NOTE TO USERS This reproduction is the best copy available. Techniques for Quantum Computing State Generation, Discrete Logarithms in Elliptic Curve Groups, Reliable Global Control Schemes and Algorithmic Cooling by Phillip R. Kaye A thesis presented to the University of Waterloo in ful¯llment of the thesis requirement for the degree of Doctor of Philosophy in Computer Science Waterloo, Ontario, Canada, 2007 °c Phillip R. Kaye, 2007 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required ¯nal revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. Phillip R. Kaye ii Abstract This thesis is about techniques for quantum computing. A common theme throughout this work is the examination of how quantum algorithms and protocols might be imple- mented in practice. I explore this question at the level of algorithmic details and computer architecture, and not at the level of speci¯c physical systems for performing quantum computation. The ¯rst problem I consider is the generation of quantum states. Many results in quantum information theory require the generation of speci¯c quantum states, such as Bell states. Some states can be e±ciently created using standard quantum computational primitives such as preparing a qubit in the state j0i and applying a sequence of quantum gates (from a ¯nite set). For example, a Bell state can be prepared from the state j0ij0i using a Hadamard gate and a controlled-not gate. However, many states cannot be e±ciently created. Chapter 1 of this thesis focusses on the generation of quantum states. In Chapter 2, I explore implementations of Shor's quantum algorithm for computing dis- crete logarithms. This algorithm is particularly signi¯cant because it threatens to un- dermine the security of widely used elliptic curve cryptosystems. I give a strategy for implementing Shor's algorithm for ¯nding discrete logarithms in groups of points on ellip- tic curves over ¯elds of characteristic 2. Chapter 3 is about globally controlled arrays, which is a paradigm for implementing quan- tum computers that may prove to be more feasible in practice than the quantum circuit model. I explore strategies for implementing error correction in such global control models, so that they might be implemented more robustly. I also cast the various global control schemes that have appeared in the literature into a uni¯ed framework so that their prop- erties can be studied somewhat independently of the di®erences in low-level details. Using this framework, I consider the main challenges and obstacles to implementing quantum computing fault tolerantly using globally controlled arrays. Finally, in Chapter 4, I consider algorithmic cooling|a technique that is potentially impor- tant for making quantum computation using nuclear magnetic resonance (NMR) feasible. Given the constraints imposed by the NMR approach to quantum computing, the most likely cooling algorithms to be practicable are those based on simple reversible polarization iii (RPC) operations acting locally on small numbers of bits. Algorithms using 2- and 3-bit RPC operations have appeared in the literature, and these are the algorithms I consider in Chapter 4. Speci¯cally, I show that the RPC operation used in all these algorithms is essentially a majority-vote of 3 bits, and prove the optimality of the best such algorithm (in a restricted setting). I go on to derive some theoretical bounds on the performance of these algorithms under some speci¯c assumptions about errors. These bounds are independent of implementation details and low-level algorithmic details. iv Acknowledgements I am indebted to my supervisor, Professor Michele Mosca, for his guidance, support and interest in my work. I would also like to thank Prof. Raymond Laflamme, Christof Zalka, Donny Cheung, Carlos Perez, Alastair Kay, Mark Saaltink and Lawrence Ioannou for the many useful conversations that have been important to the development of the work presented in this thesis. I also wish to thank my wife Janine for her love and support during my years as a graduate student. The work presented in this thesis has been supported by MITACS (Mathematics of In- formation Technology and Complex Systems), NSERC (National Science and Engineering Research Council), CSE (Communications Security Establishment), CFI (Canadian Foun- dation for Innovation), ORDCF (Ontario Research and Development Challenge Fund), and PREA (Premier's Research Excellence Awards). v Contents Preface 1 1 Quantum circuits for generating quantum states 4 1.1 Background ................................... 4 1.2 Generating the phase factors .......................... 5 1.3 Generating the state with real nonnegative amplitudes ........... 7 1.3.1 The algorithm .............................. 7 ª 1.3.2 Implementing the Uj .......................... 9 1.4 An example: symmetric states ......................... 11 1.5 Precision ..................................... 12 1.5.1 Precision in the generation of jª^ i ................... 12 1.5.2 Precision in the generation of phases . 16 1.6 Conclusions ................................... 17 2 Discrete logarithms for elliptic curve groups 18 2.1 Background ................................... 18 2.1.1 Shor's algorithm ............................ 18 2.1.2 Circuits for modular arithmetic .................... 19 2.1.3 The elliptic curve group operation ................... 20 ix 2.2 Elliptic curves over GF(2m) .......................... 21 2.3 Representations of the group elements .................... 23 2.4 The discrete-logarithm problem ........................ 24 2.5 Decomposing the group operation ....................... 25 2.6 The extended Euclidean algorithm for polynomials . 27 2.7 Naive implementation of the extended EEA . 31 2.7.1 Implementing some tools ........................ 32 2.7.2 Long division .............................. 37 2.8 The problem of synchronization ........................ 38 2.9 An optimized implementation ......................... 40 2.9.1 The implementation .......................... 40 2.9.2 Space complexity ............................ 46 2.10 Conclusions and future work .......................... 47 3 Globally controlled quantum arrays 48 3.1 Background ................................... 48 3.1.1 Quantum cellular automata and globally controlled arrays . 48 3.1.2 Between quantum circuits and simple spin chains . 50 3.2 The basic gca model .............................. 51 3.2.1 The language SPA ............................ 52 3.2.2 Implementations of SPA for some example architectures . 55 3.2.3 Implementation on lattices with a distinguished site . 66 3.2.4 SPA programs to simulate quantum circuits . 68 3.2.5 gca, qca, and error correction .................... 72 3.3 Two approaches to error correction for gca . 74 3.4 Implementation-level error correction ..................... 76 x 3.4.1 Dissipative pulses|removing unwanted entropy . 76 3.4.2 A bit-flip code for a gca memory ................... 78 3.4.3 A 9-qubit code for a gca memory ................... 81 3.4.4 A 1-dimensional implementation of the 9-qubit code . 86 3.4.5 Scaling the 9-qubit code ........................ 86 3.4.6 From a robust gca memory to a robust implementation of SPA . 90 3.4.7 A general construction for implementation-level codes on lattices . 94 3.5 gca models with parallelism .......................... 95 3.5.1 The language MPA(k) .......................... 95 3.5.2 An approach to implementing MPA(k) . 99 3.6 Data-level error correction . 103 3.6.1 The general approach . 103 3.6.2 Example: the Steane code . 104 3.6.3 Code-concatenation requires more sophisticated parallelism . 106 3.7 Unresolved problems .............................. 106 3.8 Conclusions and future work . 108 4 Cooling algorithms based on the 3-bit majority 110 4.1 Background ................................... 110 4.2 Architecture ................................... 112 4.3 The reversible polarization compression step . 114 4.3.1 The 2-bit RPC step . 115 4.3.2 The 3-bit RPC step . 116 4.3.3 Equivalence between the 2BC and 3BC operations . 118 4.4 E±ciency .................................... 119 4.4.1 The simple recursive algorithm . 119 xi 4.4.2 Algorithms using a heat bath . 120 4.4.3 Accounting for the heat bath as a computational resource . 124 4.5 Accounting for errors in an analysis of cooling . 125 4.6 The symmetric bit-flip channel . 126 4.6.1 3BC followed by a symmetric bit-flip error . 127 4.6.2 Symmetric bit-flip errors during application of 3BC . 128 4.7 Debiasing errors ................................. 130 4.7.1 3BC followed by a debiasing error . 132 4.7.2 Debiasing errors during application of 3BC . 134 4.8 More general algorithms based on 3BC . 135 4.9 Conclusions and other considerations . 136 A Proofs of correctness for sequences in Section 3.2.2.2 138 B Implementing switching stations for the architecture of Section 3.2.2.2 143 xii List of Figures 1.1 A circuit to generate jª^ i ............................ 8 1.2 A circuit to generate jªi ............................ 9 ª 1.3 A circuit implementing Uj ........................... 10 1.4 A circuit for computing the Hamming weight . 12 2.1 A circuit to compute the degree of A 2 GF (2m) . 33 2.2 A circuit to compute jki $ jk + 1i ...................... 34 2.3 The quantum swap gate ............................ 36 2.4 A cyclic left shift gate ............................. 36 2.5 A circuit for jθijsi $ jθ ¿ sijsi ....................... 37 2.6 Desynchronization example ........................... 39 2.7 The positions of A; B; a; b for register sharing . 43 2.8 Example of long division by hand ....................... 44 2.9 Example of optimized implementation of long division . 45 3.1 A fragment of a circuit to be simulated by SPA . 69 3.2 A circuit equivalent to Figure 3.1 ....................... 69 3.3 The circuit in Figure 3.2 rewritten with no gates acting in parallel . 70 3.4 A nearest-neighbour version of the circuit shown in Figure 3.3 . 70 3.5 A construction for implementing a distance-4 cnot gate . 70 xiii 3.6 Some optimizations applied to the circuit constructed in Figure 3.4 .