Optimal scheduling in probabilistic imaginary-time evolution on a quantum computer

[NadavClassiq]

Optimal scheduling in probabilistic imaginary-time evolution on a quantum computer

Abstract

Discretization plays a crucial role in the success of numerical simulations. Probabilistic Imaginary Time Evolution (PITE) is a quantum computing technique based on Imaginary Time Evolution methods, which can be used to find the ground state of a quantum Hamiltonian. The PITE algorithm by Taichi Kosugi et al. achieves ground state estimation through imaginary time evolution on a quantum computer. Optimal scheduling for this algorithm, as proposed by Hirofumi Nishi et al., improves efficiency over linear time discretization. This project challenges you to implement PITE with two time-discretization methods and analyze their resource requirements using Classiq.

Project Overview

Challenge: Implement the PITE algorithm on Classiq’s platform using two time-discretization methods from the referenced papers. Perform a quantitative analysis of the CX-gate counts for both methods over a sequence of 10 time-steps with a fixed interval. Additionally, estimate the ground state of the Hamiltonian using the quantum algorithm.

Objective

Execute the PITE algorithm on the following Ising Hamiltonian with ( N ) qubits:

$$\hat{H} = \sum_{i \ge j} h_{i,j} \, \hat{\sigma}_z^{(i)} \hat{\sigma}_z^{(j)} + \sum_{i=0}^{N-1} J_i \, \hat{\sigma}_x^{(i)}$$

where ( h_{i,j} = 0.5 ) and ( J_i = 0.7 ).

1. Implement two distinct time-discretization schedules as proposed in the paper.
2. Use Classiq’s optimization features to compare the CX-gate counts across both methods for a fixed 10-step sequence.
3. Bonus: Estimate the ground state energy of the Hamiltonian.

Deliverables

• Jupyter Notebook containing:
  • Quantum programs for the PITE algorithm with both time-discretization methods.
  • Quantitative analysis of CX-gate counts, represented graphically.
  • Bonus analysis of the lowest energy state obtained, along with its measurement probability.

Follow the Contribution Guidelines in CONTRIBUTING.md. For further assistance, contact us via GitHub or in our Slack Community.

Getting Started

1. Review Paper: Study the methods proposed by Taichi Kosugi et al. and Hirofumi Nishi et al. on PITE and optimal scheduling for imaginary-time evolution.
2. Set Up Environment: Create a new Jupyter Notebook and install the Classiq SDK; follow 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:

   • Implement the PITE algorithm using the Classiq SDK for both time-discretization methods.
   • Define the Hamiltonian ( \hat{H} ) as a list of Pauli strings, and perform time evolution using unitary() or an equivalent Hamiltonian simulation method.
   • Document steps in markdown, following the Glued Trees Example.
   • For help, contact us on GitHub or Slack.

2. Mathematical Explanation:

   • Use markdown and LaTeX to explain theoretical background, key equations, and algorithm insights.

3. Generate .qmod File:

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

4. Quality Check:

   • Proofread the notebook and verify code accuracy.
   • Use clear markdown formatting and professional presentation.

5. Submit Contribution:

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

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Resources

• Reference Papers:
  • Optimal Scheduling in Probabilistic Imaginary-Time Evolution on a Quantum Computer by Hirofumi Nishi et al.
  • Imaginary-Time Evolution Using Forward and Backward Real-Time Evolution with a Single Ancilla: First-Quantized Eigensolver Algorithm for Quantum Chemistry by Taichi Kosugi et al.
• 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!