Variational Quantum Linear Solver

[NadavClassiq]

Variational Quantum Linear Solver

Abstract

Linear systems of equations are fundamental across many fields, from physics and engineering to economics and computer science. These systems are crucial in applications like numerical solutions of differential equations, circuit analysis, and statistical regression models. The Variational Quantum Linear Solver by Carlos Bravo-Prieto et al. introduces a quantum approach to solving large linear systems, leveraging quantum machine learning methods to handle increasing system sizes beyond the capabilities of classical methods.

Project Overview

Challenge: Implement the variational quantum algorithm for solving linear systems of equations as described in Section 2.3 of the referenced paper. Your objective is to solve the following system of linear equations:

$$ \mathbf{A} \vec{x} = \vec{0} $$

where

$$\mathbf{A} = \sum_{i=1}^{10} \hat{X}_i + 0.1 \sum_{j=1}^9 \hat{Z}_j \hat{Z}_{j+1} + I$$

with $\hat{X}_j$ and $\hat{Z}_j$ representing the Pauli matrices acting on the $j$-th qubit.

Objective

1. Define the cost function and quantum ansatz to solve the linear system.
2. Implement the algorithm and execute it on a statevector simulator.
3. Analyze the CX-gate count of the quantum program for a comprehensive performance overview.

Deliverables

• Jupyter Notebook containing:
  • The quantum program that implements the variational quantum linear solver.
  • A numerical analysis of CX-gate counts in the quantum program.

Follow the Contribution Guidelines in CONTRIBUTING.md. For any questions, you can reach out via GitHub or join our Slack Community.

Getting Started

1. Review Paper: Study the Variational Quantum Linear Solver by Carlos Bravo-Prieto et al. for understanding the algorithm’s framework.
2. Set Up Environment: Create a new Jupyter Notebook and install the Classiq SDK; refer to the setup guide.
3. Guiding Materials:
   • Classiq 101 - Introduction to platform concepts.
   • Classiq Fundamentals Workshop - Hands-on Classiq basics.
   • Platform walkthrough on home page.

Implementation Steps

1. Algorithm Coding:

   • Define the cost function and quantum ansatz for solving the linear system.
   • Implement the variational quantum linear solver using Classiq SDK.
   • Document steps in markdown, following the Glued Trees Example.
   • For support, contact us on GitHub or Slack.

2. Mathematical Explanation:

   • Use markdown and LaTeX to provide theoretical explanations, key equations, and algorithm insights.

3. Generate .qmod File:

   • Use write_qmod(model, "filename.qmod") to save your models.
   • Confirm successful notebook execution and .qmod file generation.

4. Quality Check:

   • Proofread for accuracy and ensure code correctness.
   • Use clear markdown formatting and a professional presentation.

5. Submit Contribution:

   • Follow Contribution Guidelines.
   • Open a Pull Request in classiq-library/research/linear_solver.
   • Include a summary of insights and results.

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Resources

• Reference Paper: Variational Quantum Linear Solver by Carlos Bravo-Prieto et al.
• Classiq Documentation: Classiq Library
• Glued Trees Example: Link
• Contribution Guidelines: CONTRIBUTING.md

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Note: No strict deadline. Confirm with us if you start this task so we can assign it to you.

Good Luck!