Quantum Interference in Universal Linear Optical Devices for Quantum Computation and Simulation Christopher Sparrow Department of Physics Imperial College London A thesis submitted in accordance with the requirements for the degree of Doctor of Philosophy September 2017 Abstract It is believed that the exotic properties of quantum systems can be harnessed to perform certain computational tasks more efficiently than classical theories allow. The production, manipulation and detection of single photons constitutes a potential platform for perform- ing such non-classical information processing. The development of integrated quantum photonics has provided a miniaturised, monolithic architecture that is promising for the realisation of near-term analog quantum devices as well as full-scale universal quantum computers. In this thesis we investigate the viability of these photonic quantum computational approaches from an experimental and theoretical perspective. We implement the first universally reconfigurable linear optical network; a key capability for the rapid prototyping of photonic quantum protocols. We propose and demonstrate the use of these devices as a new platform for the programmable quantum simulation of molecular vibrational dynamics. We then tackle an important outstanding problem in linear optical quantum com- puting; quantifying how partial-distinguishability amongst photons affects logical error rates. Finally, we propose a series of schemes aimed at counteracting these distinguisha- bility errors in order to achieve practical quantum technologies with imperfect photonic components. 2 Acknowledgements I would first like to thank my supervisor Anthony Laing for advice, guidance and scien- tific insight throughout my PhD. I would also like to thank Jeremy O'Brien and Terry Rudolph for the inspiration to explore the many beautiful and frustrating quirks of pho- tonic quantum computing. I am hugely grateful to all those with whom the results in this thesis were obtained. Whether in the lab or at the whiteboard; Enrique Mart´ın-L´opez, Nick Russell, Jacques Carolan, Chris Harrold, Patrick Birchall, Hugo Cable, Alex Neville and Nicola Maraviglia, I learned a great deal from all of you. There are too many people to acknowledge at the Centre for Quantum Photonics (and latterly QETlabs), it provided a wonderful environment for working, both intellectually and socially. Similarly, my journey in the world of quantum information science so far would not have been the same without the learning and laughter shared with Cohort 4 of the Controlled Quantum Dynamics CDT at Imperial College London. Of course, the real thesis was the friends I made along the way... I will always look back with great happiness at my years in Bristol and this is in large part due to the many close friends and acquaintances with whom many chats, dinners and beers were shared; Alex, Allison, Beccie, Callum, Chris, Hugo, Javier, Lorraine, Nicki, Patrick, Phil, Raf and Will to name but a few. I would like to thank my whole family for a lifetime of support and providing me with the opportunities and encouragement that have led me here. Finally, this thesis is dedicated to Kirsten, for her love and for her infinite patience putting up with me while it was undertaken and written up. 4 Declaration of Originality I declare that the work in this thesis was carried out in accordance with the requirements of the University's Academic Regulations for Research Degree Programmes. The thesis was written entirely by myself and accurately reflects work carried out during my graduate studies. Work done in collaboration with, or with the assistance of, others, is indicated as such. The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work 5 Publications 1. J. Carolan, C. Harrold, C. Sparrow, E. Mart´ın-L´opez, N. J. Russell, J. W. Sil- verstone, 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, 711-716, (2015). 2. A. Holleczek, O. Barter, A. Rubenok, J. Dilley, P. B.R. Nisbet-Jones, G. Langfahl- Klabes, G. D. Marshall, C. Sparrow, J. L. O'Brien, K. Poulios, A. Kuhn, and J. C.F. Matthews, `Quantum Logic with Cavity Photons From Single Atoms', Phys. Rev. Lett., 117, 023602, (2016). 3. A. Neville, C. Sparrow, R. Clifford, E. Johnston, P. Birchall, A. Montanaro, A. Laing, `Classical Boson Sampling Algorithms with Superior Performance to Near- term Experiments', Nat. Phys., 13, 1153-1157 (2017). 4. C. Sparrow, E. Mart´ın-L´opez, N. Maraviglia, A. Neville, J. Carolan, C. Harrold, N. Matsuda, J. L. O'Brien, Y. Joglekar, D. Tew, A. Laing, `Simulating the Vibrational Quantum Dynamics of Molecules with Integrated Photonics', Submitted, (2017). 5. C. Sparrow*, P. Birchall* et al., `Linear Optical Quantum Computing with Partially- Distinguishable Photons', in prep, (2017). * authors contributed equally to this work 6 Contents Abstract 2 Acknowledgements4 Declaration of Originality5 Publications6 1 Introduction 16 1.1 Thesis Outline.................................. 18 2 Quantum Information and Computation 20 2.1 Introduction................................... 20 2.2 Quantum Theory................................ 20 2.2.1 Physical states............................. 21 2.2.2 Observables............................... 21 2.2.3 Evolution................................ 22 2.2.4 Measurements.............................. 22 2.2.5 Interpretations............................. 23 2.2.6 Operational quantum theory...................... 24 2.3 Quantum Information............................. 26 2.3.1 The Qubit................................ 27 2.3.2 Entanglement.............................. 30 2.3.3 Nonlocality............................... 32 8 CONTENTS 2.3.4 Quantum tomography......................... 34 2.3.5 Distance measures........................... 37 2.4 Quantum computing.............................. 40 2.4.1 Turing machines............................ 40 2.4.2 Computational Complexity Theory.................. 41 2.4.3 Circuit model quantum computation................. 43 2.4.4 Quantum error correction....................... 48 2.4.5 Fault tolerance............................. 50 2.4.6 Stabilizer formalism........................... 51 2.4.7 Measurement-based quantum computation.............. 53 2.4.8 Other computational models...................... 55 2.4.9 Physical platforms........................... 57 3 Quantum light 62 3.1 Introduction................................... 62 3.2 Quantisation of the electromagnetic field................... 62 3.3 Linear optics.................................. 67 3.4 Quantum Interference............................. 69 3.4.1 The Hong-Ou-Mandel effect...................... 69 3.4.2 The HOM dip and partial-distinguishability............. 71 3.5 Coherent states................................. 73 3.6 Linear optical quantum computing...................... 75 3.6.1 KLM................................... 77 3.6.2 MBLOQC................................ 80 3.6.3 Boson sampling............................. 83 3.7 Quantum photonics............................... 89 3.7.1 Photon sources............................. 89 3.7.2 Linear optical networks......................... 93 3.7.3 Detectors................................ 96 3.7.4 Experimental setup........................... 97 9 CONTENTS 4 Universal Linear Optics 100 4.1 Introduction................................... 100 4.2 The Reck et al. Scheme............................ 101 4.3 A Universal Linear Optical Processor..................... 103 4.3.1 Characterisation............................ 104 4.4 Linear Optical Gates.............................. 106 4.4.1 Process tomography experiments................... 106 4.4.2 Heralded integrated gates....................... 111 4.4.3 Characterisation of Linear Optical Networks............. 117 4.5 Discussion.................................... 120 5 Simulating Vibrational Quantum Dynamics of Molecules with Integrated Photonics 122 5.1 Introduction................................... 122 5.2 Molecular Vibrational Dynamics in the Harmonic Approximation..... 124 5.3 Experimental Procedure............................ 126 5.4 Simulating Four Atom Molecules....................... 127 5.5 Energy transfer and dephasing in NMA.................... 131 5.6 Vibrational relaxation in H2O......................... 134 5.7 Anharmonic Hamiltonian for H2O....................... 137 5.8 Adaptive feedback control in the dissociation of NH3 ............ 140 5.9 Discussion.................................... 144 6 Linear Optical Quantum Computing with Partially-Distinguishable Pho- tons 146 6.1 Introduction................................... 146 6.2 Errors in LOQC................................. 147 6.3 Photon Distinguishability........................... 149 6.3.1 Dual-path states............................ 150 6.3.2 Distinguishability models........................ 151 10 CONTENTS 6.4 Generating entanglement with partially-distinguishable photons........................ 152 6.5 Entangled measurements with partially-distinguishable photons.......................