, Y. He, Y. Li, Z. Su, B. Li, H. Huang, X. Ding, M. Chen, C. Liu, J. Qin, and J. Li, “Multi-photon boson-sampling machines beating early classical computers,” arXiv:1612.06956.

Liu, C.

H. Wang, Y. He, Y. Li, Z. Su, B. Li, H. Huang, X. Ding, M. Chen, C. Liu, J. Qin, and J. Li, “Multi-photon boson-sampling machines beating early classical computers,” arXiv:1612.06956.

Maiorino, E.

A. Crespi, R. Osellame, R. Ramponi, D. J. Bord, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, ‘multimode interferometers with arbitrary designs for photonic boson sampling,” Nat. Photonics 7, 545–549 (2013).
[Crossref]

N. Spagnolo, E. Maiorino, C. Vitelli, M. Bentivegna, A. Crespi, R. Ramponi, P. Mataloni, R. Osellame, and F. Sciarrino, “Learning an unknown transformation via a genetic approach,” arXiv:1610.03291.

Marshall, G. D.

C. M. Wilkes, X. Qiang, J. Wang, R. Santagati, S. Paesani, X. Zhou, D. A. B. Miller, G. D. Marshall, M. G. Thompson, and J. L. O’Brien, “60 dB high-extinction auto-configured Mach-Zehnder interferometer,” Opt. Lett. 41, 5318–5321 (2016).
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J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015).
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J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015).
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Mataloni, P

A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N Viggianiello, L Innocenti, P Mataloni, and F Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
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Mataloni, P.

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1(3), e1400255 (2015).
[Crossref] [PubMed]

A. Crespi, R. Osellame, R. Ramponi, D. J. Bord, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, ‘multimode interferometers with arbitrary designs for photonic boson sampling,” Nat. Photonics 7, 545–549 (2013).
[Crossref]

N. Spagnolo, E. Maiorino, C. Vitelli, M. Bentivegna, A. Crespi, R. Ramponi, P. Mataloni, R. Osellame, and F. Sciarrino, “Learning an unknown transformation via a genetic approach,” arXiv:1610.03291.

Matsuda, N.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015).
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J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015).
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Melloni, A.

D. Melati, A. Alippi, and A. Melloni, “Reconfigurable photonic integrated mode (de)multiplexer for SDM fiber transmission,” Opt. Express 24(12), 12625 (2016).
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A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D.A. B. Miller, A. Melloni, and F. Morichetti, “All-optical mode unscrambling on a silicon photonic chip,” arXiv:1512.06762

Metcalf, B. J.

W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, “An Optimal Design for Universal Multiport Interferometers,” Optica 2(8), 1–8 (2016).

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(6121), 798–801 (2013).
[Crossref]

Metcalf, B.J.

B.J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8(10), 770–774 (2014).
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Miller, D. A. B.

Miller, D.A. B.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D.A. B. Miller, A. Melloni, and F. Morichetti, “All-optical mode unscrambling on a silicon photonic chip,” arXiv:1512.06762

Morichetti, F.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D.A. B. Miller, A. Melloni, and F. Morichetti, “All-optical mode unscrambling on a silicon photonic chip,” arXiv:1512.06762

Mower, J.

J. Mower, N. C. Harris, G. R. Steinbrecher, Y. Lahini, and D. Englund, “High-fidelity quantum state evolution in imperfect photonic integrated circuits,” Phys. Rev. A 92(3),1–7 (2015).
[Crossref]

Nolte, S.

M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nat. Photonics 7(7), 540–544 (2013).
[Crossref]

O’Brien, J. L.

C. M. Wilkes, X. Qiang, J. Wang, R. Santagati, S. Paesani, X. Zhou, D. A. B. Miller, G. D. Marshall, M. G. Thompson, and J. L. O’Brien, “60 dB high-extinction auto-configured Mach-Zehnder interferometer,” Opt. Lett. 41, 5318–5321 (2016).
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J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015).
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N. J. Russell, J. L. O’Brien, and A. Laing, “Direct dialling of Haar random unitary matrices,” arXiv:1506.06220v1.

O’Brien, J.L.

N. J. Russell, L. Chakhmakhchyan, J.L. O’Brien, and A. Laing, “Direct dialling of haar random unitary matrices,” New J. Phys. 19, 033007 (2017).
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Oguma, M.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015).
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Osellame, R.

A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N Viggianiello, L Innocenti, P Mataloni, and F Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
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M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1(3), e1400255 (2015).
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A. Crespi, R. Osellame, R. Ramponi, D. J. Bord, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, ‘multimode interferometers with arbitrary designs for photonic boson sampling,” Nat. Photonics 7, 545–549 (2013).
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N. Spagnolo, E. Maiorino, C. Vitelli, M. Bentivegna, A. Crespi, R. Ramponi, P. Mataloni, R. Osellame, and F. Sciarrino, “Learning an unknown transformation via a genetic approach,” arXiv:1610.03291.

Paesani, S.

Poon, J. K. S.

Qiang, X.

Qin, J.

H. Wang, Y. He, Y. Li, Z. Su, B. Li, H. Huang, X. Ding, M. Chen, C. Liu, J. Qin, and J. Li, “Multi-photon boson-sampling machines beating early classical computers,” arXiv:1612.06956.

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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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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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A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N Viggianiello, L Innocenti, P Mataloni, and F Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
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M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1(3), e1400255 (2015).
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A. Crespi, R. Osellame, R. Ramponi, D. J. Bord, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, ‘multimode interferometers with arbitrary designs for photonic boson sampling,” Nat. Photonics 7, 545–549 (2013).
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N. Spagnolo, E. Maiorino, C. Vitelli, M. Bentivegna, A. Crespi, R. Ramponi, P. Mataloni, R. Osellame, and F. Sciarrino, “Learning an unknown transformation via a genetic approach,” arXiv:1610.03291.

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N. J. Russell, L. Chakhmakhchyan, J.L. O’Brien, and A. Laing, “Direct dialling of haar random unitary matrices,” New J. Phys. 19, 033007 (2017).
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J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015).
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N. J. Russell, J. L. O’Brien, and A. Laing, “Direct dialling of Haar random unitary matrices,” arXiv:1506.06220v1.

Sacher, W. D.

Sampietro, M.

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D.A. B. Miller, A. Melloni, and F. Morichetti, “All-optical mode unscrambling on a silicon photonic chip,” arXiv:1512.06762

Sansoni, L.

C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, “Gaussian Boson Sampling,” arXiv:1612.01199.

Santagati, R.

Sciarrino, F

A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N Viggianiello, L Innocenti, P Mataloni, and F Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
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Sciarrino, F.

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1(3), e1400255 (2015).
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A. Crespi, R. Osellame, R. Ramponi, D. J. Bord, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, ‘multimode interferometers with arbitrary designs for photonic boson sampling,” Nat. Photonics 7, 545–549 (2013).
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N. Spagnolo, E. Maiorino, C. Vitelli, M. Bentivegna, A. Crespi, R. Ramponi, P. Mataloni, R. Osellame, and F. Sciarrino, “Learning an unknown transformation via a genetic approach,” arXiv:1610.03291.

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J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015).
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C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, “Gaussian Boson Sampling,” arXiv:1612.01199.

Silverstone, J. W.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015).
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Smith, B. J.

B.J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8(10), 770–774 (2014).
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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(6121), 798–801 (2013).
[Crossref]

Smith, P. G. R.

B.J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8(10), 770–774 (2014).
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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(6121), 798–801 (2013).
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D. O. Kundys, H. C. Gates, S. Dasgupta, C. B. E. Gawith, and P. G. R. Smith, “Use of cross-couplers to decrease size of UV written photonic circuits,” IEEE Photonics Technol. Lett. 21(13), 947–949 (2009).
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Spagnolo, N.

A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N Viggianiello, L Innocenti, P Mataloni, and F Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
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M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1(3), e1400255 (2015).
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A. Crespi, R. Osellame, R. Ramponi, D. J. Bord, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, ‘multimode interferometers with arbitrary designs for photonic boson sampling,” Nat. Photonics 7, 545–549 (2013).
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N. Spagnolo, E. Maiorino, C. Vitelli, M. Bentivegna, A. Crespi, R. Ramponi, P. Mataloni, R. Osellame, and F. Sciarrino, “Learning an unknown transformation via a genetic approach,” arXiv:1610.03291.

Sparrow, C.

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015).
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Spring, J. B.

B.J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8(10), 770–774 (2014).
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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(6121), 798–801 (2013).
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Steinbrecher, G. R.

J. Mower, N. C. Harris, G. R. Steinbrecher, Y. Lahini, and D. Englund, “High-fidelity quantum state evolution in imperfect photonic integrated circuits,” Phys. Rev. A 92(3),1–7 (2015).
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Su, Z.

H. Wang, Y. He, Y. Li, Z. Su, B. Li, H. Huang, X. Ding, M. Chen, C. Liu, J. Qin, and J. Li, “Multi-photon boson-sampling machines beating early classical computers,” arXiv:1612.06956.

Szameit, A.

M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nat. Photonics 7(7), 540–544 (2013).
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Takahashi, H.

Thomas-Peter, N.

B.J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8(10), 770–774 (2014).
[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(6121), 798–801 (2013).
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N. Thomas-Peter, Quantum enhanced precision measurement and information processing with integrated photonics, (PhD Thesis, University of Oxford, 2012).

Thompson, M. G.

C. M. Wilkes, X. Qiang, J. Wang, R. Santagati, S. Paesani, X. Zhou, D. A. B. Miller, G. D. Marshall, M. G. Thompson, and J. L. O’Brien, “60 dB high-extinction auto-configured Mach-Zehnder interferometer,” Opt. Lett. 41, 5318–5321 (2016).
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J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015).
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Tillmann, M.

M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nat. Photonics 7(7), 540–544 (2013).
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van de Graaf, J.

C. A. Fuchs and J. van de Graaf, “Cryptographic Distinguishability Measures for Quantum Mechanical States,” IEEE Trans. Inf. Theory 45, 1216 (1999).
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Viggianiello, N

A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N Viggianiello, L Innocenti, P Mataloni, and F Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
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Viggianiello, N.

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1(3), e1400255 (2015).
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Vitelli, C.

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1(3), e1400255 (2015).
[Crossref] [PubMed]

A. Crespi, R. Osellame, R. Ramponi, D. J. Bord, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, ‘multimode interferometers with arbitrary designs for photonic boson sampling,” Nat. Photonics 7, 545–549 (2013).
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N. Spagnolo, E. Maiorino, C. Vitelli, M. Bentivegna, A. Crespi, R. Ramponi, P. Mataloni, R. Osellame, and F. Sciarrino, “Learning an unknown transformation via a genetic approach,” arXiv:1610.03291.

Walmsley, I. A.

W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, “An Optimal Design for Universal Multiport Interferometers,” Optica 2(8), 1–8 (2016).

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[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(6121), 798–801 (2013).
[Crossref]

Walther, P.

M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nat. Photonics 7(7), 540–544 (2013).
[Crossref]

Wang, H.

H. Wang, Y. He, Y. Li, Z. Su, B. Li, H. Huang, X. Ding, M. Chen, C. Liu, J. Qin, and J. Li, “Multi-photon boson-sampling machines beating early classical computers,” arXiv:1612.06956.

Wang, J.

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,” Science 339, 794–798 (2013).
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Wilkes, C. M.

Zeilinger, A.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73(1), 58–61 (1994).
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Zhou, X.

IEEE Photonics Technol. Lett. (1)

D. O. Kundys, H. C. Gates, S. Dasgupta, C. B. E. Gawith, and P. G. R. Smith, “Use of cross-couplers to decrease size of UV written photonic circuits,” IEEE Photonics Technol. Lett. 21(13), 947–949 (2009).
[Crossref]

IEEE Trans. Inf. Theory (1)

C. A. Fuchs and J. van de Graaf, “Cryptographic Distinguishability Measures for Quantum Mechanical States,” IEEE Trans. Inf. Theory 45, 1216 (1999).
[Crossref]

Nat. Commun. (1)

A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N Viggianiello, L Innocenti, P Mataloni, and F Sciarrino, “Suppression law of quantum states in a 3D photonic fast Fourier transform chip,” Nat. Commun. 7, 10469 (2016).
[Crossref] [PubMed]

Nat. Photonics (3)

B.J. Metcalf, J. B. Spring, P. C. Humphreys, N. Thomas-Peter, M. Barbieri, W. S. Kolthammer, X-M. Jin, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Quantum teleportation on a photonic chip,” Nat. Photonics 8(10), 770–774 (2014).
[Crossref]

A. Crespi, R. Osellame, R. Ramponi, D. J. Bord, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, ‘multimode interferometers with arbitrary designs for photonic boson sampling,” Nat. Photonics 7, 545–549 (2013).
[Crossref]

M. Tillmann, B. Dakić, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, “Experimental boson sampling,” Nat. Photonics 7(7), 540–544 (2013).
[Crossref]

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N. J. Russell, L. Chakhmakhchyan, J.L. O’Brien, and A. Laing, “Direct dialling of haar random unitary matrices,” New J. Phys. 19, 033007 (2017).
[Crossref]

Opt. Express (4)

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D. A. B. Miller, “Self-configuring universal linear optical component,” Photonics Res. 1(1), 1 (2013).
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Phys. Rev. A (2)

J. Mower, N. C. Harris, G. R. Steinbrecher, Y. Lahini, and D. Englund, “High-fidelity quantum state evolution in imperfect photonic integrated circuits,” Phys. Rev. A 92(3),1–7 (2015).
[Crossref]

A. Arkhipov, “Boson Sampling is Robust to Small Errors in the Network Matrix,” Phys. Rev. A 92, 062326 (2015).
[Crossref]

Phys. Rev. Lett. (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73(1), 58–61 (1994).
[Crossref] [PubMed]

Sci. Adv. (1)

M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galvão, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, “Experimental scattershot boson sampling,” Sci. Adv. 1(3), e1400255 (2015).
[Crossref] [PubMed]

Science (3)

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]

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(6121), 798–801 (2013).
[Crossref]

J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349(6249), 711–716 (2015).
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S. Aaronson and A. Arkhipov, “The Computational Complexity of Linear Optics,” Theory Comput. 9(4), 143–252 (2013).
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Other (8)

C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, “Gaussian Boson Sampling,” arXiv:1612.01199.

H. Wang, Y. He, Y. Li, Z. Su, B. Li, H. Huang, X. Ding, M. Chen, C. Liu, J. Qin, and J. Li, “Multi-photon boson-sampling machines beating early classical computers,” arXiv:1612.06956.

N. J. Russell, J. L. O’Brien, and A. Laing, “Direct dialling of Haar random unitary matrices,” arXiv:1506.06220v1.

All code used in this work is available at https://github.com/jrenema/RandomUnitaries

N. Spagnolo, E. Maiorino, C. Vitelli, M. Bentivegna, A. Crespi, R. Ramponi, P. Mataloni, R. Osellame, and F. Sciarrino, “Learning an unknown transformation via a genetic approach,” arXiv:1610.03291.

R. Fletcher, Practical Methods of Optimization, 2nd ed. (John Wiley & Sons, 1987).

N. Thomas-Peter, Quantum enhanced precision measurement and information processing with integrated photonics, (PhD Thesis, University of Oxford, 2012).

A. Annoni, E. Guglielmi, M. Carminati, G. Ferrari, M. Sampietro, D.A. B. Miller, A. Melloni, and F. Morichetti, “All-optical mode unscrambling on a silicon photonic chip,” arXiv:1512.06762

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Figures (5)

Fig. 1
Fig. 1 A unitary matrix can be implemented into a multiport interferometer via a mathematical decomposition. The interferometer consists of pairs of beam splitters and phase shifters (see inset). The square decomposition results in the structure of the interferometer shown.
Fig. 2
Fig. 2 Interferometers implementing Haar-random unitary matrices show a specific distribution of beam splitter reflectivities. a) shows the spatial distribution of the average reflectivity in a size 20 interferometer. b) Shows the underlying histograms for three regions in the interferometer, the first column, top row and centre. c) shows how the centre-of-interferometer histogram scales with the size of the interferometer. Note the change of scale in c).
Fig. 3
Fig. 3 The effect of unbalanced MZIs on the fidelity of the decompositions as a function of the size of the fabrication error. The triangular Reck (open symbols) decomposition and the square decomposition by Clements et al. filled symbols are used for various interferometer dimensions. a) us the fraction of the random unitaries that are affected by imperfections. b) the fidelity between the target and the effective unitary for the affected matrices when using our adapted version of the decompositions. The error bars show the standard deviation of all data points used in the average.
Fig. 4
Fig. 4 Decomposition of Fourier matrices. The top panel shows the decomposition of a 20-by-20 fourier matrix. The bottom panel shows the fraction of times a random matrox (black points) and a fourier matrix (red points) fails to implement, where we have averaged over many realizations of our model of the imperfections.
Fig. 5
Fig. 5 Numerical optimization and redundancy in a photonic network with realistic errors. We plot the fidelity enhancement, defined as (1 − Fbefore)/(1 − Fafter)). The black points show the fidelity enhancement when only numerical optimization is applied, and the red points show enhancement when an additional layer of beam splitters is added, followed by numerical optimization. The inset shows the raw fidelities, where the black squares are the unoptimized values, the red circles are the optimized values for a square network, and the blue triangles are the optimized value for a network with one redunant row of beam splitters.

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