Quantum HHL Simulation Project

This repository contains the documentation, and analysis for implementing the Harrow-Hassidim-Lloyd (HHL) algorithm to solve linear systems using quantum computing principles. The project focuses on solving a specific 2-dimensional non-Hermitian linear system, converted into a Hermitian form, and simulating the solution using the PennyLane framework.

Project Overview

The HHL algorithm is a cornerstone of quantum computation, enabling efficient solutions to linear systems of equations in ( O(\log N) ) compared to ( O(N) ) in classical computation for sparse matrices. This project builds upon prior research by:

• Transforming non-Hermitian systems into Hermitian systems.
• Implementing quantum circuits with three registers: ancilla, eigenvalue encoding, and solution registers.
• Modifying and simulating the algorithm using PennyLane.
• Comparing theoretical and simulated results.

Key Files

1. Presentation Slides.pdf

   • Provides an overview of the theoretical underpinnings, challenges, and potential of quantum algorithms for linear systems.
   • Discusses eigenvalues, eigenvectors, and implementation details for the HHL algorithm.

2. HHL_Project_Report.pdf

   • Detailed description of the ECS417 course project under Dr. Ankur Raina, IISER Bhopal.
   • Includes the mathematical formulation, quantum circuit design, and simulation results using PennyLane.
   • Highlights modifications to the original HHL algorithm for a specific non-Hermitian 2D system.

Key Results

• Efficiency: Demonstrated ( O(\log N) ) complexity for sparse matrices.
• Simulation Outcome: Normalized solution set [1, 0], deviating slightly from the expected normalized solution [0.707, 0.707].
• Challenges: Approximation errors during controlled rotation operations and the need for pre-computed oracles.

Technologies Used

• Quantum Computing Framework: PennyLane
• Mathematics Software: Mathematica (for oracle calculation)
• Quantum Circuits: Designed using quantum gates, ancilla qubits, and controlled rotations.

References

• HHL Analysis and Simulation Verification Based on Origin Quantum Platform by Liu et al.
• Using Quantum Algorithms to Solve Linear Systems by Michael C.R. Byrd Jr.

Acknowledgments

• Instructor: Dr. Ankur Raina, IISER Bhopal
• Contributors: A. A. Gandhi, Rehan

Future Scope

• Refining approximations to improve solution accuracy.
• Extending the algorithm to larger systems and exploring error mitigation techniques.
• Exploring applications of HHL in real-world quantum computing tasks.

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This repository is a demonstration of the power and limitations of quantum algorithms and a step toward harnessing quantum computing for practical problem-solving.