SOFIA UNIVERSITY \ST. KLIMENT OHRIDSKI" Faculty of Physics Design of quantum gates and entangled states for quantum information technologies Elica Sotirova Kyoseva Supervisor: Assoc. Prof. Nikolay V. Vitanov Submitted in total ful¯lment of the requirements for the degree of Doctor of Philosophy So¯a January 2009 To my mother. The candidate con¯rms that the work submitted is her own and that appropriate credit has been given where reference has been made to the work of others. Copyright © 2009. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. Acknowledgements First and foremost, I would like to express my gratitude to my PhD supervisor, Nikolay Vitanov, for his good-natured support and patient guidance over the past ¯ve years. From our ¯rst meeting he has been a continuous source of motivation and support for me. I have greatly bene¯tted from his encouragement, advice and expertise. I am very grateful for all opportunities he has given me over the years and for his good company. I would also like to thank the other members of the group, as well as many visitors with whom I have had useful discussions. A special thank you goes to Bruce Shore. His love of Physics is infectious and I have enjoyed all our discussions in So¯a and Kaiserslautern thoroughly. A milestone for me was my visit in the Quantum Information group at the University of Leeds and I would like to thank all its members for their warm welcome. I am extremely grateful to Almut Beige who supervised me during the nine months I spend there and who treated me as one of her own PhD students. I am very thankful for her great patience, her guidance, and for introducing me to the ¯eld of measurement-based quantum computing. I remain very impressed with her knowledge, broad background, and great attention to detail. I would like to say a heartfelt thank you to the leader of the group, Vlatko Vedral, for the motivating discussions I have had with him and his company. I am very grateful for his invitation to visit the Centre for Quantum Technologies in Singapore, which inspired me to continue my work in academia there. I would like to say thank you to Jonathan Busch for our work together. I want to mention also Libby Heaney, Bruno Sanguinetti and Michal Hajdusek for the good times we enjoyed in Leeds and elsewhere. I would like to thank all my family, for their love and constant support. The most special thanks goes to my mother Eli, for making me the person I am. She gave me the con¯dence to follow my heart and pursue my dreams. I felt she is behind me every step of the way, whichever path I took. This thesis is dedicated to her. A wholehearted thank you goes to my brother Vasil for taking care of his little sister throughout all her life; to my cousin Vesela, to my aunt Ani and to my grandparents Vasilka and Evtim. I love our family times together. Finally, I would like to give a loving thank you to Mark Williamson. You have been a v vi constant source of happiness in my life for the last year and a half even when we were away from each other. I am extremely grateful for your love and kind support, without you this thesis wouldn't have been completed now. So¯a, January 2009 Contents Acknowledgements v 1 Introduction 1 2 Theoretical background 11 2.1 Qubit dynamics . 13 2.2 Resonant excitation . 15 2.3 Optical Bloch equations . 17 3 E®ect of dephasing on single-qubit gates 19 3.1 Motivation . 20 3.2 Analytic model . 20 3.3 Special cases . 23 3.4 Conclusions . 26 4 Controlled design of arbitrary quNit states 29 4.1 Motivation . 30 4.2 De¯nition of the problem . 30 4.3 General solution . 32 4.4 Types of population distribution . 36 4.5 Applications to exactly soluble models . 40 4.6 Multistate Landau-Zener model . 45 4.7 Conclusions . 48 5 Engineering of arbitrary unitaries by reflections 51 5.1 Motivation . 51 5.2 Quantum Householder Reflection (QHR) . 53 5.3 QHR decomposition of U(N) ........................... 56 5.4 Quantum Fourier transform . 60 vii viii CONTENTS 5.5 Discussion and conclusions . 63 6 Physical realisation of coupled quantum reflections 65 6.1 Motivation . 66 6.2 The degenerate two-level system . 67 6.3 Exact analytical solution . 69 6.4 Quantum-state reflections . 74 6.5 Two degenerate upper states . 77 6.6 Discussion and conclusions . 82 7 Entangling distant qubits using classical interference 85 7.1 Motivation . 85 7.2 Experimental proposal . 86 7.3 Entangling scheme . 88 7.4 Evolution of the system . 91 7.5 Conclusions . 92 8 Summary and contributions 95 A Publications of the author 97 Bibliography 99 C h a p t e r 1 Introduction The observation that a classical computer can not simulate the dynamics of a quantum system in an e±cient way was made by Richard Feynman during his talk at the First Conference on the Physics of Computation held at MIT in 1981 [1]. He argued that any classical simulation of quantum evolution will involve an exponential slowdown in time as compared to the natural evolution since the amount of information required to describe the evolving quantum state in classical terms generally grows exponentially in time. Surprisingly, he considered this as an opportunity rather than a disadvantage. He thought that if it is so di±cult to compute what will happen in a multiparticle interference experiment, the very act of constructing this experiment and reading out its outcome is analogous to processing this complex simulation. In fact all multiparticle interferometers are quantum computers [2]. Peter Shor was the ¯rst one to show how quantum computers can exploit quantum inter- ference of di®erent computational paths to enhance the correct outcomes and suppress the erroneous ones for various computational tasks. He proposed an algorithm for factoring an integer number or ¯nding discrete logarithms [3] in a number of steps that are polynomial in the input size, e.g., the number of digits of the integer to be factored. Classically, the factoring problem is considered very hard to solve because it requires a number of computational steps that are exponential in the input size. The second quantum algorithm proposed - Grover's search algorithm [4], is another illustration of the speed-up that quantum computers provide. Grover showed that a quantum computer can ¯nd an element in an unstructured database of p N entries in O( N) search steps. Note, that the classical analogue of this algorithm grows as O(N=2) with N. New quantum algorithms for problems previously regarded as intractable 1 2 CHAPTER 1. INTRODUCTION are continuously being proposed and they are all based on the subtle properties of quantum interference [5]. The physical implementation of the quantum computer is based on qubits - quantum systems which like the classical logical bits have two states: j0i and j1i written in Dirac notation and known as computational basis states [6]. The di®erence between bits and qubits is that a qubit can exist in a superposition of states between j0i and j1i until it is observed. We can write down the state vector of a qubit in a pure state as a linear superposition, jÃi = ® j0i + ¯ j1i ; (1.1) We say that a qubit is in a pure state if its state can be expressed as such superposition. It is not always possible however, to ascribe a state vector to a quantum system that is not isolated but is a part of a larger compound system. In this case, assume all we know is that the system is in a mixture (not a superposition) of states jÃi, not necessary orthogonal to each other and having a di®erent expansion in the computational basis. Let pà ¸ 0 be the probability that the system is in state jÃi. Then, the density operator must be written as X ½ = pÃjÃihÃj; (1.2) à which is clearly a Hermitian operator, ½ = ½y. We can calculate that P2 P T r(½) = n=1 ½nn = à pà = 1 which is consistent with the interpretation that pà are probabilities for being in a state jÃi. Then, we say that the system is in a mixed state, i.e. a mixture of di®erent pure states, which can not be represented as a state vector. For system in a pure state its density matrix is just ½ = jÃihÃj. For convenience, we assume that the qubit which represents our logical bit is in a pure state. Then, its state has the form (1.1) where the coe±cients ® and ¯ are complex numbers. Remarkably, we can not examine a quantum state to determine the exact values of ® or ¯ provided we restrict to making measurements in the computational basis even if we have a very large number of copies of the state. The information we can retrieve by repeated measurements on identical copies of the state in di®erent bases are the values of j®j2, j¯j2 and their relative phase, j®jj0i + eiÁj¯jj1i. The real numbers j®j2 and j¯j2 are the probabilities of detecting 0 or 1 when measuring the state of the qubit. In fact, measurements in quantum computation represent probabilistic operations that convert a qubit into a classical bit M = f0; 1g. Since probabilities should sum up to one, we require that j®j2 + j¯j2 = 1: (1.3) Geometrically we can interpret the qubit state vector as a unit vector in a two-dimensional complex vector space.