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Many-Body Entanglement in Classical & Quantum Simulators

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Many-Body Entanglement in Classical & Quantum Simulators Many-Body Entanglement in Classical & Quantum Simulators Johnnie Gray A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy of University College London. Department of Physics & Astronomy University College London January 15, 2019 2 3 I, Johnnie Gray, 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 work. Abstract Entanglement is not only the key resource for many quantum technologies, but es- sential in understanding the structure of many-body quantum matter. At the interface of these two crucial areas are simulators, controlled systems capable of mimick- ing physical models that might escape analytical tractability. Traditionally, these simulations have been performed classically, where recent advancements such as tensor-networks have made explicit the limitation entanglement places on scalability. Increasingly however, analog quantum simulators are expected to yield deep insight into complex systems. This thesis advances the field in across various interconnected fronts. Firstly, we introduce schemes for verifying and distributing entanglement in a quantum dot simulator, tailored to specific experimental constraints. We then confirm that quantum dot simulators would be natural candidates for simulating many-body localization (MBL) - a recently emerged phenomenon that seems to evade traditional statistical mechanics. Following on from that, we investigate MBL from an entanglement perspective, shedding new light on the nature of the transi- tion to it from a ergodic regime. As part of that investigation we make use of the logarithmic negativity, an entanglement measure applicable to many-body mixed states. In order to tie back into quantum simulators, we then propose an experimental scheme to measure the logarithmic negativity in realistic many-body settings. This method uses choice measurements on three or more copies of a mixed state along with machine learning techniques. We also introduce a fast method for computing many-body entanglement in classical simulations that significantly increases the size of system addressable. Finally, we introduce quimb, an open-source library for interactive but efficient quantum information and many-body calculations. It contains general purpose tensor-network support alongside other novel algorithms. Impact Statement Quantum technologies have the potential to make a huge impact across many ar- eas of science and industry. One such technology is quantum simulation, which aims to mimic crucial models of interacting quantum particles with other, more controllable quantum systems. Such ‘many-body’ models are likely to underpin all sorts of advanced materials and chemicals that we expect will essentially always be impossible to fully simulate on normal, or ‘classical’, computers. One of the reasons that current computers find this such a hard task is entanglement – correlations between quantum components which are stronger than anything we are used to in our ‘classical’, day-to-day lives. Beyond this, entanglement is also a key perspective for understanding a myriad of outstanding problems in fundamental physics. Clearly then, tools to address questions of entanglement in this many-body setting are highly desirable, and these are a central part of what this thesis provides. One of these is experimentally viable schemes for both measuring and dis- tributing entanglement in quantum simulators. Another is a method for computing entanglement in much larger systems than has previously been possible. These tools are relevant for both quantum physicists working experimentally on quantum simula- tion or those wanting to perform classical simulations. We also use entanglement to reveal new features of a highly unusual phenomenon – many-body localization – that has attracted much attention in the condensed matter community. These col- laborations have resulted in various academic publications. Another part of this research has been the development of open-source software for tackling quantum entanglement and simulation problems. As well as being useful in this field, work on the underlying software has involved collaborations with quantum chemists and climate scientists among many others, and produced several software publications. Publications This thesis has resulted in the following publications, the first five of which form the basis for the main chapters: • Johnnie Gray, Abolfazl Bayat, Reuben K Puddy, Charles G Smith, and Sougato Bose. Unravelling quantum dot array simulators via singlet-triplet measure- ments. Phys. Rev. B, 94(19):195136, 2016 • Johnnie Gray, Sougato Bose, and Abolfazl Bayat. Many-body localization transition: Schmidt gap, entanglement length, and scaling. Phys. Rev. B, 97(20):201105, 2018 • Johnnie Gray, Leonardo Banchi, Abolfazl Bayat, and Sougato Bose. Machine- learning-assisted many-body entanglement measurement. Phys. Rev. Lett., 121(15):150503, 2018 • Johnnie Gray. Fast computation of many-body entanglement. arXiv preprint arXiv:1809.01685, 2018 • Johnnie Gray. quimb: a python library for quantum information and many- body calculations. Journal of Open Source Software, 3(29):819, 2018 • Daniel GA Smith and Johnnie Gray. opt einsum - a python package for optimizing contraction order for einsum-like expressions. Journal of Open Source Software, 3(26):753, 2018 Acknowledgements I would like to begin of course by thanking my advisor Sougato Bose, who has been a never ending supply of knowledge and ideas across such a huge range of interesting topics. He has completely made this thesis possible. A special thanks also to Abolfazl Bayat who has not only been a fantastic collaborator but offered so much practical guidance as well. I very much hope that we get to work together in the future! The CDT in delivering quantum technologies has been a great, structured, well- rounded way to approach a PhD and I would like to thank its heads, Andrew Fisher and Dan Browne, as well as its life-force, Lopa Murgai. It has been an pleasure to have done the course with such a nice cohort – Andrew, Cameron, Carlo, Claudia, Josh, Lorenzo, Sherif, Tim, and Tom – learning how to program a vending machine would not have been nearly as bearable without you. Going back further I would like to particularly thank two of my school physics teachers, Barbara “Bob” Justham and Duncan Nicholls, who sadly passed away in my first year at university. The last chat I had with Duncan in the pub was about how confusing I found ‘vectors’ after one term – I think he would have been proud at the profligate usage of them in this thesis. Both of them were enormously successful in managing to convey the essential excitement and magic of physics and that inspiration has clearly stuck with me. It goes without saying that there are many friends, too many to name, who at various points have offered such great advice and support – I am forever grateful to know such an amazing group of people. Finally, I would like to thank my family for ultimately providing everything that has made this thesis possible – they should probably take all the credit. Contents 1 Introduction 21 1.1 Quantum Simulation . 24 1.2 Entanglement . 28 1.3 Many-Body Quantum Models . 32 1.3.1 Hubbard Model . 32 1.3.2 Heisenberg Model . 33 1.4 Many-Body Quantum Phenomena . 35 1.4.1 Quantum Phase Transitions . 36 1.4.2 Equilibration & Thermalization . 37 1.4.3 Many-Body Localization . 38 1.5 Entanglement in Classical Simulations - Tensor Networks . 39 1.6 Experimental Realizations . 42 2 Unravelling Quantum Dot Array Simulators 45 2.1 Abstract . 45 2.2 Introduction . 46 2.3 Model . 49 2.4 Characterization of Simulator . 52 2.5 Quantum Phase Transition in the J1 − J2 Model . 55 2.6 Heralded entanglement of distant spins . 56 2.7 Imperfections . 58 2.8 Experimental Realization . 63 2.9 Relation to Many-Body Localization . 65 14 Contents 2.10 Conclusions . 67 3 Entanglement & the many-body localization transition 69 3.1 Abstract . 69 3.2 Introduction . 70 3.3 Model . 71 3.4 Characterizing the MBLT . 72 3.5 Scaling . 74 3.6 Sample Fluctuations . 76 3.7 Entanglement length . 77 3.8 Conclusions . 80 4 Machine-learning-assisted many-body entanglement measurement 81 4.1 Abstract . 81 4.2 Introduction . 82 4.3 Logarithmic Negativity . 83 TB 4.4 Measuring the Moments of ρAB ................... 85 4.5 Machine Learning Entanglement . 86 4.6 Training with Random States . 88 4.7 Numerical Results for Random States . 89 4.8 Numerical Results for Physical Statesb . 89 4.9 Conclusions . 90 5 Fast computation of many-body entanglement 93 5.1 Abstract . 93 5.2 Introduction . 94 5.3 Logarithmic Negativity . 95 5.4 Stochastic Lanczos Quadrature . 96 5.5 Tensor Networks & Graphical Notation . 99 5.6 Partial Trace States . 101 5.7 Matrix Product States . 103 5.8 Results . 107 Contents 15 5.8.1 Random pure states . 108 5.8.2 Scrambling in a Quench . 109 5.8.3 Heisenberg ground-state . 111 5.9 Error Analysis . 112 5.10 Discussion . 113 6 quimb - A library for quantum many-body calculations 117 6.1 The core module . 118 6.1.1 Basic representation and operations . 118 6.1.2 Multipartite states and partial traces . 120 6.1.3 Time Evolutions . 120 6.1.4 Distributed eigensolving . 121 6.1.5 Stochastic Lanczos Quadrature . 122 6.2 The tensor network module . 123 6.2.1 Tensors . 124 6.2.2 Tensor networks . 125 6.2.3 One-dimensional tensor networks . 125 6.2.4 DMRG . 128 6.2.5 TEBD . 129 6.2.6 TNSLQ . 130 7 General Conclusions 133 Appendices 137 A Appendix for ‘Entanglement & the many-body localization transition’ 137 A.1 Quality of Collapse . 137 A.2 Entanglement Length with Larger Block Size . 138 A.3 Detailed Behaviour of the Disjoint Entanglement vs. L ....... 139 B Appendix for ‘Machine-learning-assisted many-body entanglement measurement’ 141 B.1 Measuring Moments in spin-1=2 Systems .
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