Quantum Walks for Monte Carlo Simulation

From Pegs to Qubits: A Quantum Take on the Classic Galton Board

This repository contains the implementation, analysis, and results of our project on Quantum Galton Boards (QGBs) as Universal Statistical Simulators, extending the framework proposed by Carney & Varcoe (2022). We design scalable quantum circuits capable of generating Gaussian, Exponential, and Hadamard-walk distributions via discrete-time quantum walks, and optimize them for Noisy Intermediate-Scale Quantum (NISQ) hardware.

Key Features

• Depth-Optimized Circuits: ≤ 76 gates for a 4-layer QGB (over 2× improvement vs. reference implementations).
• N-Layer Implementations: Scalable circuit designs for both biased and unbiased QGBs, as well as a general Hadamard quantum walk.
• Bias Control: Tunable parameters to produce non-uniform distributions.
• Bias Control: Tunable parameters to produce non-uniform distributions.
• Noise Mitigation: Hardware-aware transpilation, gate cancellation, and zero-noise extrapolation.
• Multi-Metric Validation: Fidelity, Wasserstein distance, Kolmogorov–Smirnov statistics.

Example Quantum Circuit

This is the 4-layer unbiased QGB circuit generated using the n-layer function:

Performance Highlights

• Gaussian fidelity: ≈ 86%
• Exponential fidelity: ≈ 93%
• Hadamard-walk fidelity: ≈ 95%
• All above 80% fidelity under realistic IBM-Q noise models after mitigation.

Repository Contents

• notebooks/ – Two Jupyter notebooks with full circuit construction, NISQ simulation, and analysis.
• report.pdf – Full technical report with theoretical background, methodology, and results.
• images/ – Circuit diagrams
• src/ – Core Python implementations for QGB construction, noise analysis and post-processing.

References

See full reference list in report.pdf.

License

MIT License — check LICENSE file.