A PROTOTYPE QUANTUM COMPUTER USING NUCLEAR SPINS IN LIQUID SOLUTION a dissertation submitted to the department of electrical engineering and the committee on graduate studies of stanford university in partial fulfillment of the requirements for the degree of doctor of philosophy Matthias Steffen June 2003 c Copyright by Matthias Steffen 2003 All Rights Reserved ii I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a disser- tation for the degree of Doctor of Philosophy. James S. Harris (Principal adviser) I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a disser- tation for the degree of Doctor of Philosophy. Isaac L. Chuang (Co-adviser) I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a disser- tation for the degree of Doctor of Philosophy. Yoshihisa Yamamoto Approved for the University Committee on Graduate Studies. iii iv Abstract Quantum computers can potentially solve real and relevant mathematical and physical prob- lems that are intractable using classical machines. However, the experimental realization of quantum computers represents a significant challenge, because several opposing exper- imental requirements must be met. A set of coupled quantum bits must be manipulated and measured while coherently retaining entangled quantum states. Yet the manipulation and measurement processes almost inevitably lead to the decay of these fragile states. This thesis work takes significant steps towards building a practical quantum computer using nuclear spins and liquid state nuclear magnetic resonance (NMR) techniques. I present experimental results for proof of principle of quantum computing in a series of small im- plementations of quantum algorithms, culminating in the implementation of an adiabatic quantum optimization algorithm. The performance of adiabatic algorithms compared with classical optimization methods is unknown, but small quantum computers could provide crucial insight into answering this question. Furthermore, adiabatic algorithms also shed new light on the usefulness of quantum resources for computational tasks and hence they represent an important class of algorithms. The second part of this thesis presents several methods developed for improving the control over NMR quantum computing experiments. Even though liquid state NMR quan- tum computers have well known and accepted scaling limitations, the developed tools are of general use. I will show how several tools are directly transferable to two other imple- mentations of quantum computers; one using optical methods and the other using loops of superconducting material. This work provides insight into what is needed to build a practical quantum computer, and delivers several tools and techniques that will be useful in future large-scale implemen- tations of quantum computers. v Acknowledgements Equipped with a background in Physics from Emory University, I came to Stanford Univer- sity, looking for an exciting research project to join. Before I knew it, I found myself inter- ested in a wide selection of research groups including Jim Harris' group, or more specifically the quantum computing subgroup. He referred me to Lieven Vandersypen who patiently explained what quantum computing is all about and how he was involved in it. This topic truly fascinated me not just because of its enormous future applications but also because of the significant experimental work involved. Soon after, I met Isaac Chuang who initiated the entire NMR Quantum Computing project at IBM. He offered me to join the group, and this was the beginning of an exciting 4.5 year journey that started at IBM Almaden in San Jose and ended up at MIT in Boston. Jim Harris (aka coach) was my advisor at Stanford and introduced me to the right people that made my work a success. I deeply appreciate his encouragement and support over the years and his outlook on life in general. Isaac Chuang, my co-advisor guided and inspired me always at the right times. He taught me how to put my work into perspective, to think outside the box and helped me continue to develop important presentation and writing skills. After he chose to accept a position at MIT, I decided to follow him and I appreciate having learned about the research setting in industry (at IBM) and academia (at MIT). Lieven Vandersypen has been a tremendous mentor during my 2.5 years at IBM. I started as his apprentice but we soon grew together to form a very productive team. I look back at this time with great pride, and I am truly fortunate to have with him during this period. Since we worked so closely together over the years, there is substantial overlap between his and my thesis. It's hard for any one of us to fully take credit for an experiment that we worked on together. Most of our ideas and work has been a combined effort, but there are different areas of concentrations. Lieven worked a lot on the early simulation code, the vi decoherence model, did a significant portion of the writing of the papers, and worked out some of the quantum circuits used in chapter 6. I subsequently simplified several of these circuits, and focused on the software reference frame used for the seven spin framework (shown in the Appendix) as well as the solutions to several artifacts that arose throughout our experiments (shown in chapter 5). Plenty of thanks to the entire IBM crew. Nino Yannoni (Nino) found most of the molecules we ended up using - very spectacular molecules I might add. I enjoyed his words of wisdom which he was very willing to share with us during the most enjoyable lunch hour. Mark Sherwood always offered helpful hints and comments pertaining to NMR techniques and molecule choices, while ensuring adequate coffee supplies. Greg Breyta synthesized all of the molecules that Nino discovered during his free time - a commitment he did not have to make. My other colleagues, Anne Verhulst and Oskar Liivak, contributed to a very pleasant working atmosphere and provided numerous useful discussions. Of course, I also wish to thank the new group that was formed at MIT. Andrew Houck was the first student that joined our group. He continuously found ways to bring humor into the laboratory (Maxwell's Demon). Steve Huang, our electronics guru, followed soon after and I had some great discussions with him. Many thanks also to (and this is in no particular order) Teri, Aram, Francois, Andrew (Cross), Josh, Joshua, and Zilong for being such spectacular group members. Murali Kota, a fine enthusiastic student, started as my apprentice at MIT, but unfor- tunately I was not able to convince him to pursue NMR quantum computation \full time". Nonetheless, he had several very interesting experimental proposals, which are now part of chapters 4, 6 and 7 of my thesis. Bin is an extraordinary undergraduate student who worked with us for his undergrad- uate thesis. He learned the spectrometer details at an amazing pace, and started running experiments soon after he joined the group. Him, Murali and I together worked on the connection between the EIT effect and quantum computing as outlined in section 7.5. Gail Chun Creech was extremely helpful in coordinating a lot of the complex adminis- trative tasks resulting from my move to Boston. The same goes for the rest of the admin- istrational staff of Stanford and MIT (Maureen, Mike, and the Ex's Murray and Claire). Thanks to everyone else who enriched my life throughout my graduate school career. The Neil Gershenfeld gang (Jason, Yael, Ben), Wim van Dam and Tad Hogg from HP, who vii helped me tremendously on one of my later projects. To my friends and \the boys" who supported me when needed. My girlfriend Jennifer Padgett was always there for me when I had concerns or just whenever I needed support. Last but certainly not least, thank you to my parents, brother, uncle and aunt who have always believed in me and my abilities, and taught me how to succeed in life. viii Contents Abstract v Acknowledgements vi 1 Introduction 1 1.1 Historical background . 1 1.2 Related fields and additional literature . 5 1.3 Goals of my work . 6 1.4 Organization of the dissertation . 7 2 Theory of quantum computing 10 2.1 Quantum computing vs. classical computing . 10 2.1.1 Quantum bits vs. classical bits . 10 2.1.2 Quantum computing subsumes classical computing . 15 2.1.3 Quantum parallelism . 19 2.1.4 Complexity theory . 22 2.1.5 Quantum error correction . 23 2.2 Quantum gates . 25 2.2.1 Universal quantum gates . 26 2.2.2 Single qubit gates . 26 2.2.3 Two-qubit gates . 29 2.2.4 Remarks on unitary gates . 31 2.2.5 Multiple (n > 2) qubit gates . 32 2.3 Quantum algorithms . 34 2.3.1 The Deutsch-Jozsa algorithm . 35 ix 2.3.2 Grover's algorithm . 38 2.3.3 Order-finding and Shor's algorithm . 42 2.3.4 Adiabatic quantum algorithms . 50 2.3.5 Quantum simulations . 52 2.3.6 Perspectives . 53 2.4 Decoherence . 54 2.4.1 Energy dissipation . 54 2.4.2 Phase randomization . 56 2.4.3 Remarks on amplitude and phase damping . 58 2.4.4 Other sources of decoherence . 58 2.5 Summary . 59 3 Implementation of quantum computers 61 3.1 Requirements . 61 3.1.1 System of qubits . 63 3.1.2 Quantum gates . 64 3.1.3 Initialization . 67 3.1.4 Measurement . 70 3.1.5 Coherence . 72 3.2 State of the art . 73 3.3 Summary . 74 4 Liquid-state NMR quantum computing 75 4.1 System of qubits .