Exact and approximate continuous-variable gate decompositions

Timjan Kalajdzievski and Nicolás Quesada

Xanadu, Toronto, ON, M5G 2C8, Canada

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We gather and examine in detail gate decomposition techniques for continuous-variable quantum computers and also introduce some new techniques which expand on these methods. Both exact and approximate decomposition methods are studied and gate counts are compared for some common operations. While each having distinct advantages, we find that exact decompositions have lower gate counts whereas approximate techniques can cover decompositions for all continuous-variable operations but require significant circuit depth for a modest precision.

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[1] Hoi-Kwan Lau, Raphael Pooser, George Siopsis, and Christian Weedbrook. Quantum machine learning over infinite dimensions. Phys. Rev. Lett, 118: 080501, 2017. 10.1103/​PhysRevLett.118.080501.

[2] 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.

[3] Juan Miguel Arrazola, Timjan Kalajdzievski, Christian Weedbrook, and Seth Lloyd. Quantum algorithm for non-homogeneous linear partial differential equations. Phys. Rev. A, 100: 032306, 201908. 10.1103/​PhysRevA.100.032306.

[4] Seckin Sefi, Vishal Vaibhav, and Peter van Loock. A measurement-induced optical kerr interaction. Phys. Rev. A, 88: 012303, 2013. 10.1103/​PhysRevA.88.012303.

[5] Christopher M. Dawson and Michael A. Nielsen. The Solovay-Kitaev algorithm. Quantum Inf. Comput., 6: 1, 2006. 10.5555/​2011679.2011685.

[6] Matthew Amy, Dmitri Maslov, Michele Mosca, and Martin Roetteler. A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst., 32 (6): 818–830, 2013. 10.1109/​TCAD.2013.2244643.

[7] Seth Lloyd. Almost any quantum logic gate is universal. Phys. Rev. Lett., 75 (2): 346, 1995. 10.1103/​PhysRevLett.75.346.

[8] David P DiVincenzo. Two-bit gates are universal for quantum computation. Phys. Rev. A, 51 (2): 1015, 1995. 10.1103/​PhysRevA.51.1015.

[9] Adriano Barenco, Charles H Bennett, Richard Cleve, David P DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A Smolin, and Harald Weinfurter. Elementary gates for quantum computation. Phys. Rev. A, 52 (5): 3457, 1995. 10.1103/​PhysRevA.52.3457.

[10] Seth Lloyd. Universal quantum simulators. Science, 23: 1073, 1996. 10.1126/​science.273.5278.1073.

[11] Seth Lloyd and Samuel L. Braunstein. Quantum computation over continuous variables. Phys. Rev. Lett, 82: 1784, 1999. 10.1103/​PhysRevLett.82.1784.

[12] Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal Clifford gates and noisy ancillas. Phys. Rev. A, 71 (2): 022316, 2005. 10.1103/​PhysRevA.71.022316.

[13] Seckin Sefi and Peter van Loock. How to decompose arbitrary continuous-variable quantum operations. Phys. Rev. Lett., 107: 170501, 2011. 10.1103/​PhysRevLett.107.170501.

[14] Timjan Kalajdzievski and Juan Miguel Arrazola. Exact gate decompositions for photonic quantum computing. Phys. Rev. A, 99: 022341, 2019. 10.1103/​PhysRevA.99.022341.

[15] A Yu Kitaev. Quantum computations: algorithms and error correction. Russ. Math. Surv., 52 (6): 1191–1249, 1997. 10.1070/​RM1997v052n06ABEH002155.

[16] Vadym Kliuchnikov, Dmitri Maslov, and Michele Mosca. Asymptotically optimal approximation of single qubit unitaries by Clifford and T circuits using a constant number of ancillary qubits. Phys. Rev. Lett., 110 (19): 190502, 2013. 10.1103/​PhysRevLett.110.190502.

[17] Vadym Kliuchnikov, Alex Bocharov, and Krysta M Svore. Asymptotically optimal topological quantum compiling. Phys. Rev. Lett., 112 (14): 140504, 2014. 10.1103/​PhysRevLett.112.140504.

[18] Vadym Kliuchnikov and Jon Yard. A framework for exact synthesis. arXiv:1504.04350, 2015.

[19] Alex Bocharov, Martin Roetteler, and Krysta M Svore. Efficient synthesis of universal repeat-until-success quantum circuits. Phys. Rev. Lett., 114 (8): 080502, 2015. 10.1103/​PhysRevLett.114.080502.

[20] Nicolas C. Menicucci, Peter van Loock, Mile Gu, Christian Weedbrook, Timothy C. Ralph, and Michael A. Nielsen. niversal quantum computation with continuous-variable cluster states. Phys. Rev. Lett, 97: 110501, 2006. 10.1103/​PhysRevLett.97.110501.

[21] Mile Gu, Christian Weedbrook, Nicolas C. Menicucci, Timothy C. Ralph, and Peter van Loock. Quantum computing with continuous-variable clusters. Phys. Rev. A, 79: 062318, 2009a. 10.1103/​PhysRevA.79.062318.

[22] 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: 621, 2012. 10.1103/​RevModPhys.84.621.

[23] Tomasz Sowinski, Omjyoti Dutta, Philipp Hauke, Luca Tagliacozzo, and Maciej Lewenstein. Dipolar molecules in optical lattices. Phys. Rev. Lett., 108: 115301, 2012. 10.1103/​PhysRevLett.108.115301.

[24] CR Myers and TC Ralph. Coherent state topological cluster state production. New J. Phys., 13 (11): 115015, 2011. 10.1088/​1367-2630/​13/​11/​115015.

[25] Timothy C Ralph, Alexei Gilchrist, Gerard J Milburn, William J Munro, and Scott Glancy. Quantum computation with optical coherent states. Phys. Rev. A, 68 (4): 042319, 2003. 10.1103/​PhysRevA.68.042319.

[26] Giacomo Pantaleoni, Ben Q Baragiola, and Nicolas C Menicucci. Modular bosonic subsystem codes. Phys. Rev. Lett., 125 (4): 040501, 2020. 10.1103/​PhysRevLett.125.040501.

[27] 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.

[28] Naomichi Hatano and Masuo Suzuki. Finding exponential product formulas of higher orders. In A. Das and B.K. Chakrabarti, editors, Quantum Annealing and Other Optimization Methods, pages 37–68. Springer, Berlin, 2005. 10.1007/​11526216_2.

[29] Nathan Wiebe, Dominic W. Berry, Peter Hoyer, and Barry C. Sanders. Higher order decompositions of ordered operator exponentials. J. Phys. A: Math. Theor., 43: 065203, 2010. 10.1088/​1751-8113/​43/​6/​065203.

[30] Samuel L Braunstein. Squeezing as an irreducible resource. Phys. Rev. A, 71 (5): 055801, 2005. 10.1103/​PhysRevA.71.055801.

[31] Biswadeb Dutta, N Mukunda, R Simon, et al. The real symplectic groups in quantum mechanics and optics. Pramana, 45 (6): 471–497, 1995. 10.1007/​BF02848172.

[32] Timjan Kalajdzievski. Exact Gate Decompositions For Photonic Quantum Computers. PhD thesis, York University, 2020. URL https:/​/​yorkspace.library.yorku.ca/​xmlui/​handle/​10315/​37435.

[33] 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 operation. Phys. Rev. Lett., 124: 240503, 2020. 10.1103/​PhysRevLett.124.240503.

[34] Mitsuyoshi Yukawa, Kazunori Miyata, Hidehiro Yonezawa, Petr Marek, Radim Filip, and Akira Furusawa. Emulating quantum cubic nonlinearity. Phys. Rev. A, 88 (5): 053816, 2013. 10.1103/​PhysRevA.88.053816.

[35] 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, 2009b. 10.1103/​PhysRevA.79.062318.

[36] 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, 2015. 10.1103/​PhysRevA.91.032321.

[37] 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.

[38] 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.

[39] Petr Marek, Radim Filip, Hisashi Ogawa, Atsushi Sakaguchi, Shuntaro Takeda, Jun ichi Yoshikawa, and Akira Furusawa. General implementation of arbitrary nonlinear quadrature phase gates. Phys. Rev. A, 97: 022329, 2018. 10.1103/​PhysRevA.97.022329.

[40] Timo Hillmann, Fernando Quijandría, Göran Johansson, Alessandro Ferraro, Simone Gasparinetti, and Giulia Ferrini. Universal gate set for continuous-variable quantum computation with microwave circuits. Phys. Rev. Lett., 125 (16): 160501, 2020. 10.1103/​PhysRevLett.125.160501.

[41] Yaakov S. Weinstein, Seth Lloyd, and David G. Cory. Implementation of the quantum Fourier transform. Phys. Rev. Lett., 86: 1889, 2001. 10.1103/​PhysRevLett.86.1889.

[42] Manas K. Patra and Samuel L. Braunstein. Quantum Fourier transform, heisenberg groups and quasiprobability distributions. New J. Phys., 13: 063013, 2011. 10.1088/​1367-2630/​13/​6/​063013.

[43] Wilhelm Magnus. On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math., 7 (4): 649–673, 1954. 10.1002/​cpa.3160070404.

[44] Hale F Trotter. On the product of semi-groups of operators. Proc. Am. Math. Soc., 10 (4): 545–551, 1959. 10.2307/​2033649.

[45] Masuo Suzuki. Generalized Trotter's formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems. Commun. Math. Phys., 51: 183, 1976. 10.1007/​BF01609348.

[46] Andrew M Childs, Dmitri Maslov, Yunseong Nam, Neil J Ross, and Yuan Su. Toward the first quantum simulation with quantum speedup. Proc. Natl. Acad. Sci. U.S.A., 115: 9456–9461, 2018. 10.1073/​pnas.1801723115.

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

[48] Michael Reck, Anton Zeilinger, Herbert J Bernstein, and Philip Bertani. Experimental realization of any discrete unitary operator. Phys. Rev. Lett., 73 (1): 58, 1994. 10.1103/​PhysRevLett.73.58.

[49] William R Clements, Peter C Humphreys, Benjamin J Metcalf, W Steven Kolthammer, and Ian A Walmsley. Optimal design for universal multiport interferometers. Optica, 3 (12): 1460–1465, 2016. 10.1364/​OPTICA.3.001460.

[50] Hubert de Guise, Olivia Di Matteo, and Luis L Sánchez-Soto. Simple factorization of unitary transformations. Phys. Rev. A, 97 (2): 022328, 2018. 10.1103/​PhysRevA.97.022328.

[51] Daiqin Su, Ish Dhand, Lukas G Helt, Zachary Vernon, and Kamil Brádler. Hybrid spatiotemporal architectures for universal linear optics. Phys. Rev. A, 99 (6): 062301, 2019. 10.1103/​PhysRevA.99.062301.

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

[53] Jaromír Fiurášek. Unitary-gate synthesis for continuous-variable systems. Phys. Rev. A, 68 (2): 022304, 2003. 10.1103/​PhysRevA.68.022304.

[54] Chris Sparrow, Enrique Martín-López, Nicola Maraviglia, Alex Neville, Christopher Harrold, Jacques Carolan, Yogesh N Joglekar, Toshikazu Hashimoto, Nobuyuki Matsuda, Jeremy L O’Brien, et al. Simulating the vibrational quantum dynamics of molecules using photonics. Nature, 557 (7707): 660, 2018. 10.1038/​s41586-018-0152-9.

[55] Patrick Rebentrost, Brajesh Gupt, and Thomas R. Bromley. Photonic quantum algorithm for Monte Carlo integration. arXiv:1809.02579, 2018.

[56] Raymond Kan. From moments of sum to moments of product. J. Multivar. Anal., 99: 542, 2008. 10.1016/​j.jmva.2007.01.013.

[57] 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, 2019. 10.22331/​q-2019-03-11-129.

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