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Gaussian State Preparation

Introduction

State Preparation is a cornerstone of quantum algorithms and applications, enabling the initialization of quantum systems into specific states as a starting point for subsequent operations. Its efficiency directly impacts the accuracy and scalability of quantum computations.

A particular interest is the Gaussian state preparation, which is essential for simulating physical systems and tackling problems in quantum chemistry, machine learning, and optimization. Gaussian states, characterized by their Gaussian-shaped wavefunctions, are powerful tools for encoding probability distributions and modeling quantum systems.

This project leverages Classiq's quantum computing SDK to implement a Gaussian state preparation in the interval [-2, 2).

Target State: The Gaussian State Representation

The Gaussian state is defined as:

$$ |x_0\rangle_N = |0\rangle_N \longrightarrow \sum_{|x\rangle_N} \sqrt{G(x)} |x\rangle_N $$

Where $( G(x) )$ is represented as a vector:

$$ {G}(x_i) = \frac{\exp(-\lambda \cdot {x_i}^2)}{\sum_{x' \in \text{domain}} \exp(-\lambda \cdot (x')^2)} \text{, for } x_i \in \text{domain} $$

  • $( G(x) )$ is the normalized Gaussian vector across the discrete domain.
  • $( \vec{x} )$ represents the set of discrete points in the domain.
  • $( \lambda )$ represents the decay rate, controlled by the variable EXP_RATE, which determines the spread of the Gaussian.
  • The denominator ensures normalization across the entire domain.

Installation

To use this project, ensure you have the required dependencies installed:

pip install classiq

Authentication

If running for the first time, authenticate with Classiq:

import classiq
# classiq.authenticate()  # Uncomment to authenticate

Running the Code

Run the Jupyter Notebook to execute the Gaussian state preparation algorithm. The implementation initializes and prepares the quantum system in a Gaussian state within the specified interval.

Setting Up the Environment

This project includes a requirements.txt file that lists all the Python packages and their versions used in this project. You can use this file to recreate the exact environment required to run the code.

License

This project is intended for educational and research purposes. Please ensure compliance with any relevant licensing agreements for the Classiq SDK.