Principles, Designs and Analysis of Quantum Computing Devices Goong Chen David A. Church Berthold-Georg Englert Carsten Henkel Bernd Rohwedder Marlan O. Scully ii Contents 1 Introduction to Quantum Computation 1 1.1 Introduction. Turing machines and binary logic gates . .... 1 1.2 Quantum mechanical systems . ................ 5 1.3 Hilbert spaces . ........................... 13 1.4 Universality of elementary quantum gates . ............ 18 1.5 Quantum algorithms . ....................... 26 1.5.1 The Deutsch–Jozsa problem ([15]) . ............ 26 1.5.2 Bernstein–Vazirani problem ([6]) . ............ 28 1.5.3 Simon’s problem ([50]) . ................ 30 1.5.4 Grover’s quantum search algorithm ([25]) . ........ 32 1.5.5 Shor’s factorization algorithm ([49]) ............ 34 1.6 Quantum adder and multiplier . ................ 37 1.7 Quantum error correction codes . ................ 41 1.8 Quantum computing devices and requirements . ........ 41 2 Two-Level Atoms and Cavity QED 41 2.1 Two-Level atoms ........................... 41 2.1.1 Atom-light interaction . ................ 41 2.1.2 Reduction to a two-level atom . ............ 44 2.1.3 Single atom qubit rotation . ................ 46 2.1.4 Two-level atom hardware . ................ 48 2.2 Quantization of the electromagnetic field . ............ 51 2.2.1 Normal mode expansion . ................ 51 2.2.2 Field mode quantization . ................ 52 2.2.3 Energy spectrum and stationary states . ........ 53 2.2.4 Cavity hardware ....................... 56 2.3 Cavity QED for the quantum phase gate . ............ 59 2.3.1 Atom-cavity Hamiltonian . ................ 60 2.3.2 Large detuning limit . ................ 62 2.3.3 Two-qubit operation . ................ 64 2.3.4 Atom-cavity hardware . ................ 66 iii iv CONTENTS 3 Imperfect Quantum Operations 71 3.1 Fidelity . ................................ 71 3.2 Density matrices ............................ 72 3.3 Time evolution of density matrices ................. 75 3.3.1 The von Neumann equation ................. 75 3.3.2 Quantum operations . ................. 75 3.3.3 The Kraus representation theorem ............. 76 3.3.4 Quantum Markov processes ................. 78 3.3.5 Non-Markovian environments . ............. 81 3.4 Examples of master equations . ................. 82 3.4.1 Leaky cavity . ........................ 82 3.4.2 Unstable two-level system . ................. 83 3.4.3 Dephasing . ........................ 85 3.5 Fidelity calculations . ........................ 86 3.5.1 Fluctuating gate parameters ................. 86 3.5.2 Spontaneous decay . ................. 88 4 Ion Traps 95 4.1 Introduction . ............................ 95 4.2 Ion confinement, cooling, and condensation . ......... 98 4.2.1 Confinement. Several types of ion traps . ......... 98 4.2.2 Ion cooling and condensation . ............. 105 4.3 Ion qubits . ............................ 111 4.4 Summary of ion preparation . ................. 116 4.5 Coherence . ............................ 118 4.5.1 Coherence of the motional qubit . ............. 119 4.5.2 Coherence of the internal qubit . ............. 122 4.5.3 Coherence in logic operations . ............. 123 4.5.4 Studies of decoherence through coupling to engineered reservoirs . ........................ 126 4.5.5 Summary of ion coherence ................. 127 4.6 Quantum gates ............................ 128 4.6.1 General considerations . ................. 129 4.6.2 Cirac–Zoller CNOT gate . ................. 140 4.6.3 Wave packet or Debye–Waller CNOT gate . 146 4.6.4 Sørensen–Mølmer gate . ................. 147 4.6.5 Geometrical phase gate . ................. 151 4.6.6 Summary of quantum gates ................. 154 4.7 A vision of a large scale confined-ion quantum computer . 156 4.8 Trap architecture and performance ................. 158 4.9 Teleportation of coherent information . ............. 159 4.10 Experimental DFS logic gates . ................. 160 4.11 Quantum error correction by ion traps . ............. 161 4.12 Summary of ion quantum computation . ............. 162 4.12.1 Assessment . ........................ 162 4.12.2 Qubits . ............................ 162 CONTENTS v 4.12.3 Coherence ........................... 164 4.12.4 Gates . ........................... 165 4.12.5 Computation . ....................... 166 4.12.6 Summary ........................... 168 4.12.7 Outlook . ........................... 169 5 Quantum Dots Quantum Computing Gates 175 5.1 Introduction . ........................... 175 5.1.1 QD properties and fabrication: from quantum wells, wires to quantum dots . ................ 176 5.1.2 QD-based single-electron devices and single-photon sources . ................ 181 5.1.3 A simple quantum dot for quantum computing . .... 184 5.1.4 Spintronics ........................... 186 5.1.5 Three major designs of QD-based quantum gates .... 186 5.1.6 Universality of 1-bit and 2-bit gates in quantum computing188 5.2 Electrons in quantum dots microcavity . ............ 190 5.2.1 Resonance, 1-bit and CNOT gates . ............ 192 5.2.2 Decoherence and measurement . ............ 194 5.3 Coupled electron spins in an array of quantum dots . .... 195 5.3.1 Electron spin . ....................... 195 5.3.2 The design due to D. Loss and D. DiVincenzo . .... 197 5.3.3 Model of two identical laterally coupled quantum dots . 199 5.3.4 More details of the QD arrangements: Laterally coupled and vertically coupled arrays ................ 206 5.3.5 Decoherence and measurement . ............ 209 5.3.6 New advances . ....................... 211 5.4 Biexciton in a single quantum dot . ................ 211 5.4.1 Derivation of the unitary rotation matrix and the condi- tional rotation gate . ................ 214 5.4.2 Decoherence and measurement . ............ 218 5.4.3 Proposals for coupling of two or more biexciton QD . 218 5.5 Conclusions . ........................... 219 6 Linear Optics Computers 185 6.1 Classical electrodynamics – Classical computers . ....................... 186 6.1.1 Light beam manipulation with four degrees of freedom . 187 6.1.2 Optical circuits and examples ................ 196 6.1.3 Complexity issues of LOCC and alternatives . .... 200 6.2 Quantum electrodynamics – Quantum computers . ....................... 204 6.2.1 Quantum optical states . ................ 205 6.2.2 Quantum operations and gates . ............ 210 6.2.3 The approach of Knill, Laflamme and Milburn . .... 216 6.2.4 Quantum teleportation . ................ 224 vi CONTENTS 6.2.5 Application of quantum teleportation to LOQC . 228 6.3 Summary and outlook ........................ 237 7 Nuclear Magnetic Resonance (Optional and Tentative) 245 A 247 A.1 The Fock–Darwin States . ................. 247 B 251 B.1 Evaluation of the exchange energy ................. 251 Chapter 1 Introduction to Quantum Computation 1.1 Introduction. Turing machines and binary logic gates The earliest ideas of simulating and utilizing quantum systems to do compu- tation, i.e., quantum computation, can be attributed to P. Benioff ([5, 1980]) and R. Feynman ([23, 1982]). In his 1980 paper, Benioff introduced a quantum Turing machine model. Deutsch ([12, 13]) further developed more concrete proposals and introduced the quantum circuit model of computation. Later, Yao [61] showed that the quantum circuit model of computation is equivalent to the quantum Turing machine model. The foundation of modern computer science is built on the theory of Tur- ing machines. Alan Turing (1912–1954) ([43]) published his ideas of an ab- stract computing machine (now called a “Turing machine” (TM) in 1936, which moved from one state to another using a precise finite set of rules, given by a finite table, and depending on a single symbol it read from a tape ([45]). A TM, at the time Alan Turing invented it in his 1936 paper [54], was a hypothetical computer. It consists of the following: (i) an infinite tape on which symbols may be read and written. (ii) The machine travels right or left along the tape, following a program. (iii) At each step the machine writes to the tape, travels either left or right and changes state, according to a set of internal states. (iii) The set of symbols and the set of internal states are both finite sets. Turing wrote [54]: 1 2 CHAPTER 1. INTRODUCTION TO QUANTUM COMPUTATION “::: Some of the symbols written down will form the sequences of figures which is the decimal of the real number which is being computed. The others are just rough notes to ”assist the memory”. It will only be these rough notes which will be liable to erasure. :::” Further, he established that a universal Turing machine existed [54]: “:::which can be made to do the work of any special-purpose ma- chine, that is to say to carry out any piece of computing, if a tape bearing suitable ”instructions” is inserted into it. :::” The tape referred above in his description of a universal Turing machine was the “computer memory” and the instructions on the tape constituted the “computer program.” We may formalize the notion of a TM as follows. Definition 1.1.1. A (classical) Turing machine (TM) is a 6-tuple (Q; A; δ; q0;qa;qr), where Q = {q1;q2;:::;qm}is a finite set of control states; A = {α1,α2,...,αn}; the alphabet, is a finite set of distinct symbols; q0;qa;qr ∈Qare, respectively, the initial, accepting and rejecting states; and δ : Q × A −→ A × Q ×{L; R}; is a transition function mapping from each “square” (q; α) ∈ Q×A on a “tape” to (q0,α0;L)or (q0,α0;R), left (L) or right (R) of that square on the tape. A simple TM satisfying the above definition is exemplified below.