COPYRIGHT AND USE OF THIS THESIS This thesis must be used in accordance with the provisions of the Copyright Act 1968. Reproduction of material protected by copyright may be an infringement of copyright and copyright owners may be entitled to take legal action against persons who infringe their copyright. Section 51 (2) of the Copyright Act permits an authorized officer of a university library or archives to provide a copy (by communication or otherwise) of an unpublished thesis kept in the library or archives, to a person who satisfies the authorized officer that he or she requires the reproduction for the purposes of research or study. The Copyright Act grants the creator of a work a number of moral rights, specifically the right of attribution, the right against false attribution and the right of integrity. You may infringe the author’s moral rights if you: - fail to acknowledge the author of this thesis if you quote sections from the work - attribute this thesis to another author - subject this thesis to derogatory treatment which may prejudice the author’s reputation For further information contact the University’s Director of Copyright Services sydney.edu.au/copyright Quantum computational phases of matter Andrew S. Darmawan A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy School of Physics University of Sydney August 11, 2014 Statement of contribution This thesis is mainly comprised of three research papers, presented as Chapters 2, 3, and 4. Note that they are not presented in the order in which they were published. On all three papers, I am the primary (first) author as well as the sole student author. Measurement-based quantum computation in a two-dimensional phase of matter, Chapter 2 This chapter contains the published paper Darmawan, Brennen, and Bartlett (2012) with minor typographical error in Eq. (2.3) and Eq. (2.4) removed. This work was a project undertaken by me, and supervised by Gavin Brennen and Stephen Bartlett. The research and calculations were undertaken by me and paper was written by me with edits from GB and SB. Graph states as ground states of two-body frustration-free Hamiltonians, Chapter 3 This chapter contains the published paper Darmawan and Bartlett (2014), with some small modifications to remove typographical errors and improve clarity. This work was a project undertaken by me, and supervised by Stephen Bartlett. The research and calculations were undertaken by me and paper was written by me with edits from SB. Optical spin-1 chain and its use as a quantum computa- tional wire, Chapter 4 This chapter contains the published paper Darmawan and Bartlett (2010), with minor typographical errors removed. The original idea for this project is due to Stephen Bartlett. The research was undertaken by me and the paper was written by me with edits from SB. 1 Acknowledgements I would like to sincerely thank members of the Quantum Science Group at the University of Sydney for their support throughout my PhD. Thank you to my fellow students and office mates, especially Courtney Brell, Matt Palmer, Matthew Wardrop, Joel Wallman, Rafael Alexander and Jacob Bridgeman for always having time to chat. I also thank staff members in the group including Aroon O'Brien, Steve Flammia and Andrew Doherty for helpful discussions and the administrative staff especially Leanne Price for the help with paperwork and with travel planning. I thank the research groups I visited overseas. In particular I thank Jiannis Pachos, Terry Rudolph, Tobias Osborne and Jens Eisert for their hospitality during these stays. I thank the experts Tzu-Chieh Wei, Akimasa Miyake, and Robert Raussendorf for stimulating discussions during conferences and visits to the University of Sydney. I thank my family for their unwavering support over these last few years. Everyday I am grateful for the opportunities, assistance and security they pro- vide. Finally, a huge thanks goes to my associate supervisor Gavin Brennen and to my supervisor Stephen Bartlett. Their support and inspiration over the years of my PhD has been immeasurable. Acknowledgments for specific papers are as follows. Measurement-based quantum computation in a two-dimensional phase of matter, Chapter 2 We thank Akimasa Miyake for helpful comments and Andrew Darmawan thanks Tzu-Chieh Wei for helpful discussions. This research was supported by the ARC via the Centre of Excellence in Engineered Quantum Systems (EQuS), project number CE110001013. Graph states as ground states of two-body frustration-free Hamiltonians, Chapter 3 We thank Tzu-Chieh Wei for helpful discussions and for verifying numerical re- sults and Gavin Brennen for helpful comments. This research was supported by 2 the ARC via the Centre of Excellence in Engineered Quantum Systems (EQuS), project number CE110001013. Optical spin-1 chain and its use as a quantum computa- tional wire, Chapter 4 We thank Andrew Doherty, Rainer Kaltenbaek, Anthony Laing, Jonathan Lavoie, Jeremy O'Brien, Geoff Pryde, Kevin Resch, Terry Rudolph, Howard Wiseman, and Bei Zeng for discussions. SDB acknowledges the support of the Australian Research Council. 3 Contents 1 Introduction 6 1.1 Quantum computation ........................ 6 1.1.1 Measurement-based quantum computation ........ 7 1.2 Quantum many-body physics .................... 10 1.2.1 The AKLT model and tensor network ground states ... 10 1.2.2 MBQC with ground states .................. 11 1.2.3 Quantum computational phases of matter ......... 12 1.3 Summary of thesis .......................... 13 2 Measurement-based quantum computation in a two-dimensional phase of matter 15 2.1 Introduction .............................. 15 2.2 Model definitions ........................... 17 2.3 MBQC using ground states in the AKLT phase .......... 19 2.3.1 Protocol at AKLT point ................... 19 2.3.2 Generalized reduction scheme ................ 20 2.3.3 Statistical model ....................... 21 2.3.4 Identifying computationally powerful ground states .... 22 2.4 Exploring the phase ......................... 23 2.5 Conclusion .............................. 26 2.6 Appendix: Ground states as PEPS ................. 27 2.7 Appendix: Distribution of measurement outcomes ........ 28 2.8 Appendix: Monte Carlo sampling .................. 29 3 Graph states as ground states of two-body frustration-free Hamil- tonians 30 3.1 Introduction .............................. 31 3.2 Model definitions and notation ................... 33 3.2.1 Local conversion to graph states .............. 33 3.2.2 Frustration-free Hamiltonians with unique ground states close to graph states ..................... 34 3.3 Universality of ground states for MBQC .............. 38 3.3.1 The spin-3/2 AKLT model ................. 38 4 3.3.2 General approach for driving frustration-free models into computational phases .................... 47 3.4 Spectral properties .......................... 48 3.4.1 Gap vs. fidelity ........................ 49 3.4.2 The spin-1 AKLT model ................... 50 3.4.3 Generalising spectral properties ............... 55 3.5 Conclusions .............................. 57 4 Optical spin-1 chain and its use as a quantum computational wire 59 4.1 Introduction .............................. 59 4.2 The AKLT state ........................... 61 4.2.1 The AKLT state as a quantum computational wire .... 64 4.3 Optical implementation ....................... 64 4.3.1 Creating a photonic AKLT state .............. 65 4.3.2 Quantum computational wire operations .......... 66 4.4 Conclusion .............................. 74 5 Conclusion 75 5 Chapter 1 Introduction Computers are machines that solve problems. Like all machines, they are built and operate according to well-understood physical principles. Present-day com- puters are built from physical systems that can be described using classical physics. However, in recent years our ability to manipulate quantum mechani- cal systems has improved substantially, and with this has come the potential of building a computer which operates according to quantum physics, rather than classical physics. It has been shown that a quantum computer, if it could be built, could efficiently solve a variety of problems that currently have no efficient solution on classical computers. To obtain a performance improvement over a classical computer, quantum computers must exploit entanglement in large numbers of quantum mechanical particles. In this thesis we will investigate how the necessary entanglement for quantum computing may be present in natural phases of quantum matter. In this chapter we will introduce key concepts and tools from quantum computation and many-body physics and summarise our main results. 1.1 Quantum computation Here we will provide a brief introduction to quantum computation, with a par- ticular emphasis on measurement-based quantum computation (MBQC). The concept of MBQC is central to the research presented in this thesis, and in the following section we will show how it may be applied to various systems systems in many-body physics. Our intention here is to provide the reader with a broad overview of essential concepts, without including all the detail required for full rigor. We will assume basic knowledge of quantum physics. A more thorough overview of quantum computation is provided in Nielsen and Chuang (2004). We will start with a brief explanation of the circuit model of quantum compu- tation, and some preliminary definitions. A qubit is a generic two-level quantum system that represents the basic unit of quantum information.