The Amazing World of Quantum Computing (Undergraduate Lecture Notes in Physics)

✍ Scribed by Rajendra K. Bera

• Tongue: English
• Leaves: 278
• Category: Library

✦ Table of Contents

Preface
Our World Consists of Both Real and Imagined Things
Acknowledgements
My Expectations from the Reader
Contents
About the Author
1 Quantum Cryptography and Quantum Teleportation
1.1 Introduction
1.2 Hello to Some Weirdness in Quantum Mechanics
1.3 Time for Some Mathematics
1.3.1 Quantum Operators that Act on a Qubit
1.3.2 A Quantum Operator that Acts on a Qubit Pair
1.4 Encryption and Key Distribution
1.5 Teleportation
1.6 Concluding Remarks
References
2 Distinguishing Features and Axioms of Quantum Mechanics
2.1 Introduction
2.2 Two-Layer Description of the World
2.2.1 The Observer in Physics
2.2.2 Complementarity (Wave-Particle Duality)
2.2.3 Causality and Determinism
2.3 Superposition, Measurement, and Entanglement
2.4 Classical Mechanics Powers Our Intuition
2.5 The Birth of Modern Quantum Mechanics
2.5.1 Serendipity at Work
2.6 Cautionary Note on Notations in Quantum Mechanics
2.7 Postulates of Quantum Mechanics Formally Stated
2.7.1 A Quantum System’s State Space Is a Hilbert Space
2.7.2 A Quantum System Evolves via Unitary Transformations
2.7.3 A Quantum System Collapses When Measured
2.7.4 Hilbert Space Grows Rapidly with the Size of a Quantum System
2.7.5 Born’s Probabilistic Interpretation
2.7.6 Heisenberg’s Uncertainty Principle
2.8 Observables and Operators
2.8.1 Observables in Quantum Mechanics Are Operators
2.8.2 The Need for Observable-Operators
2.8.3 Remarks on Vector Spaces
2.9 Weirdness of Quantum Mechanics (In Summary)
2.10 Interpretations of Quantum Mechanics
2.10.1 Copenhagen Interpretation
2.10.2 Everett’s Many-World Interpretation
2.10.3 Bohm’s Interpretation
2.11 From Galileo–Newton to Schrödinger–Born
2.12 Concluding Remarks
References
3 Mathematical Elements Needed to Compute
3.1 Introduction
3.1.1 Propositional Calculus (Propositional Logic)
3.1.2 First-Order Predicate Calculus (First Order Logic)
3.2 Elements of Linear Algebra
3.2.1 Various Representations of a State Vector
3.2.2 Bases and Linear Independence
3.3 Linear Operators and Matrices
3.3.1 Inner Product
3.3.2 Outer Product
3.3.3 Tensor Product
3.4 Eigenvalue, Eigenvector, Spectral Decomposition, Trace
3.4.1 Eigenvalues and Eigenvectors
3.4.2 Diagonal Representation of an Operator or Orthonormal Decomposition
3.4.3 Normal Operators and Spectral Decomposition
3.4.4 Unitary Operators
3.4.5 Positive Operator
3.4.6 Trace of a Matrix
3.4.7 Commutator and Anti-Commutator
3.4.8 Polar and Singular Value Decompositions
3.4.9 Completeness Relation
3.5 Cauchy–Schwarz Inequality
3.6 Pauli Matrices
3.7 Concluding Remarks
References
4 Some Mathematical Consequences of the Postulates
4.1 Introduction
4.2 No-Cloning Theorem
4.2.1 Consequences of the No-Cloning Theorem
4.3 No-Deleting Theorem
4.4 No-Hiding Theorem
4.5 EPR Paradox and Bell Inequalities
4.5.1 An Analogy for Factorizable States
4.5.2 Einstein, Podolsky, Rosen Pose a Paradox
4.5.3 What Does Hidden Variable Theory Mean?
4.5.4 Bell Inequality
4.5.5 An Intriguing Question
4.5.6 Returning to the Bell Inequality
4.5.7 Would Newton Have Approved of Entanglement?
4.6 Superposition and Indeterminacy
4.7 Mathematical Consequences
4.8 Concluding Remarks
References
5 Waves and Fourier Analyses
5.1 Introduction
5.2 Waves
5.2.1 The Wave Equation
5.2.2 Travelling Waves
5.2.3 Standing or Stationary Waves
5.2.4 Wave Packets
5.2.5 Probability Waves
5.3 Fourier Analysis
5.4 Wave Packets in Some Detail
5.4.1 Group and Phase Velocities
5.5 Concluding Remarks
References
6 Getting a Hang of Measurement
6.1 Introduction
6.2 Measurement of Quantum Systems
6.2.1 Cascaded Measurements Are Single Measurements
6.2.2 Projective Measurements; Observable-Operators
6.2.3 Distinguishing Quantum States
6.2.4 When Measurement Basis States Differ from Computational Basis States
6.2.5 Positive Operator-Valued Measure (POVM) Measurements
6.2.6 The Effect of Phase on Measurement
6.2.7 Can Every Observable Be Measured?
6.2.8 Measurement with Photons and Electrons
6.2.9 Whither Causality?
6.3 Heisenberg’s Uncertainty Principle (Revisited)
6.4 Concluding Remarks
References
7 Quantum Gates
7.1 Introduction
7.2 Operators (A Summary)
7.3 The Qubit
7.3.1 Global Phase Factor
7.3.2 Relative Phase Factor
7.3.3 Unitary Operators
7.3.4 Hermitian Operators
7.4 Important Qubit Gates
7.4.1 Pauli Gates and Other 1-Qubit Gates
7.4.2 2-Qubit Controlled-not Gate
7.4.3 Creating Entangled Bell States
7.4.4 Bit Copying—An Application of the Controlled-not Gate
7.4.5 3-Qubit Toffoli Gate
7.4.6 3-Bit Fredkin Gate
7.4.7 Controlled-U Gate
7.5 Universal Set of Gates
7.5.1 Universal Set of Classical Gates
7.5.2 Universal Set of Quantum Gates
7.6 Some Basic Quantum Operations
7.6.1 Random Number Generation
7.6.2 n-Qubit Hadamard Gate
7.6.3 A 3-Qubit Gate for AND and NOT Operations
7.7 Taking Stock of Gates
7.8 Concluding Remarks
References
8 Unusual Solutions of Usual Problems
8.1 Introduction
8.1.1 Mach–Zehnder Interferometer
8.2 Some Simple Quantum Algorithms
8.2.1 Computing x ∧ y
8.2.2 Computing x + y
8.2.3 Swapping States
8.2.4 The Deutsch Algorithm
8.2.5 The Deutsch–Jozsa Algorithm
8.2.6 Computing f(x) in Parallel
8.2.7 Hardy’s Reprieve
8.2.8 The Elitzur–Vaidman Bomb Problem
8.2.9 Securing Banknotes
8.3 Concluding Remarks
References
9 Fundamental Limits to Computing
9.1 Introduction
9.2 Hilbert’s Second Problem
9.2.1 Recursive Set
9.3 Hilbert’s Tenth Problem
9.4 Turing and the Entscheidungsproblem
9.4.1 Turing’s Halting Problem
9.4.2 The Church–Turing Thesis
9.4.3 Deutsch on the Church–Turing Thesis
9.4.4 Can Quantum Computers Prove Theorems?
9.5 Thermodynamic Considerations
9.5.1 The One-Molecule Gas
9.5.2 Knowledge and Entropy
9.5.3 Information Is Physical
9.5.4 Toffoli Gate
9.5.5 Bennett’s Solution for Junk Bits
9.5.6 Reversible Classical Computation Set the Stage for Quantum Computing
9.5.7 Maxwell’s Demon
9.6 Computational Complexity
9.6.1 Classification of Complexity
9.6.2 NP-Complete Problems Stand or Fall Together
9.7 Concluding Remarks
References
10 The Crown Jewels of Quantum Algorithms
10.1 Introduction
10.2 General Remarks on Quantum Algorithms
10.3 Modulo Arithmetic
10.3.1 Some Important Properties of Congruence
10.3.2 Congruence Classes
10.3.3 Modulo 2 Arithmetic
10.4 Bits and Qubits
10.4.1 Bitwise Operators
10.4.2 String Manipulation Leads to Algorithms
10.5 UTM, DTM, PTM, and QTM
10.5.1 Are Quantum Computers More Powerful?
10.6 The Quantum Fourier Transform
10.6.1 Background
10.6.2 Quantum Fourier Transform
10.7 Computing the Period of a Sequence
10.8 Shor’s Factoring Algorithm
10.8.1 Shor’s Algorithm Implemented
10.8.2 Computational Complexity of Shor’s Algorithm
10.9 Phase Estimation Problem
10.10 Grover’s Search Algorithm
10.10.1 Grover’s Algorithm Verified
10.10.2 Computational Complexity of Grover’s Algorithm
10.10.3 Remarks on Grover’s Algorithm
10.11 Dense Coding and Teleportation
10.11.1 Dense Coding
10.11.2 Teleportation
10.12 Concluding Remarks
References
11 Quantum Error Corrections
11.1 Introduction
11.2 Protecting the Computational Hilbert Space
11.2.1 Dissipation
11.2.2 Decoherence
11.2.3 Algorithmic Error Correction Is Possible
11.3 Calderbank–Shor–Steane Error Correction
11.3.1 Encoding-Decoding
11.3.2 Steps of Error Correction
11.4 Decoherence-Free Subspace
11.5 Concluding Remarks
References
12 Time-Multiplexed Interpretation of Measurement
12.1 Introduction
12.2 A Conjectured Sub-planck Mechanism
12.3 Application of the Basic Model
12.3.1 Measurement of a Two-Particle Entangled System
12.3.2 Quantum Adder
12.4 Teleporting a Qubit of an Unknown State
12.5 Concluding Remarks
References
Index