Quantum 101

Table of Contents Introduction What Is Superdense Coding? Classical Communication Limits Quantum Advantage in Communication Ingredients Required for Superdense Coding The Entangled Resource: Bell States Step-by-Step Protocol Mathematical Derivation Encoding Operations by Alice Bell...

Table of Contents Introduction What Is Quantum Teleportation? Misconceptions About Teleportation Why Teleportation Matters Basic Ingredients of Quantum Teleportation The Quantum Circuit Step-by-Step Protocol The Role of Entanglement Mathematical Derivation Measurement and Classical Communication Reconstruction...

Table of Contents Introduction What is Quantum Entanglement? Historical Context and EPR Paradox Entanglement vs Classical Correlations Mathematical Representation of Entanglement Bell States and Maximally Entangled States Schmidt Decomposition Criteria for Entanglement Local...

Table of Contents Introduction What is Decoherence? Decoherence vs Relaxation Mathematical Description of Decoherence Open Quantum Systems Density Matrix Formalism The Lindblad Master Equation Loss of Coherence in Superposition States Environmental Coupling Sources of...

Table of Contents Introduction What Are Quantum Channels? Why Study Noise in Quantum Systems? Classical vs Quantum Channels Mathematical Framework: CPTP Maps Kraus Operators Stinespring Dilation Theorem Types of Quantum Noise Depolarizing Channel Bit-Flip...

Table of Contents Introduction Quantum Noise and Fragility What Is Fault Tolerance? Fault Tolerance vs Error Correction Motivation for Fault-Tolerant Quantum Computing Types of Faults in Quantum Circuits Quantum Circuit Fault...

Table of Contents Introduction Why Quantum Error Correction (QEC) is Needed Quantum Errors vs Classical Errors Principles of Quantum Error Correction Qubit Redundancy and Codewords Quantum Error Models The Three-Qubit Bit-Flip...

Table of Contents Introduction Motivation for Steane Code Classical Foundations: Hamming Code Quantum Extension of Hamming Code What is the Steane Code? Logical Qubit Encoding Correcting Arbitrary Single-Qubit Errors Codewords of the...

Table of Contents Introduction Why We Need the Shor Code Basics of Quantum Error Correction The Concept Behind Shor Code What Errors Can Shor Code Correct? Encoding in Shor Code Step-by-Step...

Table of Contents Introduction Quantum Errors and the Need for Error Correction Overview of Quantum Error Correcting Codes The Bit-Flip Code: Concept Bit-Flip Code Encoding Bit-Flip Code Detection and Correction Bit-Flip...

Table of Contents Introduction Why Error Models Are Crucial Types of Quantum Errors Bit Flip Error Phase Flip Error Bit-Phase Flip Error Amplitude Damping Phase Damping (Dephasing) Depolarizing Channel General Quantum Noise Channels Kraus Operators...

Table of Contents Introduction Motivation and Context Classical Copying vs Quantum Copying Statement of the No-Cloning Theorem Mathematical Proof of the No-Cloning Theorem Implications of the Theorem Linearity and Unitarity in...