Simulating Quantum Walks & Monte Carlo Using a Galton Box

Project Name

Simulating Quantum Walks & Monte Carlo Using a Galton Box

Team Name

Quanto_Gladiators

Team Members

• Nayan Jadhav: WISER Enrollment ID: gst-AmKO7kPeqNZy6xI
• Arham Sabadra: WISER Enrollment ID: gst-U6ySL3CkHLSbHdy

Note: WISER Enrollment IDs are placeholders and will be updated upon receipt.

Quantum Galton Box Analysis

This project explores the potential of quantum computing in simulating complex systems using a Galton Box-style Monte Carlo approach, also known as a quantum walk simulation. Inspired by the concept of the Universal Statistical Simulator, the project highlights how quantum circuits can model probabilistic behaviors efficiently, and serves as an educational exploration into both classical and quantum stochastic systems.

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Objective

The main objective of this project is to simulate, analyze, and compare different versions of the Galton Box — classical, quantum, noisy quantum, and biased quantum — across multiple layers. The goal is to investigate:

• How quantum superposition and interference affect output distributions,
• The impact of noise on quantum circuits,
• Differences between biased and unbiased systems,
• And how quantum results deviate or converge to classical expectations under various conditions.

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Background

A Galton Box (Plinko) is a statistical device where balls drop through several layers of pegs, taking random left or right turns, and accumulate into a distribution that closely resembles a binomial (or normal) curve. This randomness makes it an ideal candidate for Monte Carlo simulation. In the quantum version, Hadamard gates represent a quantum coin flip, allowing superpositions of left/right states.

The quantum Galton Box is relevant to:

• Quantum walks and quantum Fourier transforms
• Simulation of high-dimensional probabilistic systems
• Particle transport and quantum dynamics
• Benchmarking noise models and exploring quantum error resilience

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Tasks Breakdown

Task 1 – Classical Simulation

Simulates the Galton Box using random binary decisions. Repeats this process for 5, 10, 15, 20, and 25 layers.

Task 2 – Quantum Galton Box

Implements quantum circuits using Hadamard gates and measurements, analyzing how quantum superposition distributes probabilities.

Task 3 – Biased Quantum Galton Box

Applies a bias to the quantum flips, altering probability distributions (e.g., bias toward '1' with 70% probability).

Task 4 – Noise Simulation

Uses Qiskit Aer noise models to introduce depolarizing errors into the quantum circuits, simulating realistic, noisy environments.

Task 5 – Method-Wise Comparison

Compares distributions using:

• Total Variation Distance (TVD)
• Kullback–Leibler Divergence (KL)
• (Optional) Wasserstein Distance

Task 6 – Report

Summarizes results, includes plots, analysis, and learning outcomes from all simulations.

Task 7 – Hardware Execution

Attempted real hardware execution using IBM Quantum, but due to Open Plan limitations (only simulator access), full execution on real devices was skipped.

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Tools Used

• Qiskit (Quantum circuits, noise simulation)
• NumPy, Matplotlib, SciPy (Data processing, visualization, statistical comparisons)
• Python 3.x in Jupyter Notebooks
• IBM Quantum Platform (simulators used)

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Outcomes

This project deepened understanding of how quantum systems behave under uncertainty, and how they compare to classical randomness. It showed the strengths of quantum simulations in reproducing statistical phenomena and highlighted current challenges, especially hardware access under free plans.

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Repository Structure

Quanto_Gladiators/
│
├── Summary Report/
│    ├── summary.pdf
│    └── presentation.pdf
│
├── 1_Classical GB/               <--  Task 1: Classical Galton Box
│   ├── classical_galton_box.ipynb
│   ├── results/
│   │   ├── galton_box_5_layers.png
│   │   ├── galton_box_10_layers.png
│   │   ├── galton_box_15_layers.png
│   │   ├── galton_box_20_layers.png
│   │   ├── galton_box_25_layers.png
│   └── Classical_README.md
│
├── 2_Quantum GB/                 <--  Task 2: Quantum Galton Box
│   ├── quantum_galton_box.ipynb
│   ├── results_quantum/
│   │   ├── quantum_galton_box_5_layers.png
│   │   ├── quantum_galton_box_10_layers.png
│   │   ├── quantum_galton_box_15_layers.png
│   │   ├── quantum_galton_box_20_layers.png
│   │   ├── quantum_galton_box_25_layers.png
│   └── Quantum_README.md
│
├── task_3_Biased GB/         <--  Task 3: Biased Quantum Box
│   └── biased_quantum_galton_box.ipynb
│   ├── results_quantum_biased/
│   │   ├── biased_quantum_galton_box_5_layers.png
│   │   ├── biased_quantum_galton_box_10_layers.png
│   │   ├── biased_quantum_galton_box_15_layers.png
│   │   ├── biased_quantum_galton_box_20_layers.png
│   │   ├── biased_quantum_galton_box_25_layers.png
│   └── Biased_README.md 
│   
├── 4_Noise/         <--   Task 4: Noise Simulation (Qiskit Runtime)
│   └── noisy_galton_box.ipynb
│   ├── results\noise
│   │   ├── noisy_biased_galton_box_5_layers.png
│   │   ├── noisy_biased_galton_box_10_layers.png
│   │   ├── noisy_biased_galton_box_15_layers.png
│   │   ├── noisy_biased_galton_box_20_layers.png
│   │   ├── noisy_biased_galton_box_25_layers.png
│   └── Noise_README.md  
│
├── 5_Comparison/             <--  Task 5: Method-wise Comparison
│   ├── galton_box_comparison_5_layers.ipynb
│   ├── galton_box_comparison_10_layers.ipynb
│   ├── galton_box_comparison_15_layers.ipynb
│   ├── galton_box_comparison_20_layers.ipynb
│   ├── galton_box_comparison_25_layers.ipynb
│   ├── results/
│   │   ├── 5_layers/
│   │   │   └── galton_box_comparison_5_layers.png
│   │   ├── 10_layers/
│   │   │   └── galton_box_comparison_10_layers.png
│   │   ├── 15_layers/
│   │   │   └── galton_box_comparison_15_layers.png
│   │   ├── 20_layers/
│   │   │   └── galton_box_comparison_20_layers.png
│   │   └── 25_layers/ 
│   │       └── galton_box_comparison_25_layers.png     
│   └── Comparison_README.md
│
├── 6_Educational_toolkit/
│   ├── results/
│   ├── education_toolkit_README.md
│   ├── requirements.txt
│   └── voila quantum_dashboard_simulator.ipynb
│   
├── README.md                        <-- Project overview & task descriptions
├── requirements.txt                 <-- All required Python packages
└── .gitignore                       <-- Files to ignore (e.g., .ipynb_checkpoints)