Fast optimization of parametrized quantum optical circuits

Filippo M. Miatto; Nicolás Quesada

doi:10.22331/q-2020-11-30-366

Fast optimization of parametrized quantum optical circuits

Filippo M. Miatto^1,2 and Nicolás Quesada³

¹Institut Polytechnique de Paris
²Télécom Paris, LTCI, 19 Place Marguerite Perey 91120 Palaiseau
³Xanadu, Toronto, ON, M5G 2C8, Canada

Published:	2020-11-30, volume 4, page 366
Eprint:	arXiv:2004.11002v5
Doi:	https://doi.org/10.22331/q-2020-11-30-366
Citation:	Quantum 4, 366 (2020).

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Abstract

Parametrized quantum optical circuits are a class of quantum circuits in which the carriers of quantum information are photons and the gates are optical transformations. Classically optimizing these circuits is challenging due to the infinite dimensionality of the photon number vector space that is associated to each optical mode. Truncating the space dimension is unavoidable, and it can lead to incorrect results if the gates populate photon number states beyond the cutoff. To tackle this issue, we present an algorithm that is orders of magnitude faster than the current state of the art, to recursively compute the exact matrix elements of Gaussian operators and their gradient with respect to a parametrization. These operators, when augmented with a non-Gaussian transformation such as the Kerr gate, achieve universal quantum computation. Our approach brings two advantages: first, by computing the matrix elements of Gaussian operators directly, we don't need to construct them by combining several other operators; second, we can use any variant of the gradient descent algorithm by plugging our gradients into an automatic differentiation framework such as TensorFlow or PyTorch. Our results will find applications in quantum optical hardware research, quantum machine learning, optical data processing, device discovery and device design.

Featured image: The matrix elements of an optical Gaussian transformation can be expressed recursively as linear combinations of neighbouring elements. This gives us an extremely fast method to generate the exact matrix representation of general Gaussian transformations up to any desired dimension. As the recurrence relation is differentiable, we can also perform backpropagation and apply gradient-descent optimisation of quantum photonic circuits up to 100 times faster than the previous state of the art.

Popular summary

Quantum photonic circuits are a promising platform for quantum computing. In this work the authors introduce a new method to simulate and optimize photonic circuits using machine learning methods, up to 100 times faster than the previous state of the art. This new algorithm will allow researchers to perform more accurate simulations and design larger circuits with fewer computational resources.

► BibTeX data

@article{Miatto2020fastoptimizationof,
  doi = {10.22331/q-2020-11-30-366},
  url = {https://doi.org/10.22331/q-2020-11-30-366},
  title = {Fast optimization of parametrized quantum optical circuits},
  author = {Miatto, Filippo M. and Quesada, Nicol{\'{a}}s},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {366},
  month = nov,
  year = {2020}
}

► References

[1] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. Evaluating analytic gradients on quantum hardware. Phys. Rev. A, 99 (3): 032331, 2019. 10.1103/PhysRevA.99.032331.
https://doi.org/10.1103/PhysRevA.99.032331

[2] Marcello Benedetti, Erika Lloyd, Stefan Sack, and Mattia Fiorentini. Parameterized quantum circuits as machine learning models. Quantum Sci. Technol., 4 (4): 043001, 2019. 10.1088/2058-9565/ab4eb5.
https://doi.org/10.1088/2058-9565/ab4eb5

[3] Vedran Dunjko, Jacob M Taylor, and Hans J Briegel. Quantum-enhanced machine learning. Phys. Rev. Lett., 117 (13): 130501, 2016. 10.1103/PhysRevLett.117.130501.
https://doi.org/10.1103/PhysRevLett.117.130501

[4] Iordanis Kerenidis, Jonas Landman, Alessandro Luongo, and Anupam Prakash. q-means: A quantum algorithm for unsupervised machine learning. In Advances in Neural Information Processing Systems, pages 4136–4146, 2019.

[5] Marco Cerezo, Alexander Poremba, Lukasz Cincio, and Patrick J Coles. Variational quantum fidelity estimation. Quantum, 4: 248, 2020. 10.22331/q-2020-03-26-248.
https://doi.org/10.22331/q-2020-03-26-248

[6] Ryan LaRose, Arkin Tikku, Étude O’Neel-Judy, Lukasz Cincio, and Patrick J Coles. Variational quantum state diagonalization. npj Quantum Inf., 5 (1): 1–10, 2019. 10.1038/s41534-019-0167-6.
https://doi.org/10.1038/s41534-019-0167-6

[7] Carlos Bravo-Prieto, Ryan LaRose, Marco Cerezo, Yigit Subasi, Lukasz Cincio, and Patrick J Coles. Variational quantum linear solver: A hybrid algorithm for linear systems. arXiv preprint arXiv:1909.05820, 2019.
arXiv:1909.05820

[8] Juan Miguel Arrazola, Thomas R Bromley, Josh Izaac, Casey R Myers, Kamil Brádler, and Nathan Killoran. Machine learning method for state preparation and gate synthesis on photonic quantum computers. Quantum Sci. Technol., 4 (2): 024004, 2019. 10.1088/2058-9565/aaf59e.
https://doi.org/10.1088/2058-9565/aaf59e

[9] Nathan Killoran, Thomas R Bromley, Juan Miguel Arrazola, Maria Schuld, Nicolás Quesada, and Seth Lloyd. Continuous-variable quantum neural networks. Phys. Rev. Research, 1 (3): 033063, 2019a. 10.1103/PhysRevResearch.1.033063.
https://doi.org/10.1103/PhysRevResearch.1.033063

[10] Nathan Killoran, Josh Izaac, Nicolás Quesada, Ville Bergholm, Matthew Amy, and Christian Weedbrook. Strawberry Fields: A software platform for photonic quantum computing. Quantum, 3: 129, 2019b. 10.22331/q-2019-03-11-129.
https://doi.org/10.22331/q-2019-03-11-129

[11] Gregory R Steinbrecher, Jonathan P Olson, Dirk Englund, and Jacques Carolan. Quantum optical neural networks. npj Quantum Inf., 5 (1): 1–9, 2019. 10.1038/s41534-019-0174-7.
https://doi.org/10.1038/s41534-019-0174-7

[12] N Quesada, LG Helt, J Izaac, JM Arrazola, R Shahrokhshahi, CR Myers, and KK Sabapathy. Simulating realistic non-Gaussian state preparation. Phys. Rev. A, 100 (2): 022341, 2019. 10.1103/PhysRevA.100.022341.
https://doi.org/10.1103/PhysRevA.100.022341

[13] Jonathan Romero, Jonathan P Olson, and Alan Aspuru-Guzik. Quantum autoencoders for efficient compression of quantum data. Quantum Sci. Technol., 2 (4): 045001, 2017. 10.1088/2058-9565/aa8072.
https://doi.org/10.1088/2058-9565/aa8072

[14] Maria Schuld, Alex Bocharov, Krysta M Svore, and Nathan Wiebe. Circuit-centric quantum classifiers. Phys. Rev. A, 101 (3): 032308, 2020. 10.1103/PhysRevA.101.032308.
https://doi.org/10.1103/PhysRevA.101.032308

[15] Maria Schuld and Nathan Killoran. Quantum machine learning in feature Hilbert spaces. Phys. Rev. Lett., 122 (4): 040504, 2019. 10.1103/PhysRevLett.122.040504.
https://doi.org/10.1103/PhysRevLett.122.040504

[16] Vojtěch Havlíček, Antonio D Córcoles, Kristan Temme, Aram W Harrow, Abhinav Kandala, Jerry M Chow, and Jay M Gambetta. Supervised learning with quantum-enhanced feature spaces. Nature, 567 (7747): 209–212, 2019. 10.1038/s41586-019-0980-2.
https://doi.org/10.1038/s41586-019-0980-2

[17] Peter JJ O’Malley, Ryan Babbush, Ian D Kivlichan, Jonathan Romero, Jarrod R McClean, Rami Barends, Julian Kelly, Pedram Roushan, Andrew Tranter, Nan Ding, et al. Scalable quantum simulation of molecular energies. Phys. Rev. X, 6 (3): 031007, 2016. 10.1103/PhysRevX.6.031007.
https://doi.org/10.1103/PhysRevX.6.031007

[18] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J Love, Alán Aspuru-Guzik, and Jeremy L O'Brien. A variational eigenvalue solver on a photonic quantum processor. Nature Commun., 5: 4213, 2014. 10.1038/ncomms5213.
https://doi.org/10.1038/ncomms5213

[19] Alberto Politi, Jonathan CF Matthews, Mark G Thompson, and Jeremy L O'Brien. Integrated quantum photonics. IEEE J. Sel. Top. Quantum Electron., 15 (6): 1673–1684, 2009. 10.1109/JSTQE.2009.2026060.
https://doi.org/10.1109/JSTQE.2009.2026060

[20] Y Zhang, M Menotti, K Tan, VD Vaidya, DH Mahler, L Zatti, M Liscidini, B Morrison, and Z Vernon. Single-mode quadrature squeezing using dual-pump four-wave mixing in an integrated nanophotonic device. arXiv preprint arXiv:2001.09474, 2020.
arXiv:2001.09474

[21] VD Vaidya, B Morrison, LG Helt, R Shahrokshahi, DH Mahler, MJ Collins, K Tan, J Lavoie, A Repingon, M Menotti, et al. Broadband quadrature-squeezed vacuum and nonclassical photon number correlations from a nanophotonic device. Sci. Adv., 6 (39): eaba9186, 2020. 10.1126/sciadv.aba9186.
https://doi.org/10.1126/sciadv.aba9186

[22] Seth Lloyd and Samuel L Braunstein. Quantum computation over continuous variables. In Quantum information with continuous variables, pages 9–17. Springer, 1999. 10.1007/978-94-015-1258-9_2.
https://doi.org/10.1007/978-94-015-1258-9_2

[23] Alessio Serafini. Quantum continuous variables: a primer of theoretical methods. CRC Press, 2017.

[24] Christian Weedbrook, Stefano Pirandola, Raúl García-Patrón, Nicolas J Cerf, Timothy C Ralph, Jeffrey H Shapiro, and Seth Lloyd. Gaussian quantum information. Rev. Mod. Phys., 84 (2): 621, 2012. 10.1103/RevModPhys.84.621.
https://doi.org/10.1103/RevModPhys.84.621

[25] Stephen Barnett and Paul M Radmore. Methods in theoretical quantum optics, volume 15. Oxford University Press, 2002.

[26] Kevin E Cahill and Roy J Glauber. Ordered expansions in boson amplitude operators. Phys. Rev., 177 (5): 1857, 1969. 10.1103/PhysRev.177.1857.
https://doi.org/10.1103/PhysRev.177.1857

[27] P Král. Displaced and squeezed Fock states. J. Mod. Opt., 37 (5): 889–917, 1990. 10.1080/09500349014550941.
https://doi.org/10.1080/09500349014550941

[28] Xin Ma and William Rhodes. Multimode squeeze operators and squeezed states. Phys. Rev. A, 41 (9): 4625, 1990. 10.1103/PhysRevA.41.4625.
https://doi.org/10.1103/PhysRevA.41.4625

[29] N. Quesada. Very Nonlinear Quantum Optics. PhD thesis, University of Toronto, 2015.

[30] Ish Dhand, Barry C Sanders, and Hubert de Guise. Algorithms for SU(n) boson realizations and D-functions. J. Math. Phys., 56 (11): 111705, 2015. 10.1063/1.4935433.
https://doi.org/10.1063/1.4935433

[31] EV Doktorov, IA Malkin, and VI Man'ko. Dynamical symmetry of vibronic transitions in polyatomic molecules and the Franck-Condon principle. J. Mol. Spectrosc, 64 (2): 302–326, 1977. 10.1016/0022-2852(75)90199-X.
https://doi.org/10.1016/0022-2852(75)90199-X

[32] Daniel Gruner and Paul Brumer. Efficient evaluation of harmonic polyatomic Franck-Condon factors. Chem. Phys. Lett., 138 (4): 310–314, 1987. 10.1016/0009-2614(87)80389-5.
https://doi.org/10.1016/0009-2614(87)80389-5

[33] R Berger, C Fischer, and M Klessinger. Calculation of the vibronic fine structure in electronic spectra at higher temperatures. 1. benzene and pyrazine. J. Phys. Chem. A, 102 (36): 7157–7167, 1998. 10.1021/jp981597w.
https://doi.org/10.1021/jp981597w

[34] Vadim Mozhayskiy, Samer Gozem, and Anna I. Krylov. ezspectrum v3.0. http://iopenshell.usc.edu/downloads/, 2016.
http://iopenshell.usc.edu/downloads/

[35] Joonsuk Huh. Unified description of vibronic transitions with coherent states. PhD thesis, Johann Wolfgang Goethe-Universität in Frankfurt am Main, 2011.

[36] Scott M Rabidoux, Victor Eijkhout, and John F Stanton. A highly-efficient implementation of the Doktorov recurrence equations for Franck–Condon calculations. J. Chem. Theory Comput., 12 (2): 728–739, 2016. 10.1021/acs.jctc.5b00560.
https://doi.org/10.1021/acs.jctc.5b00560

[37] Marcos Moshinsky and Christiane Quesne. Linear canonical transformations and their unitary representations. J. Math. Phys., 12 (8): 1772–1780, 1971. 10.1063/1.1665805.
https://doi.org/10.1063/1.1665805

[38] Kurt Bernardo Wolf. Canonical transforms. i. complex linear transforms. J. Math. Phys., 15 (8): 1295–1301, 1974. 10.1063/1.1666811.
https://doi.org/10.1063/1.1666811

[39] P Kramer, Marcos Moshinsky, and TH Seligman. Complex extensions of canonical transformations and quantum mechanics. In Group theory and its applications, pages 249–332. Elsevier, 1975. 10.1016/B978-0-12-455153-4.50011-3.
https://doi.org/10.1016/B978-0-12-455153-4.50011-3

[40] Brajesh Gupt, Josh Izaac, and Nicolás Quesada. The Walrus: a library for the calculation of hafnians, Hermite polynomials and Gaussian boson sampling. J. Open Source Softw., 4 (44): 1705, 2019. 10.21105/joss.01705.
https://doi.org/10.21105/joss.01705

[41] Andreas Björklund, Brajesh Gupt, and Nicolás Quesada. A faster hafnian formula for complex matrices and its benchmarking on a supercomputer. ACM J. Exp. Algorithmics, 24 (1): 11, 2019. 10.1145/3325111.
https://doi.org/10.1145/3325111

[42] Nicolás Quesada. Franck-Condon factors by counting perfect matchings of graphs with loops. J. Chem. Phys., 150 (16): 164113, 2019. 10.1063/1.5086387.
https://doi.org/10.1063/1.5086387

[43] Leonardo Banchi, Nicolás Quesada, and Juan Miguel Arrazola. Training Gaussian boson sampling distributions. Phys. Rev. A, 102: 012417, 2020. 10.1103/PhysRevA.102.012417.
https://doi.org/10.1103/PhysRevA.102.012417

[44] Claude Bloch and Albert Messiah. The canonical form of an antisymmetric tensor and its application to the theory of superconductivity. Nucl. Phys., 39: 95–106, 1962. 10.1016/0029-5582(62)90377-2.
https://doi.org/10.1016/0029-5582(62)90377-2

[45] Filippo M Miatto. Recursive multivariate derivatives of $e^{f(x_1,\dots,x_n)}$ of arbitrary order. arXiv preprint arXiv:1911.11722, 2019.
arXiv:1911.11722

[46] Gianfranco Cariolaro and Gianfranco Pierobon. Bloch-Messiah reduction of Gaussian unitaries by Takagi factorization. Phys. Rev. A, 94 (6): 062109, 2016. 10.1103/PhysRevA.94.062109.
https://doi.org/10.1103/PhysRevA.94.062109

[47] Andrei B Klimov and Sergei M Chumakov. A group-theoretical approach to quantum optics: models of atom-field interactions. John Wiley & Sons, 2009.

[48] Raphael Hunger. An introduction to complex differentials and complex differentiability. Technical report, Munich University of Technology, Inst. for Circuit Theory and Signal Processing, 2007.

[49] Chu Guo and Dario Poletti. A scheme for automatic differentiation of complex loss functions. arXiv preprint arXiv:2003.04295, 2020.
arXiv:2003.04295

[50] Daniel Gottesman, Alexei Kitaev, and John Preskill. Encoding a qubit in an oscillator. Phys. Rev. A, 64 (1): 012310, 2001. 10.1103/PhysRevA.64.012310.
https://doi.org/10.1103/PhysRevA.64.012310

[51] Joonsuk Huh. Multimode Bogoliubov transformation and Husimi's q-function. arXiv preprint arXiv:2004.05766, 2020. https://doi.org/10.1088/1742-6596/1612/1/012015.
https://doi.org/10.1088/1742-6596/1612/1/012015
arXiv:2004.05766

[52] Stéfan van der Walt, S Chris Colbert, and Gael Varoquaux. The NumPy array: a structure for efficient numerical computation. Comput. Sci. Eng., 13 (2): 22–30, 2011. 10.1109/MCSE.2011.37.
https://doi.org/10.1109/MCSE.2011.37

[53] Pauli Virtanen, Ralf Gommers, Travis E Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, et al. SciPy 1.0: fundamental algorithms for scientific computing in Python. Nat. Methods, 17 (3): 261–272, 2020. 10.1038/s41592-019-0686-2.
https://doi.org/10.1038/s41592-019-0686-2

[54] Thomas Kluyver, Benjamin Ragan-Kelley, Fernando Pérez, Brian E Granger, Matthias Bussonnier, Jonathan Frederic, Kyle Kelley, Jessica B Hamrick, Jason Grout, Sylvain Corlay, et al. Jupyter notebooks-a publishing format for reproducible computational workflows. In ELPUB, pages 87–90, 2016. 10.3233/978-1-61499-649-1-87.
https://doi.org/10.3233/978-1-61499-649-1-87

[55] John D Hunter. Matplotlib: A 2d graphics environment. Comput. Sci, Eng., 9 (3): 90–95, 2007. 10.1109/MCSE.2007.55.
https://doi.org/10.1109/MCSE.2007.55

[56] Siu Kwan Lam, Antoine Pitrou, and Stanley Seibert. Numba: A LLVM-based Python JIT compiler. In Proceedings of the Second Workshop on the LLVM Compiler Infrastructure in HPC, pages 1–6, 2015. 10.1145/2833157.2833162.
https://doi.org/10.1145/2833157.2833162

[57] S Berkowitz and FJ Garner. The calculation of multidimensional Hermite polynomials and Gram-Charlier coefficients. Math. Comput., 24 (111): 537–545, 1970. 10.2307/2004829.
https://doi.org/10.2307/2004829

[58] Pieter Kok and Samuel L Braunstein. Multi-dimensional Hermite polynomials in quantum optics. J. Phys. A: Math. Gen., 34 (31): 6185, 2001. 10.1088/0305-4470/34/31/312.
https://doi.org/10.1088/0305-4470/34/31/312

[59] Maurice M Mizrahi. Generalized Hermite polynomials. J. Comput. Appl. Math., 1 (3): 137–140, 1975. 10.1016/0771-050X(75)90031-5.
https://doi.org/10.1016/0771-050X(75)90031-5

[60] Samuel L Braunstein and Peter Van Loock. Quantum information with continuous variables. Rev. Mod. Phys., 77 (2): 513, 2005. 10.1103/RevModPhys.77.513.
https://doi.org/10.1103/RevModPhys.77.513

[61] Timjan Kalajdzievski, Christian Weedbrook, and Patrick Rebentrost. Continuous-variable gate decomposition for the Bose-Hubbard model. Phys. Rev. A, 97 (6): 062311, 2018. 10.1103/PhysRevA.97.062311.
https://doi.org/10.1103/PhysRevA.97.062311

[62] Timjan Kalajdzievski and Juan Miguel Arrazola. Exact gate decompositions for photonic quantum computing. Phys. Rev. A, 99 (2): 022341, 2019. 10.1103/PhysRevA.99.022341.
https://doi.org/10.1103/PhysRevA.99.022341

[63] Ryotatsu Yanagimoto, Tatsuhiro Onodera, Edwin Ng, Logan G Wright, Peter L McMahon, and Hideo Mabuchi. Engineering a Kerr-based deterministic cubic phase gate via Gaussian operations. Phys. Rev. Lett., 124 (24): 240503, 2020. 10.1103/PhysRevLett.124.240503.
https://doi.org/10.1103/PhysRevLett.124.240503

[64] Mile Gu, Christian Weedbrook, Nicolas C Menicucci, Timothy C Ralph, and Peter van Loock. Quantum computing with continuous-variable clusters. Phys. Rev. A, 79 (6): 062318, 2009. 10.1103/PhysRevA.79.062318.
https://doi.org/10.1103/PhysRevA.79.062318

[65] Kevin Marshall, Raphael Pooser, George Siopsis, and Christian Weedbrook. Repeat-until-success cubic phase gate for universal continuous-variable quantum computation. Phys. Rev. A, 91 (3): 032321, 2015a. 10.1103/PhysRevA.91.032321.
https://doi.org/10.1103/PhysRevA.91.032321

[66] Krishna Kumar Sabapathy and Christian Weedbrook. ON states as resource units for universal quantum computation with photonic architectures. Phys. Rev. A, 97 (6): 062315, 2018. 10.1103/PhysRevA.97.062315.
https://doi.org/10.1103/PhysRevA.97.062315

[67] Krishna Kumar Sabapathy, Haoyu Qi, Josh Izaac, and Christian Weedbrook. Production of photonic universal quantum gates enhanced by machine learning. Phys. Rev. A, 100 (1): 012326, 2019. 10.1103/PhysRevA.100.012326.
https://doi.org/10.1103/PhysRevA.100.012326

[68] Milton Abramowitz and Irene A Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume 55. US Government printing office, 1948.

[69] Vallee Olivier and Soares Manuel. Airy functions and applications to physics. World Scientific Publishing Company, 2010. https://doi.org/10.1142/p709.
https://doi.org/10.1142/p709

[70] Kevin Marshall, Raphael Pooser, George Siopsis, and Christian Weedbrook. Quantum simulation of quantum field theory using continuous variables. Phys. Rev. A, 92: 063825, 2015b. 10.1103/PhysRevA.92.063825.
https://doi.org/10.1103/PhysRevA.92.063825

[71] Anthony Zee. Quantum field theory in a nutshell, volume 7. Princeton university press, 2010.

[72] Laura García-Álvarez, Cameron Calcluth, Alessandro Ferraro, and Giulia Ferrini. Efficient simulatability of continuous-variable circuits with large Wigner negativity. arXiv preprint arXiv:2005.12026, 2020.
arXiv:2005.12026

[73] Izrail Solomonovich Gradshteyn and Iosif Moiseevich Ryzhik. Table of integrals, series, and products. Academic press, 1980.

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