Quantum Observables Simulation

Basic Quantum Theory, Observables and Measurements Workshop

Author: Andersson Programming

Course: Quantum Computing and Information Theory (CNYT)
Institution: Escuela Colombiana de Ingeniería Julio Garavito
Repository: https://github.com/AnderssonProgramming/quantum-observables-simulation

Project Overview

This project implements a comprehensive quantum system simulator based on fundamental quantum mechanics principles from Chapter 4. The simulator addresses quantum observables, measurements, and system dynamics through practical implementations of theoretical concepts.

Key Features

Section 4.1 Basic Quantum System

• Discrete Position Modeling: Particle confined to discrete positions on a line
• Position Probability Calculation: Find probability of particle at specific position
• State Transition Probability: Calculate transition probability between quantum states

Chapter 4 Programming Challenges

• Transition Amplitude Calculation: Compute amplitude and probability between two state vectors
• Observable Analysis: Hermitian verification, mean value, and variance calculations
• Eigenvalue Probabilities: Calculate probabilities of transitioning to eigenvector states
• System Dynamics: Evolution through series of unitary transformations

Specific Exercise Solutions

• Exercise 4.3.1: Post-measurement state transitions and eigenvector analysis
• Exercise 4.3.2: Probability calculations and statistical distributions
• Exercise 4.4.1: Unitary matrix verification and composition properties
• Exercise 4.4.2: Quantum system evolution over multiple time steps
• Exercise 4.5.2: Multi-particle state vector generalization
• Exercise 4.5.3: Quantum state separability analysis

Getting Started

Prerequisites

# Required Python packages
numpy>=1.20.0
scipy>=1.7.0
matplotlib>=3.4.0
jupyter>=1.0.0
sympy>=1.8.0

Installation

1. Clone the repository:

git clone https://github.com/AnderssonProgramming/quantum-observables-simulation.git
cd quantum-observables-simulation

2. Install dependencies:

pip install -r requirements.txt

3. Launch Jupyter notebook:

jupyter notebook quantum_observables_simulation.ipynb

Usage

Running the Complete Simulation

Execute all notebook cells sequentially to run the complete quantum observables simulation:

1. Environment Setup: Install and configure required packages
2. Library Imports: Load scientific computing libraries
3. Section 4.1 Basic System: Test discrete quantum system with position probabilities
4. Programming Challenges: Execute all four Chapter 4 programming challenges
5. Specific Exercises: Run exercises 4.3.1, 4.3.2, 4.4.1, 4.4.2, 4.5.2, 4.5.3
6. Results Analysis: Review comprehensive outputs and mathematical explanations

Individual Exercise Execution

Each exercise is implemented as a complete section in the notebook and can be run independently.

Mathematical Framework

Quantum State Representation

States are represented as complex-valued vectors in Hilbert space:

|ψ⟩ = Σᵢ cᵢ|i⟩, where Σᵢ |cᵢ|² = 1

Observable Measurements

Observables are Hermitian matrices with real eigenvalues:

Ω = Ω†, Ω|eᵢ⟩ = λᵢ|eᵢ⟩

Quantum Dynamics

Evolution is governed by unitary transformations:

|ψ(t+1)⟩ = U|ψ(t)⟩, where U†U = I

File Structure

quantum-observables-simulation/
├── quantum_observables_simulation.ipynb  # Main notebook with all solutions
├── transcription_quantum_chapter.py      # Theory content and exercise descriptions
├── requirements.txt                      # Python dependencies
├── README.md                           # Project documentation
└── LICENSE                             # GNU GPL v3 license

Built With

• Python 3.8+ - Core programming language
• NumPy - Numerical computing and linear algebra
• SciPy - Advanced mathematical functions
• Matplotlib - Data visualization and plotting
• Jupyter - Interactive notebook environment
• SymPy - Symbolic mathematics

License

This project is licensed under the GNU General Public License v3.0 - see the LICENSE file for details.

Authors

• Andersson Programming - Complete implementation - AnderssonProgramming

Acknowledgments

• Quantum Computing and Information Theory course materials
• Chapter 4 theoretical framework from course textbook
• NumPy and SciPy communities for scientific computing tools

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For questions or support, please open an issue in the GitHub repository.