## Background

The study of differential equations with the aid of quantum algorithms has gained significant attention with the recent advancements in quantum computing. A particularly notable contribution is the work by Yuki Sato et al.,

Hamiltonian Simulation for Hyperbolic Partial Differential Equations using Scalable Quantum Circuits, which presents a novel method for solving a class of partial differential equations. This approach reformulates differential equations as a Schrödinger equation and leverages Hamiltonian simulation techniques to solve them on a quantum computer.

## The Advection Equation

The advection equation is a first-order hyperbolic partial differential equation used to describe the transport of a quantity or material by bulk motion of a fluid medium, without involving diffusion or dispersion. It represents the movement of this quantity due to the flow of the medium and is often a simplified version of more complex fluid dynamics models. The advection equation is commonly used to model the transport of pollutants by water currents.

The advection equation can be solved using various quantum computing approaches, including Variational Quantum Algorithms (VQA), Quantum Linear Systems Algorithms (QLSA, such as the HHL algorithm), and Hamiltonian simulation. The advection equation considered in this challenge is of the form:

$\frac{\partial \phi}{\partial t}+\frac{\partial \phi}{\partial x}=0,$
with the following Dirichlet boundary conditions:

$\phi(0,x) = \begin{cases}
1, & \text{if } 1 < x < 2 \\
0, & \text{otherwise}
\end{cases}$
## Challenge Description

In this challenge, your objective is to apply the technique developed by Yuki Sato et al. to solve the advection equation with Dirichlet boundary conditions using the Classiq platform, which enables users to conceptualize and design complex algorithms at a high functional level.

For the Hamiltonian simulation step, you may choose the method that best suits your approach and then compare it to the circuit implementation proposed in the article.

## Goals

The goal of this challenge is to develop a parametrized quantum program as a function of time, using the Classiq platform.

A complete and thorough solution should include the implementation of the quantum program with Classiq, a single execution of the quantum algorithm using the Classiq state-vector simulator for $𝑡=1.0$, and a comparison of the CX-gate counts between the quantum circuit presented in the paper and the circuit generated using Hamiltonian simulation on the Classiq platform, for various time values.

## Deliverables - a python notebook containing the following:

- A quantum program implemented using Classiq that solves the advection equation with Dirichlet boundary conditions
- A graphical visualization of the resulting function $\phi(x,t=1.0)$
- A quantitative analysis of CX-gate counts, presented in a graphical format

## Resources

## Project-Specific Resources

## Classiq Resources: