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Quantum Computing.. A Gentle Introduction
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QUANTUM COMPUTING A Gentle Introduction Eleanor Rieffel and Wolfgang Polak ©2011 Massachusetts Institute of Technology The book is concerned with theory: what changes when the classical model underpinning conventional computing is replaced with a quantum one. It contains only a brief discussion of the ongoing efforts to build quantum computers, an active area which is still so young that it is impossible even for experts to predict which approaches will be most successful. While this book is about theory, it is important to ground the discussion of quantum computation in the physics that motivates it. For this reason, the text includes discussions of quantum physics and experiments that illustrate why the theory is defined the way it is. 1 Introduction 1 I QUANTUM BUILDING BLOCKS 7 2 Single-Qubit Quantum Systems 9 2.1 The Quantum Mechanics of Photon Polarization 9 2.1.1 A Simple Experiment 10 2.1.2 A Quantum Explanation 11 2.2 Single Quantum Bits 13 2.3 Single-Qubit Measurement 16 2.4 A Quantum Key Distribution Protocol 18 2.5 The State Space of a Single-Qubit System 21 2.5.1 Relative Phases versus Global Phases 21 2.5.2 Geometric Views of the State Space of a Single Qubit 23 2.5.3 Comments on General Quantum State Spaces 25 2.6 References 25 2.7 Exercises 26 3 Multiple-Qubit Systems 31 3.1 Quantum State Spaces 32 3.1.1 Direct Sums of Vector Spaces 32 3.1.2 Tensor Products of Vector Spaces 33 3.1.3 The State Space of an n-Qubit System 34 3.2 Entangled States 38 3.3 Basics of Multi-Qubit Measurement 41 3.4 Quantum Key Distribution Using Entangled States 43 3.5 References 44 3.6 Exercises 44 4 Measurement of Multiple-Qubit States 47 4.1 Dirac’s Bra/Ket Notation for Linear Transformations 47 4.2 Projection Operators for Measurement 49 4.3 Hermitian Operator Formalism for Measurement 53 4.3.1 The Measurement Postulate 55 4.4 EPR Paradox and Bell’s Theorem 60 4.4.1 Setup for Bell’s Theorem 62 4.4.2 What Quantum Mechanics Predicts 62 4.4.3 Special Case of Bell’s Theorem: What Any Local Hidden Variable Theory Predicts 63 4.4.4 Bell’s Inequality 64 4.5 References 65 4.6 Exercises 66 5 Quantum State Transformations 71 5.1 Unitary Transformations 72 5.1.1 Impossible Transformations: The No-Cloning Principle 73 5.2 Some Simple Quantum Gates 74 5.2.1 The Pauli Transformations 75 5.2.2 The Hadamard Transformation 76 5.2.3 Multiple-Qubit Transformations from Single-Qubit Transformations 76 5.2.4 The Controlled-NOT and Other Singly Controlled Gates 77 5.3 Applications of Simple Gates 80 5.3.1 Dense Coding 81 5.3.2 Quantum Teleportation 82 5.4 Realizing Unitary Transformations as Quantum Circuits 84 5.4.1 Decomposition of Single-Qubit Transformations 84 5.4.2 Singly-Controlled Single-Qubit Transformations 86 5.4.3 Multiply-Controlled Single-Qubit Transformations 87 5.4.4 General Unitary Transformations 89 5.5 A Universally Approximating Set of Gates 91 5.6 The Standard Circuit Model 93 5.7 References 93 5.8 Exercises 94 6 Quantum Versions of Classical Computations 99 6.1 From Reversible Classical Computations to Quantum Computations 99 6.1.1 Reversible and Quantum Versions of Simple Classical Gates 101 6.2 Reversible Implementations of Classical Circuits 103 6.2.1 A Naive Reversible Implementation 103 6.2.2 A General Construction 106 6.3 A Language for Quantum Implementations 110 6.3.1 The Basics 111 6.3.2 Functions 112 6.4 Some Example Programs for Arithmetic Operations 115 6.4.1 Efficient Implementation of AND 115 6.4.2 Efficient Implementation of Multiply-Controlled Single-Qubit Transformations 116 6.4.3 In-Place Addition 117 6.4.4 Modular Addition 117 6.4.5 Modular Multiplication 118 6.4.6 Modular Exponentiation 119 6.5 References 120 6.6 Exercises 121 II QUANTUM ALGORITHMS 123 7 Introduction to Quantum Algorithms 125 7.1 Computing with Superpositions 126 7.1.1 The Walsh-Hadamard Transformation 126 7.1.2 Quantum Parallelism 128 7.2 Notions of Complexity 130 7.2.1 Query Complexity 131 7.2.2 Communication Complexity 132 7.3 A Simple Quantum Algorithm 132 7.3.1 Deutsch’s Problem 133 7.4 Quantum Subroutines 134 7.4.1 The Importance of Unentangling Temporary Qubits in Quantum Subroutines 134 7.4.2 Phase Change for a Subset of Basis Vectors 135 7.4.3 State-Dependent Phase Shifts 138 7.4.4 State-Dependent Single-Qubit Amplitude Shifts 139 7.5 A Few Simple Quantum Algorithms 140 7.5.1 Deutsch-Jozsa Problem 140 7.5.2 Bernstein-Vazirani Problem 141 7.5.3 Simon’s Problem 144 7.5.4 Distributed Computation 145 7.6 Comments on Quantum Parallelism 146 7.7 Machine Models and Complexity Classes 148 7.7.1 Complexity Classes 149 7.7.2 Complexity: Known Results 150 7.8 Quantum Fourier Transformations 153 7.8.1 The Classical Fourier Transform 153 7.8.2 The Quantum Fourier Transform 155 7.8.3 A Quantum Circuit for Fast Fourier Transform 156 7.9 References 158 7.10 Exercises 159 8 Shor’s Algorithm 163 8.1 Classical Reduction to Period-Finding 164 8.2 Shor’s Factoring Algorithm 164 8.2.1 The Quantum Core 165 8.2.2 Classical Extraction of the Period from the Measured Value 166 8.3 Example Illustrating Shor’s Algorithm 167 8.4 The Efficiency of Shor’s Algorithm 169 8.5 Omitting the Internal Measurement 170 8.6 Generalizations 171 8.6.1 The Discrete Logarithm Problem 172 8.6.2 Hidden Subgroup Problems 172 viii Contents 8.7 References 175 8.8 Exercises 176 9 Grover’s Algorithm and Generalizations 177 9.1 Grover’s Algorithm 178 9.1.1 Outline 178 9.1.2 Setup 178 9.1.3 The Iteration Step 180 9.1.4 How Many Iterations? 181 9.2 Amplitude Amplification 183 9.2.1 The Geometry of Amplitude Amplification 185 9.3 Optimality of Grover’s Algorithm 188 9.3.1 Reduction to Three Inequalities 189 9.3.2 Proofs of the Three Inequalities 191 9.4 Derandomization of Grover’s Algorithm and Amplitude Amplification 193 9.4.1 Approach 1: Modifying Each Step 194 9.4.2 Approach 2: Modifying Only the Last Step 194 9.5 Unknown Number of Solutions 196 9.5.1 Varying the Number of Iterations 197 9.5.2 Quantum Counting 198 9.6 Practical Implications of Grover’s Algorithm and Amplitude Amplification 199 9.7 References 200 9.8 Exercises 201 III ENTANGLED SUBSYSTEMS AND ROBUST QUANTUM COMPUTATION 203 10 Quantum Subsystems and Properties of Entangled States 205 10.1 Quantum Subsystems and Mixed States 206 10.1.1 Density Operators 207 10.1.2 Properties of Density Operators 213 10.1.3 The Geometry of Single-Qubit Mixed States 215 10.1.4 Von Neumann Entropy 216 10.2 Classifying Entangled States 218 10.2.1 Bipartite Quantum Systems 218 10.2.2 Classifying Bipartite Pure States up to LOCC Equivalence 222 10.2.3 Quantifying Entanglement in Bipartite Mixed States 224 10.2.4 Multipartite Entanglement 225 10.3 Density Operator Formalism for Measurement 229 10.3.1 Measurement of Density Operators 230 10.4 Transformations of Quantum Subsystems and Decoherence 232 10.4.1 Superoperators 233 10.4.2 Operator Sum Decomposition 234 10.4.3 A Relation Between Quantum State Transformations and Measurements 238 10.4.4 Decoherence 239 10.5 References 240 10.6 Exercises 240 Contents ix 11 Quantum Error Correction 245 11.1 Three Simple Examples of Quantum Error Correcting Codes 246 11.1.1 A Quantum Code That Corrects Single Bit-Flip Errors 246 11.1.2 A Code for Single-Qubit Phase-Flip Errors 251 11.1.3 A Code for All Single-Qubit Errors 252 11.2 Framework for Quantum Error Correcting Codes 253 11.2.1 Classical Error Correcting Codes 254 11.2.2 Quantum Error Correcting Codes 257 11.2.3 Correctable Sets of Errors for Classical Codes 258 11.2.4 Correctable Sets of Errors for Quantum Codes 259 11.2.5 Correcting Errors Using Classical Codes 261 11.2.6 Diagnosing and Correcting Errors Using Quantum Codes 264 11.2.7 Quantum Error Correction across Multiple Blocks 268 11.2.8 Computing on Encoded Quantum States 268 11.2.9 Superpositions and Mixtures of Correctable Errors Are Correctable 269 11.2.10 The Classical Independent Error Model 270 11.2.11 Quantum Independent Error Models 271 11.3 CSS Codes 274 11.3.1 Dual Classical Codes 274 11.3.2 Construction of CSS Codes from Classical Codes Satisfying a Duality Condition 275 11.3.3 The Steane Code 278 11.4 Stabilizer Codes 280 11.4.1 Binary Observables for Quantum Error Correction 280 11.4.2 Pauli Observables for Quantum Error Correction 282 11.4.3 Diagnosing and Correcting Errors 283 11.4.4 Computing on Encoded Stabilizer States 285 11.5 CSS Codes as Stabilizer Codes 289 11.6 References 290 11.7 Exercises 291 12 Fault Tolerance and Robust Quantum Computing 293 12.1 Setting the Stage for Robust Quantum Computation 294 12.2 Fault-Tolerant Computation Using Steane’s Code 297 12.2.1 The Problem with Syndrome Computation 297 12.2.2 Fault-Tolerant Syndrome Extraction and Error Correction 298 12.2.3 Fault-Tolerant Gates for Steane’s Code 300 12.2.4 Fault-Tolerant Measurement 303 12.2.5 Fault-Tolerant State Preparation of |π/4 304 12.3 Robust Quantum Computation 305 12.3.1 Concatenated Coding 306 12.3.2 A Threshold Theorem 308 12.4 References 310 12.5 Exercises 310 13 Further Topics in Quantum Information Processing 311 13.1 Further Quantum Algorithms 311 13.2 Limitations of Quantum Computing 313 x Contents 13.3 Further Techniques for Robust Quantum Computation 314 13.4 Alternatives to the Circuit Model of Quantum Computation 316 13.4.1 Measurement-Based Cluster State Quantum Computation 317 13.4.2 Adiabatic Quantum Computation 318 13.4.3 Holonomic Quantum Computation 319 13.4.4 Topological Quantum Computation 320 13.5 Quantum Protocols 320 13.6 Insight into Classical Computation 321 13.7 Building Quantum Computers 322 13.8 Simulating Quantum Systems 325 13.9 Where Does the Power of Quantum Computation Come From? 326 13.10 What if Quantum Mechanics Is Not Quite Correct? 327 |
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