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Quantum Computation Beyond the Unitary Circuit Model

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Quantum Computation Beyond the Unitary Circuit Model Quantum Computation Beyond the Unitary Circuit Model Na¨ıri Usher A thesis submitted to University College London for the degree of Doctor of Philosophy Department of Physics and Astronomy University College London September 2016 I, Na¨ıri Usher, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. Signed: Date: 1 Publications The majority of the work presented in this thesis contains materials from the following publications: N. Usher, and D.E. Browne, Noise in Measurement-Based Quantum Computing, arXiv:1704.07298. N. Usher, M.J. Hoban, and D.E. Browne, Constructing Local Hamiltonians from Non-unitary Quan- tum Computation, arXiv:1703:08118. M.J. Hoban, J.J. Wallman, H. Anwar, N. Usher, R. Raussendorf, and D.E. Browne, Measurement- Based Classical Computation, Phys. Rev. Lett. 112, 140505 (2014). 2 Acknowledgements I am thankful for my time at UCL which has brought to me more than words can convey. First and foremost, I am thankful to Dan for having supported and guided me throughout my PhD with endless patience and kindness. I am truly grateful for his encouragement to explore and investigate questions I found interesting as well as for everything I had the chance to learn from him. I wish to thank my examiners Shashank and Sougato for having taken the time to read this thesis as well as for offering valuable feedback for its improvement. I was fortunate to collaborate with Matty, with whom I shared many both interesting and entertaining discussions, and I am thankful for his invaluable insights and encouragement. I am in particular thankful to Simone for his continued support and guidance. From my start at UCL, the quantum group meetings were a source of joy, each week introducing me to new ideas, and granting me the privilege of learning from exceptional people, and for this I am thankful to Dan, Fernando, Aram, Jonathan, Simone and Toby. I have made many friends throughout both my time at UCL and travelling the world to con- ferences: to all those with whom I have shared coffees, drinks, dances and walks, thank you for enriching my life. I found within the physics department a space in which questions were always welcome, and I thank all those with whom I spent time discussing and debating all matter of ideas. I look back with a particular fondness to both my first ever summer school in Waterloo and a trip to Malawi, for which I am greatly thankful to Terry for the brilliant organisation and experience. The K3 has been a bastion of sanity, and have proven themselves to be thoroughly solid vertices throughout all these adventures. Thank you to Chris for walking, benching and impromptu singing, and to Pete for his skill at fixing all things, endless laughter and activities. To my co-author and friend, Hussain, I am thankful for arguments, coffees and most of all, his kindness. To my friends Noor and Justin, I am grateful for your true friendship, and look forward to times yet to come. And finally, thank you to my mother, my friend, Victoria Gaspari, for showing me, ever since the day I was born, all the beauty and wonder to be discovered within the world. 3 Abstract This thesis considers various paradigms of quantum computation in an attempt to understand the nature of the underlying physics. A standard approach is to consider unitary computation on pure input states, such that the outcome of the computation is determined by single computational basis measurement on the output state. It has been shown that there exists equivalent models of computa- tion, such as measurement based quantum computing (MBQC), which provide insight into the role of entanglement and measurement. Furthermore, constraining or relaxing available resources can directly impacts the power of the computation, allowing one to gauge their role in the process. Here, we first extend known constructions such as Matrix Product States, MBQC and the one- clean qubit model to a mixed state formalism, in an attempt to develop computational models where noise acting on the physical resources, as might be experienced in laboratory settings, may be mapped to logical noise on the computation. Next, we introduce Measurement-Based Classical Computing, an essentially classical model of computation, wherein the complexity hard wired into probability distributions generated via quantum means yields surprising non classical results. Fi- nally, we consider postselection the ability to discard displeasing measurement outcomes and ar- gue that it may be used in a tame way, which does not provide a dramatic increase in computational power. From here, we develop a new Hamiltonian, based on a circuit to Hamiltonian construction, presenting evidence of QMA-hardness. 4 Contents 1 Introduction 10 1.1 Physics, Computation, Information . 10 1.2 Quantum Physics . 15 1.2.1 Quantum States . 16 1.2.2 Tensor Network States: Matrix Product States . 17 1.2.3 Mixed States . 22 1.2.4 Quantum State Evolution . 23 1.2.5 Measurement . 24 1.3 Quantum Computing . 26 1.3.1 The Quantum Circuit Model . 27 1.3.2 Measurement Based Quantum Computing . 35 1.4 Computational Complexity . 41 1.4.1 The Language of Computational Complexity . 42 1.4.2 Complexity Classes . 44 1.4.3 The k-LOCAL HAMILTONIAN Problem . 53 1.4.4 Postselection . 63 1.4.5 Simulatability . 67 2 Noisy Measurement-Based Quantum Computation 72 2.1 Models of Noise . 74 2.1.1 Pauli Channels . 74 2.1.2 Unital Channel . 76 2.1.3 Dephasing Channel . 76 2.2 Motivation: Teleportation with Mixed States . 77 2.2.1 Mixed resource states . 78 2.2.2 Diagonal Bell States as Resource . 78 2.2.3 Noisy teleportation: examples . 79 5 2.3 Noise in MBQC via One Bit Teleportation protocol . 80 2.3.1 The Noiseless Model . 81 2.3.2 Examples: Pauli Noise . 83 2.3.3 General Local Noise . 84 2.3.4 Examples . 94 2.4 Matrix Product Operators . 96 2.4.1 Motivation . 97 2.4.2 Matrix Product Operators . 98 2.4.3 MPO Representation: Examples . 102 2.5 Noise in MBQC in the MPO Framework . 103 2.5.1 Error Propagation: Framework . 103 2.5.2 Noise in One-dimensional MBQC . 106 2.6 Conclusion . 108 3 Measurement Based Classical Computation 110 3.1 Classical MBQC . 111 3.1.1 From Quantum to Classical Resource States . 111 3.1.2 Measurement Based Classical Computing . 112 3.2 IQP∗ Circuit Families . 112 3.2.1 IQP∗ ........................................ 113 3.2.2 The C omplexity of IQP∗ ............................. 115 3.2.3 The Complexity of IQP∗ Zero Input Families . 117 3.3 The Complexity of MBCC . 118 3.3.1 MBQC and IQP∗ ................................. 118 3.3.2 Fixed Bases MBQC . 119 3.3.3 Non-classicality of MBCC . 119 3.4 Conclusion . 120 3.4.1 Results . 120 3.4.2 Future Directions . 120 4 Non-Unitary Quantum Computation in the Ground Space of Local Hamiltonians 121 4.1 Hamiltonians, Postselection and Renormalised Projectors . 124 4.1.1 Measurements, Postselection and Renormalised Projectors . 124 4.1.2 A General Evolution . 128 4.1.3 Non-Unitary Evolution via Renormalised Measurement . 129 4.2 Hamiltonians from Postselected Quantum Circuits . 131 6 4.2.1 General postselected circuits . 131 4.2.2 IQP circuits . 132 4.2.3 Hamiltonian associated to a Hadamard gadget . 133 4.3 Numerical Simulation . 136 4.3.1 Cascaded Gadgets: the Exponential Case . 137 4.3.2 Sequential Gadgets: the Polynomial Case . 140 4.4 Conclusion . 142 4.4.1 Results . 143 4.4.2 postQMA . 143 4.4.3 Future Directions . 144 7 List of Figures 1.1 Tensor networks. 18 1.2 Tensor network index contraction. 18 1.3 One-dimensional system on a line. 19 1.4 One-dimensional system on a ring. 21 1.5 Bloch sphere. 29 1.6 Circuit for Bell state preparation. 31 1.7 Teleportation. 33 1.8 One bit teleportation. 35 1.9 One bit teleportation as a quantum channel. 35 1.10 Hadamard gadget . 35 1.11 Measurement in the equatorial plane of the Bloch sphere. 37 1.12 Measurement in the X basis. 37 1.13 Two-dimensional cluster state. 38 1.14 One-dimensional cluster state. 39 1.15 MPS evolution. 41 1.16 Relationship between complexity classes P and NP................... 46 1.17 A hierarchy of complexity classes . 52 2.1 Single qubit quantum channel. ..
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