Introduction to Quantum Information Science Lecture Notes Scott Aaronson1 Fall 2018 1With crucial help from: Corey Ostrove and Paulo Alves Contents Page 1 Course Introduction and The Extended Church-Turing Thesis 7 2 Probability Theory and Quantum Mechanics 11 2.1 Linear Algebra Approach to Probability Theory . 15 3 Basic Rules of Quantum Mechanics 19 3.1 Quantum States and The Ket Notation . 19 3.2 Transforming Quantum States . 21 3.3 Quantum Interference . 22 3.3.1 Global and Relative Phase . 23 4 Quantum Gates and Circuits, Quantum Zeno and The Elitzur-Vaidman Bomb 25 4.1 Quantum Gates . 25 4.1.1 Generalized Born Rule . 26 4.1.2 General Properties of Quantum Gates and Measurements 27 4.2 Quantum Circuit Notation . 29 4.3 Quantum Zeno Effect . 32 4.4 The Elitzur-Vaidman Bomb . 32 5 The Coin Problem, Distinguishability, Multi-Qubit States and Entanglement 35 5.1 The Coin Problem . 35 5.2 Distinguishability of Quantum States . 36 5.3 Multi-Qubit States and Operations . 38 5.3.1 Multi-Qubit Operations . 38 5.3.2 Entanglement . 40 2 CONTENTS 3 6 Mixed States 43 6.1 Mixed States . 43 6.1.1 Density Matrices . 44 6.1.2 Properties of Density Matrices . 47 6.1.3 Partial Trace and Reduced Density Matrices . 48 7 The Bloch Sphere, No-Cloning Theorem and Wiesner's Quantum Money Scheme 51 7.1 The Bloch Sphere . 51 7.1.1 Quantum Gates in the Bloch Sphere Representation . 54 7.2 The No-Cloning Theorem . 55 7.3 Quantum Money . 57 7.3.1 Wiesner's Quantum Money Scheme . 57 8 Quantum Money and Quantum Key Distribution 59 8.1 Quantum Money Attacks . 59 8.1.1 Interactive Attacks . 60 8.1.2 Public-Key Quantum Money . 62 8.2 Quantum Key Distribution . 62 9 Superdense Coding 66 9.1 Superdense Coding . 66 10 Teleportation, Entanglement Swapping, GHZ State and The Monogamy of Entanglement 68 10.1 Quantum Teleportation . 68 10.1.1 Multi-Qubit Teleportation and Entanglement Swapping 71 10.1.2 The GHZ State and Monogamy of Entanglement . 73 11 Quantifying Entanglement 76 11.1 Schmidt Decomposition . 77 11.2 Von Neumann Entropy . 77 11.2.1 Entanglement Entropy . 78 11.3 Mixed State Entanglement . 80 12 Interpretations of Quantum Mechanics 83 12.1 The Copenhagen Interpretation . 84 12.1.1 \Shut Up and Calculate" . 84 12.2 Schr¨odinger'sCat and Wigner's Friend . 85 12.3 Dynamical Collapse . 86 12.3.1 Ghirardi-Rimini-Weber (GRW) Theory . 88 4 CONTENTS 12.3.2 Penrose Theory . 88 12.4 The Many-Worlds Interpretation . 91 13 Hidden Variables and Bell's Inequality 97 13.1 Hidden Variable Theories . 97 13.1.1 Bohmian Mechanics . 97 13.1.2 Local Hidden Variable Theories . 99 13.2 The CHSH Game . 100 14 Nonlocal Games 104 14.1 CHSH Game: Quantum Strategy . 104 14.1.1 Analysis of Protocol . 105 14.1.2 CHSH Game: Interpretations and Local Realism . 107 14.1.3 Tsirelson's Inequality . 108 14.1.4 Experimental Tests of Bell's Inequalities . 110 14.2 The Odd Cycle Game . 112 14.3 The Magic Square Game . 114 15 Einstein-Certified Randomness 116 15.1 Guaranteed Random Numbers . 116 15.1.1 Leashing Quantum Systems . 120 16 Quantum Computing and Universal Gate Sets 121 16.1 Complexity of General Unitaries: Counting Argument . 124 16.2 Universal Gate Sets . 126 16.2.1 Classical Universality . 126 16.2.2 Quantum Universality . 128 16.2.3 The Solovay-Kitaev Theorem . 130 17 Quantum Query Complexity and The Deutsch-Josza Problem132 17.1 Quantum Query Complexity . 133 17.2 Quantum Garbage Collection . 135 17.3 Deutsch's Algorithm . 138 17.4 Deutsch-Josza Algorithm . 139 18 Bernstein-Vazirani and Simon's Algorithm 142 18.1 The Bernstein-Vazirani Problem . 142 18.1.1 Quantum Algorithm . 142 18.2 Simon's Problem . 144 18.2.1 Classical Lower Bound . 146 18.2.2 Quantum Algorithm . 147 CONTENTS 5 19 RSA and Shor's Algorithm 151 19.1 RSA Encryption . 151 19.2 Period Finding . 153 19.2.1 Factoring to Period-Finding Reduction . 154 19.2.2 Quantum Algorithm for Period-Finding . 157 20 Quantum Fourier Transform 159 20.1 Quantum Fourier Transform . 159 20.1.1 Implementing the QFT . 162 20.1.2 Period Finding Using the QFT . 165 21 Continued Fractions and Shor's Algorithm Wrap-Up 169 21.1 Continued Fraction Algorithm . 170 21.2 Applications of Shor's Algorithm . 172 21.2.1 Graph Isomorphism . 173 21.2.2 Lattice-Based Cryptography . 173 22 Grover's Algorithm 175 22.1 The Algorithm . 177 22.1.1 Implementing the Diffusion Operator . 180 22.1.2 Geometric Interpretation . 181 22.1.3 Analysis . 182 22.1.4 Multiple Marked Items . 185 23 BBBV Theorem and Applications of Grover's Algorithm 189 23.1 The BBBV Theorem . 189 23.2 Applications of Grover's Algorithm . 195 23.2.1 OR of ANDs . 195 24 More Grover Applications and Quantum Complexity Theory199 24.1 More Applications of Grover's Algorithm . 199 24.1.1 The Collision Problem . 199 24.1.2 Element Distinctness . 201 24.2 Parity Lower Bound . 203 24.3 Quantum Complexity Theory . 204 25 Hamiltonians 207 25.1 Quantum Algorithms for NP-complete Problems . 207 25.2 Hamiltonians . 208 25.2.1 Matrix Exponentiation . 209 25.2.2 Energy . 212 6 CONTENTS 25.2.3 Tensor Products of Hamiltonians . 216 25.2.4 Addition of Hamiltonians . 216 26 The Adiabatic Algorithm 221 26.1 Local Hamiltonians . 221 26.2 The Adiabatic Algorithm . 223 27 Quantum Error Correction 234 27.1 Classical Error Correction . 235 27.1.1 Classical Fault-Tolerance . 237 27.2 Quantum Error Correction . 238 27.2.1 The Shor 9-Qubit Code . 242 27.2.2 Quantum Fault Tolerance . 247 28 The Stabilizer Formalism 249 28.1 The Gottesman-Knill Theorem . 253 28.1.1 The Gottesman-Knill Algorithm . 254 28.2 Stabilizer Codes . 257 28.2.1 Transversal Gates . 259 Lecture 1: Course Introduction and The Extended Church-Turing Thesis I Quantum Information Science is an inherently interdisciplinary field (Physics, CS, Math, Engineering, Philosophy) I It's not just about inventing useful devices and algorithms, but also about clarifying the workings of quantum mechanics. { We use it to ask questions about what you can and can't do with quantum mechanics { It can help us better understand the nature of quantum mechanics itself. I Professor Aaronson is very much on the theoretical end of research. I Theorists inform what experimentalists make, which in turn informs the- orists' queries Today we'll articulate several \self-evident" statements about the physical world. We'll then see that quantum mechanics leaves some of these statements in place, but overturns others|with the distinctions between the statements it upholds and the ones it overturns often extremely subtle! To start with. Probability (P 2 [0; 1]) is the standard way of representing uncertainty in the world. Probabilities have to follow certain axioms such as: I Given a set of n mutually exclusive exhaustive events, the sum of the probabilities satisfies P1 + P2 + ··· + Pn = 1 I The probability of any particular event satisfies Pi ≥ 0 ... 7 8 LECTURE 1. COURSE INTRODUCTION AND THE ECT There's a view that \probabilities are all in our heads." Which is to say that if we knew everything about the uni- verse (let's say position/velocity of all atoms in the solar system) that we could just crunch the equations and see that things either happen or they don't. Let's suppose we have two points separated by a barrier with an open slit, and we want to measure the probability that a particle goes from one point to the other. It seems obviously true that increasing the number of paths (say, by opening another slit) should increase, or at any rate not decrease, the likelihood that it will reach the other end. We refer to this property by saying that probabilities are monotone. Locality is the idea that things can only propagate through the universe at a certain speed. When we update the state of a little patch of space, it should only require knowledge of a small neighborhood around it. Conway's Game Of Life (left) is a good model here: changes you make to the system can affect it, but since each cell only directly interacts with its nearest neighbors the changes only propagate at a certain speed. In physics, locality naturally emerges due to Einstein's Special Theory of Relativity which im- plies that no signal can propagate faster than the (finite) speed of light; this simple princi- ple can explain a large number of physical phe- nomena. In special relativity anything travel- ing faster than the speed of light would be ef- fectively traveling backwards in time, from some observer's standpoint. Local Realism is the principle that any in- stantaneous update in knowledge about faraway events can be explained by correlations of random variables. For example, if you in Austin and a friend in San Francisco both subscribe to the same newspaper then when you read your copy in the morning your knowledge of the headline on your friend-in- San-Francisco's copy instantly collapses to whatever your copy's headline is. Before picking up the copy in the morning your knowledge of the headline for yourself and your friend may have been best described by a probability 9 distribution over various possibilities, but since the outcomes are perfectly correlated, as soon as you learn the headline on your copy you instantly know that your friend's must be the same.