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Lecture Notes on Quantum Information and Computation

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Lecture Notes on Quantum Information and Computation Draft for Internal Circulations: v1: Fall Semester, 2012, v2: Fall Semester, 2013 v3: Fall Semester, 2014, v4: Fall Semester, 2015 Lecture Notes on Quantum Information and Computation Yong Zhang1 School of Physics and Technology, Wuhan University (Fall 2015) Abstract These lectures notes are written for both advanced undergraduate students and first-year graduate students in the School of Physics and Technology, University Wuhan. They are mainly based on both online lecture notes of John Preskill from Caltech and the standard textbook of Michael Nielsen and Issac Chuang, so I do not claim any originality. These notes certainly have all kinds of typos or errors, so they will be updated from time to time. I do take the full responsibility for all kinds of typos or errors (besides errors in English writing), and please let me know of them. * The second version of these notes are typeset by Yi Peng2 Undergraduate student (Id: 2010301020027), Participant in the Fall semester, 2013. * The third version of these notes are typeset by Kun Zhang3 Graduate student (Id: 2013202020002), Participant in the Fall semester, 2012-2013-2014. 1 yong [email protected] 2 pengyi [email protected] 3 kun [email protected] Draft for Internal Circulations: v1: Fall Semester, 2012, v2: Fall Semester, 2013 v3: Fall Semester, 2014, v4: Fall Semester, 2015 Acknowledgements I thank all participants in class including advanced undergraduate students, first-year graduate students and French students for their patience and persis- tence and for their various enlightening questions. I especially thank students who are willing to devote their precious time to the typewriting of these lecture notes in Latex. Main References to Lecture Notes * [Preskill] John Preskill (Caltech): online lecture notes on QIC (1997-present), http://www.theory.caltech.edu/ preskill/ph229/ http://www.theory.caltech.edu/ preskill/ph229/ lecture ∼ * [Nilesen & Chuang] Michael A.∼ Nielsen and Isaac♯ L. Chuang, Quantum Computation and Quantum Information (Cambridge, 2000&2010) * [KLM] Phillip Kaye, Raymond Laflamme and Michele Mosca, An Introduction to Quantum Computing (Oxford, 2007). * [Zhang] Yong-De Zhang (University of Science & Technology of China), Principles of Quantum Information Physics (in Chinese) (Science Press, 2009). Main References to Homeworks * [Preskill] John Preskill (Caltech): online lecture notes on QIC (1997-present), http://www.theory.caltech.edu/ preskill/ph229/ lecture Research∼ Projects♯ * See Yong Zhang's English and Chinese homepages. 2 To our parents and our teachers! To be the best researcher is to be the best person first of all: Respect and listen to our parents and our teachers always! 3 This course focuses on fundamental principles of quantum mechanics. The aim of this course is to study the simulation of both quantum field theory and quantum gravity on quantum computer. 4 Contents I Introduction to Quantum Information and Computation 10 1 Overview 1 1.1 Reasons to learn Quantum Information and Computation . .1 1.2 What's Quantum Information and Computation? . .1 1.3 Research topics . .2 2 Thermodynamics and Statistical Mechanics 4 3 Quantum Mechanics (I): Axioms 5 3.1 Axioms of quantum mechanics for closed system . .5 3.1.1 State . .5 3.1.2 Observable . .6 3.1.3 Projective measurement . .7 3.1.4 Schr¨odinger equation . .8 3.1.5 Composite system . .9 4 Qubit, Quantum Gates and Bell States 10 4.1 Overview . 10 4.2 Pure state formalism of a qubit . 11 4.2.1 Single-qubit gate SU 2 in the spin-1 2 case . 12 4.2.1.1 Spin-1 2 operator and Pauli matrices . 12 4.2.1.2 Spinor representation( ) of SU ~2 group . 12 4.2.2 Properties . .~ . 13 4.2.3 Physical realization of qubit . .( .) . 16 4.3 Bell states . 16 4.3.1 Notation . 17 4.3.2 Parity-bit (i) and Phase-bit (j)...................... 21 4.3.3 Orthonormal basis of the two-qubit Hilbert space . 22 4.3.4 How to distinguish (create) i; j .................... 24 4.4 Quantum circuit model of Bell states . 25 4.4.1 Quantum circuit model . .S .( . .)⟩ . 25 4.4.2 Hadamard gate: H ............................. 25 4.4.3 CNOT gate . 28 4.4.4 Quantum circuit model of Bell states . 32 5 No-Cloning, Dense Coding, Teleportation and Cryptography 35 5.1 No-cloning theorem . 35 5.2 Dense coding . 38 5.3 Quantum teleportation . 40 5 5.4 The quantum teleportation using continuous variables . 44 5.5 Quantum cryptography (information security) . 47 5.5.1 Classical cryptography . 47 5.5.2 Quantum key distribution (QKD) . 48 5.5.3 BB84 quantum key distribution . 49 6 Bell Inequalities 51 6.1 Einstein's quantum mechanics: local hidden variable theory (LHV) . 51 6.1.1 What hidden variable (HV) theory? . 51 6.1.2 What local theory? . 52 6.1.3 The rule to justify the rightful theory . 52 6.2 Bell's inequality in the local hidden variable theory . 52 6.3 Bell's inequality in quantum mechanics . 54 6.4 The CHSH inequality . 58 6.5 Hints for the violation of the Bell inequality . 60 6.6 Hardy's theorem . 60 6.7 The GHZ theorem . 65 II Quantum Computing and Quantum Algorithm 67 7 Classical Circuit Model 68 7.1 Classical circuit . 68 7.1.1 Elementary logical gates . 69 7.1.2 Universal gate set . 70 7.2 Reversible classical computation . 70 7.2.1 Irreversible computation . 70 7.2.2 Classical reversible gate . 71 7.2.3 Three-bit Toffoli gate . 72 7.2.4 Three-bit Fredkin gate . 73 7.3 The construction of an n-bit Toffoli gate using the 3-bit Toffoli gate . 74 8 Quantum Circuit Model 77 8.1 Definition of quantum circuit . 77 8.1.1 One-qubit gates . 77 8.1.2 Controlled two-qubit gates and controlled three-qubit gates . 81 8.1.2.1 Quantum Toffoli gate and Fredkin gate . 82 8.1.3 Quantum circuit model of GHZ states . 82 8.2 Universal quantum computation . 86 8.2.1 Quantum universal gate set . 86 8.2.2 Universal quantum gate set of two-qubit gates . 87 8.2.3 Deutsch's gate is a universal quantum gate . 90 9 Physical Realization of Quantum Computers 92 10 Quantum Algorithms 93 10.1 Classical and quantum algorithm . 93 10.2 Oracle model . 94 10.3 Deutsch's algorithm . 95 10.3.1 Definitions . 95 6 10.3.2 Classical algorithm . 96 10.3.3 Deutsch's problem . 96 10.3.4 Phase kick-back . 96 10.3.5 Deutsch's algorithm . 98 10.4 Deutsch-Jozsa's algorithm . 99 10.4.1 Constant and balanced function in n-qubit . 99 10.4.2 Constant or balanced function? . 99 10.4.3 Notation and lemma . 100 10.4.4 Deutsch-Jozsa's algorithm . 101 10.5 Bernstein-Vazirani's algorithm . 103 10.6 Simon's algorithm . 104 10.7 Grover's algorithm . 104 10.7.1 Overview of the problem . 105 10.7.2 Grover's algorithm . 105 10.7.3 Example: N 4 ............................... 108 10.7.4 Quantum circuit model of Grover's algorithm . 110 = 11 Quantum Circuit Complexity 113 11.1 Circuit complexity . 113 11.1.1 Definitions . 113 11.1.2 Complexity class . 114 11.1.3 Quantum complexity . 114 11.1.4 Accuracy . 115 12 Quantum Simulation 116 III Density Matrix and Quantum Entanglement 117 13 Quantum Mechanics (II): Density Matrix 118 13.1 Density matrix as state of quantum open system . 118 13.1.1 State ensemble formalism of density matrix . 119 13.1.2 Operator formalism of density matrix . 121 13.1.3 Reduced density matrix (State for subsystem) . 122 13.2 Mixed state formalism of a qubit . 124 13.2.1 Why polarization vector? . 125 13.2.2 Pure state and mixed state in two-dimensional Hilbert space H2 .. 126 13.3 Convexity of density matrix . 128 13.4 Two-qubit system and its subsystem . 129 13.4.1 Example: EPR pair (Bell state) . 129 13.4.2 Maximally entangled two-qubit pure states . 130 13.4.3 Monogamy of maximal entanglement . 131 14 Schmidt Decomposition, Purification and GHJW Theorem 133 14.1 Introduction . 133 14.2 Schmidt decomposition and quantum entanglement . 133 14.2.1 Schmidt decomposition . 133 14.2.2 Quantum entanglement . 134 14.2.3 Proof for the theorem of the Schmidt decomposition . 135 14.3 Example for the Schmidt decomposition . ..
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