Improved circuit implementation of the HHL algorithm — Paper Review

Project: Improved circuit implementation of the HHL algorithm and its simulations on QISKIT

Overview

This repository contains a review of the research paper on the HHL algorithm (Harrow–Hassidim–Lloyd), an analysis of its improvements, and a practical Qiskit-based implementation and simulation. The goal is to implement an improved circuit construction, reproduce key experiments, and compare results with the claims in the paper.

Contents

• paper_review.md — concise summary & critical analysis of the research paper.
• hhl.ipynb/ — implementation of HHL-related circuit modules.
• research-paper/ — extract research paper.

Requirements

• Python 3.9+
• Qiskit, Qiskit-aer (recommended stable release)
• numpy, scipy
• matplotlib
• jupyter (for notebooks)

Install dependencies (recommended inside a virtualenv):

python -m venv .venv
source .venv/bin/activate    # macOS / Linux
.venv\Scripts\activate      # Windows
pip install -r requirements.txt

requirements.txt should include (example):

qiskit>=0.43
numpy
scipy
matplotlib
jupyter

Quick start — run a simulation

1. Run the main simulation script:

2. The script will:

   • prepare the input state corresponding to b,
   • build the improved HHL circuit (state preparation, phase estimation, controlled rotations, inverse QPE),
   • run the circuit on a Qiskit simulator (local Aer or statevector),
   • post-process measurement outcomes to reconstruct the approximate solution vector, and

Notes on reproducibility

• Use controlled seeds for random matrix generation to allow reproducible comparison with the paper.
• Document the backend (Aer vs statevector vs noisy simulator) and Qiskit version used when reporting results.

References

• Harrow, Hassidim & Lloyd, Quantum algorithm for linear systems of equations, PRL (2009).
• [The paper being reviewed] — include full citation and URL in paper_review.md.

HHL_Algorithm_Quantum

HHL_Algorithm_Quantum