Tag: Quantum Computing

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...

Table of Contents Introduction Measurement in Quantum Mechanics Quantum Measurement Postulates Projective Measurement Measurement in the Computational Basis Quantum Measurement and Collapse Measurement Probability and Born Rule Measurement Circuits in Quantum Computing Non-Destructive...

Table of Contents Introduction Classical Random Walks Overview Motivation for Quantum Walks What Are Quantum Walks? Discrete-Time vs Continuous-Time Quantum Walks Mathematical Framework Coined Quantum Walks Quantum Coin Operators Shift (Step) Operators Unitary Evolution...

Table of Contents Introduction Motivation and Background What Is Amplitude Amplification? Relationship to Grover's Algorithm General Form of Amplitude Amplification Oracle and Reflection Operators Mathematical Foundation Amplitude Amplification Operator The Grover Iterate as...

Table of Contents Introduction Motivation and Applications Problem Statement Mathematical Background Quantum Phase Estimation: Goal Overview of the Algorithm Unitary Operators and Eigenstates Phase Encoding via Controlled Operations Quantum Fourier Transform in Phase...

Table of Contents Introduction Classical Fourier Transform Background Motivation for Quantum Fourier Transform Definition of QFT Mathematical Formulation Action on Quantum Basis States Inverse Quantum Fourier Transform Comparison to Classical FFT Circuit Construction Hadamard...

Table of Contents Introduction Background: Integer Factorization Problem Classical Complexity Quantum Advantage and Shor’s Breakthrough Problem Statement High-Level Overview of Shor's Algorithm Mathematical Foundations Role of Modular Arithmetic Reduction to Order Finding Quantum Order...

Table of Contents Introduction What Is a Quantum Oracle? Role in Quantum Algorithms Oracle as a Black Box Mathematical Definition Oracle for Boolean Functions Oracle Types: Phase and Bit Flip Oracle in...

Table of Contents Introduction The Unstructured Search Problem Classical Complexity Quantum Speedup with Grover’s Algorithm Problem Statement Oracle Design in Grover’s Algorithm Amplitude Amplification Intuition High-Level Steps of the Algorithm Initial State Preparation Applying...

Table of Contents Introduction Problem Overview Classical vs Quantum Approach Mathematical Formulation The Oracle Promise Simon's Problem Statement High-Level Idea of Simon’s Algorithm Oracle Structure and Functionality Input and Output Qubit Registers Quantum Circuit...