## TECHNICAL NOTE

## Quantum Solver Explanation for Time-Evolution Partial Differential Equations (2): On the Advection-Diffusion-Reaction Equation

In Part 2, based on the content from Part 1, we explain what the advection-diffusion-reaction equation is and introduce the quantum solver using PITE®. Additionally, we compare it with other quantum algorithms and discuss its application in fluid calculations as well as future challenges.

INTRODUCTION

## WHAT IS

## What is the Advection-Diffusion-Reaction Equation?

The advection-diffusion-reaction equation is a mathematical model that describes the temporal changes in substance concentration within a given domain, and it is a type of fluid calculation. Mathematically, it involves second-order derivatives (also called the diffusion term), first-order derivatives (also called the advection term), and zero-order derivatives (also called the reaction term), which represent the spreading, movement, and increase or decrease of the substance concentration in space, respectively.

## METHOD

## Quantum Solver Using PITE®

For simplicity, let us consider periodic boundary conditions. In this case, we can use the grid method mentioned in Part 1. This results in a Hamiltonian matrix (with the advection term added) that is similar to that in Part 1. Therefore, the quantum solver for the advection-diffusion-reaction equation computes the matrix product of the imaginary time evolution operator (the matrix exponential) and the vector representing the initial state with respect to this Hamiltonian matrix. Here, since the imaginary time evolution operator (matrix) is not a unitary operation (matrix), it cannot be directly implemented on a quantum computer. However, by introducing auxiliary qubits for measurement, it becomes possible to implement non-unitary operations like the imaginary time evolution operator on a quantum computer. In particular, the matrix exponential can be constructed using the quantum circuit of the probabilistic imaginary time evolution operator PITE® introduced in explanation article 005. In our paper arXiv:2409.18559, we propose a new approximate quantum circuit for PITE® to achieve exponential acceleration concerning the matrix size and establish a quantum algorithm with the first quantization HS from Part 1 as a subroutine.

## EXAMPLES

## Simulations in One and Two Dimensions

As a validation of the quantum algorithm, we reported examples of simulations in one-dimensional and two-dimensional spaces in arXiv:2409.18559.

Simulation Example 2: Advection-Diffusion (Reaction) Equation

## COMPARISONS

## Comparison with Other Quantum Algorithms

In particular, the quantum algorithms for the advection-diffusion-reaction equation that we would like to focus on this time include the well-known non-variational quantum algorithm HHL (Harrow-Hassidim-Lloyd) and the variational quantum algorithm VQA (Variational Quantum Approximation) proposed in the preceding research Computers & Fluids 281, 2024, 106369 (based on the finite difference method). However, compared to these prior quantum algorithms, we found that our quantum algorithm has shorter computation times and smaller errors relative to the true solution (see Figure 3). Additionally, in the case of periodic boundary conditions, we clarified that the grid method outperforms the finite difference method. However, when using the same discretization method, we showed that our proposed method has similar computation times (i.e., circuit depth) to the traditional Harrow-Hassidim-Lloyd (HHL) method while having the advantage of using significantly fewer auxiliary qubits (only one, excluding the HS part).

Figure3: Comparison of Other Methods in Terms of Mean Squared Error (MSE)

## coming soon ...

NEXT

## PROSPECTS & CHALLENGES

## Application to Fluid Calculations and Future Challenges

In our paper arXiv:2409.18559, we reported on advection-diffusion reaction systems (coupled equations). Essentially, the proposed method can be generally applied to linear evolution equation systems. However, if it is possible to prepare or read the state vector (obtain the solution) at each time step, we can leverage the information from the previous step's solution to update the potential, thereby enabling us to handle nonlinear equation systems. In particular, by eliminating the advection term and considering the reaction term as the spatial first derivative of the solution, we can derive the (viscous) Burgers' equation, which describes nonlinear waves in fluid dynamics. Additionally, in fluid calculations such as thermal convection, it is standard practice to solve the advection-diffusion equation alongside other equations iteratively, and if the solution can be obtained at each iterative step, it is possible to utilize the proposed method. However, obtaining the solution at all grid points from a quantum computer generally requires O(N) measurements with respect to the mesh size N, which masks the advantage of achieving O(log2 N) in the solution calculation part. Moving forward, developing efficient algorithms for solving nonlinear equation systems using iterative methods will be a key challenge.

Figure 4: Simulation Example 3 = Two-Dimensional Burgers' Equation

The numerical examples presented in this article utilize IBM's Qiskit, a gate-based quantum emulator.