- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Decomposing large unitaries into multimode devices of arbitrary size

Christian Arends1,2, Lasse Wolf3, Jasmin Meinecke4,5,6,7, Sonja Barkhofen2,8, Tobias Weich3,2, and Tim J. Bartley2,8

  • 1Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark
  • 2PhoQS, Universität Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany
  • 3Institute of Mathematics, Universität Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany
  • 4Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
  • 5Department für Physik, Ludwig-Maximilians-Universität, 80539 München, Germany
  • 6Munich Center for Quantum Science and Technology (MCQST), 80799 München, Germany
  • 7Institute of Solid State Physics, Technische Universität Berlin, 10623 Berlin, Germany
  • 8Department of Physics, Universität Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany

PDF

Phys. Rev. Research 6, L012043 – Published 29 February, 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.L012043

Abstract

Decomposing complex unitary evolution into a series of constituent components is a cornerstone of practical quantum information processing. While the decomposition of an n×n unitary into a product of 2×2 subunitaries (which can for example be realized by beam splitters and phase shifters in linear optics) is well established, we show how for any m>2 this decomposition can be generalized into a product of m×m subunitaries (which can then be realized by a more complex device acting on m modes). If the cost associated with building each m×m multimode device is less than constructing with m(m−1)2 individual 2×2 devices, we show that the decomposition of large unitaries into m×m submatrices is more resource efficient and exhibits a higher tolerance to errors, than its 2×2 counterpart. This allows larger-scale unitaries to be constructed with lower errors, which is necessary for various tasks, not least boson sampling, the quantum Fourier transform, and quantum simulations.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (35)

  1. We use the expression “mode” in the context of optics; mathematically speaking, a mode then corresponds to a one-dimensional subspace of a given Hilbert space and an m-mode device is a linear transformation acting on an m-dimensional subspace.
  2. S. Aaronson and A. Arkhipov, The computational complexity of linear optics, in Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing, STOC ‘11 (Association for Computing Machinery, New York, 2011), pp. 333–342.
  3. C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, Gaussian boson sampling, Phys. Rev. Lett. 119, 170501 (2017).
  4. P. W. Shor, Algorithms for quantum computation: Discrete logarithms and factoring, in Proceedings of the 35th Annual Symposium on Foundations of Computer Science (IEEE, New York, 1994), pp. 124–134.
  5. D. Coppersmith, An approximate Fourier transform useful in quantum factoring, arXiv:quant-ph/0201067.
  6. A. Aspuru-Guzik and P. Walther, Photonic quantum simulators, Nat. Phys. 8, 285 (2012).
  7. B. J. Shastri, A. N. Tait, T. Ferreira de Lima, W. H. P. Pernice, H. Bhaskaran, C. D. Wright, and P. R. Prucnal, Photonics for artificial intelligence and neuromorphic computing, Nat. Photonics 15, 102 (2021).
  8. V. Ramakrishna, R. Ober, X. Sun, O. Steuernagel, J. Botina, and H. Rabitz, Explicit generation of unitary transformations in a single atom or molecule, Phys. Rev. A 61, 032106 (2000).
  9. M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Experimental realization of any discrete unitary operator, Phys. Rev. Lett. 73, 58 (1994).
  10. W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, Optimal design for universal multiport interferometers, Optica 3, 1460 (2016).
  11. B. A. Bell and I. A. Walmsley, Further compactifying linear optical unitaries, APL Photonics 6, 070804 (2021).
  12. 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, 711 (2015).
  13. W. Bogaerts, D. Pérez, J. Capmany, D. A. B. Miller, J. Poon, D. Englund, F. Morichetti, and A. Melloni, Programmable photonic circuits, Nature (London) 586, 207 (2020).
  14. N. C. Harris, D. Bunandar, M. Pant, G. R. Steinbrecher, J. Mower, M. Prabhu, T. Baehr-Jones, M. Hochberg, and D. Englund, Large-scale quantum photonic circuits in silicon, Nanophotonics 5, 456 (2016).
  15. J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, Boson sampling on a photonic chip, Science 339, 798 (2013).
  16. A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, Integrated multimode interferometers with arbitrary designs for photonic boson sampling, Nat. Photonics 7, 545 (2013).
  17. L. S. Madsen, F. Laudenbach, M. F. Askarani, F. Rortais, T. Vincent, J. F. F. Bulmer, F. M. Miatto, L. Neuhaus, L. G. Helt, M. J. Collins, A. E. Lita, T. Gerrits, S. W. Nam, V. D. Vaidya, M. Menotti, I. Dhand, Z. Vernon, N. Quesada, and J. Lavoie, Quantum computational advantage with a programmable photonic processor, Nature (London) 606, 75 (2022).
  18. C. Sparrow, E. Martín-López, N. Maraviglia, A. Neville, C. Harrold, J. Carolan, Y. N. Joglekar, T. Hashimoto, N. Matsuda, J. L. O'Brien, D. P. Tew, and A. Laing, Simulating the vibrational quantum dynamics of molecules using photonics, Nature (London) 557, 660 (2018).
  19. R. van der Meer, Z. Huang, M. C. Anguita, D. Qu, P. Hooijschuur, H. Liu, M. Han, J. J. Renema, and L. Cohen, Experimental simulation of loop quantum gravity on a photonic chip, npj Quantum Inf. 9, 32 (2023).
  20. F. H. B. Somhorst, R. van der Meer, M. Correa Anguita, R. Schadow, H. J. Snijders, M. de Goede, B. Kassenberg, P. Venderbosch, C. Taballione, J. P. Epping, H. H. van den Vlekkert, J. Timmerhuis, J. F. F. Bulmer, J. Lugani, I. A. Walmsley, P. W. H. Pinkse, J. Eisert, N. Walk, and J. J. Renema, Quantum simulation of thermodynamics in an integrated quantum photonic processor, Nat. Commun. 14, 3895 (2023).
  21. J. M. Shainline, S. M. Buckley, R. P. Mirin, and S. W. Nam, Superconducting optoelectronic circuits for neuromorphic computing, Phys. Rev. Appl. 7, 034013 (2017).
  22. J. Feldmann, N. Youngblood, C. D. Wright, H. Bhaskaran, and W. H. P. Pernice, All-optical spiking neurosynaptic networks with self-learning capabilities, Nature (London) 569, 208 (2019).
  23. L. B. Soldano and E. C. M. Pennings, Optical multi-mode interference devices based on self-imaging: Principles and applications, J. Lightwave Technol. 13, 615 (1995).
  24. G. Weihs, M. Reck, H. Weinfurter, and A. Zeilinger, Two-photon interference in optical fiber multiports, Phys. Rev. A 54, 893 (1996).
  25. 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).
  26. N. Spagnolo, C. Vitelli, L. Aparo, P. Mataloni, F. Sciarrino, A. Crespi, R. Ramponi, and R. Osellame, Three-photon bosonic coalescence in an integrated tritter, Nat. Commun. 4, 1606 (2013).
  27. A. J. Menssen, A. E. Jones, B. J. Metcalf, M. C. Tichy, S. Barz, W. S. Kolthammer, and I. A. Walmsley, Distinguishability and many-particle interference, Phys. Rev. Lett. 118, 153603 (2017).
  28. M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O'Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, High-dimensional quantum cryptography with twisted light, New J. Phys. 17, 033033 (2015).
  29. A. Babazadeh, M. Erhard, F. Wang, M. Malik, R. Nouroozi, M. Krenn, and A. Zeilinger, High-dimensional single-photon quantum gates: Concepts and experiments, Phys. Rev. Lett. 119, 180510 (2017).
  30. M. Kues, C. Reimer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azaña, and R. Morandotti, On-chip generation of high-dimensional entangled quantum states and their coherent control, Nature (London) 546, 622 (2017).
  31. B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, Photon temporal modes: A complete framework for quantum information science, Phys. Rev. X 5, 041017 (2015).
  32. V. Ansari, J. M. Donohue, B. Brecht, and C. Silberhorn, Tailoring nonlinear processes for quantum optics with pulsed temporal-mode encodings, Optica 5, 534 (2018).
  33. S. De, V. Ansari, J. Sperling, S. Barkhofen, B. Brecht, and C. Silberhorn, Realization of high-fidelity unitary operations on up to 64 frequency bins, arXiv:2206.06059.
  34. L. Serino, J. Gil-Lopez, M. Stefszky, R. Ricken, C. Eigner, B. Brecht, and C. Silberhorn, Realization of a multi-output quantum pulse gate for decoding high-dimensional temporal modes of single-photon states, PRX Quantum 4, 020306 (2023).
  35. G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, 3rd ed. (Johns Hopkins University Press, Baltimore, 1996).

Outline

Information

Phys. Rev. Research 6, L012043– Published 29 February, 2024
  • Received 29 September 2023
  • Accepted 23 January 2024
Crossmark - Check for updates button
Creative Commons Logo

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

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation