## TECHNICAL NOTE

## Explanation of the Quantum Solver for Time-Evolution Partial Differential Equations (1) On First-Quantized Hamiltonian Simulation (HS)

Partial differential equations are recognized as model equations that describe various physical phenomena. Obtaining their numerical solutions correctly and quickly is essential for clarifying and predicting phenomena, making it an extremely important task. Quemix has recently researched and developed a quantum algorithm for solving time-evolution partial differential equations (hereafter simply referred to as "evolution equations"). This research confirmed the potential for acceleration using quantum computers, and the results suggest that in the future, large-scale numerical calculations, such as those for materials and fluid dynamics, could be significantly accelerated. This article is divided into two parts, focusing on applications to materials and fluid dynamics, to explain first-quantized Hamiltonian Simulation (HS) and the advection-diffusion-reaction equation. In the first part, we will explain what first-quantized HS is and discuss why and to what extent it can be accelerated on a quantum computer. We will also introduce some of the challenges faced when applying it to materials calculations.

INTRODUCTION

## WHAT IS

## What is First-Quantized HS?

The Hamiltonian of a multi-particle system is composed of kinetic energy, which depends on the position and momentum of each particle, and potential energy. First-quantized Hamiltonian Simulation (HS) involves solving the Schrödinger equation governed by such a Hamiltonian, meaning it calculates the time evolution of the wave function. Through this, physical quantities are computed, forming the fundamental calculations in materials simulations.

## REASON

## Why Can Quantum Algorithms Achieve Speedup?

To numerically solve equations, it is necessary to discretize the operator into a sufficiently large matrix and perform matrix calculations with the discretized vectors of initial and boundary conditions. On classical computers, arithmetic operations are performed element-wise on the matrix, so even for sparse matrices, the computation time is at least proportional to the matrix size (denoted as N). On the other hand, quantum computers replace arithmetic operations with unitary matrix operations from a set of basic gate operations (or fundamental operations) specified by the type of quantum hardware. To explain in more detail, quantum computers utilize the concept of qubits, where n (＝log2(N)) qubits can represent a large state space of 2^n =N dimensions (see Technical Note 001). Consequently, in the best case, an N-dimensional unitary matrix can be constructed using O(n) basic gates (single and two-qubit gate operations). This is the key to quantum speedup and the reason why quantum computers are said to offer exponential acceleration compared to classical methods. However, in terms of degrees of freedom, it is important to note that representing a matrix with O(N) elements requires at least O(N) basic gate operations, even for quantum computers. (In other words, quantum computers do not always have an inherent advantage.)

## SPEEDUP

## How Much Speedup Can Be Achieved?

Regarding the first quantization Hamiltonian simulation (HS), we employ the grid method (also referred to as the real-space method or real-space grid method), which reduces to the computation of unitary matrices of size N. Specifically, the kinetic energy part corresponds to the quantum Fourier transform, its inverse operation, and the first diagonal unitary matrix that sandwiches these operations, while the potential part corresponds to the second diagonal unitary matrix. Since the phases of the elements in the first and second diagonal unitary matrices are determined by a quadratic function and a pre-given potential function, respectively, our research has shown that these can be implemented with a polynomial order of basic gate operations in terms of n. In our paper X. Huang et al., arXiv:2407.16345 (2024), we propose a detailed method for constructing approximate circuits using the spline method by introducing sparse partition parameters for a given potential function. This indicates that exponential acceleration compared to classical methods is expected for matrix sizes N when the potential function is favorable.

## EXAMPLES

## Simulation of One-Particle and Two-Electron Systems

As a validation of our quantum algorithms, we have presented simulation examples of one-particle and multi-particle systems in one-dimensional space, with our research findings published in J. Phys. Soc. Jpn. 93, 054002 (2024).

Simulation Example 1: Electron Density of One-Particle and Two-Electron Systems

## CHALLENGES

## Challenges in Applying to Materials Calculations

When performing materials calculations (for example, structure optimization), one of the important challenges is to obtain the ground state of quantum systems. The probabilistic imaginary-time evolution method (Probabilistic Imaginary-Time Evolution, PITE®), which we at Quemix have been developing independently, is considered an effective approach to solving this challenge. What is important here is that the quantum circuit of HS is used as a subroutine in the PITE® algorithm. In other words, to actually implement the PITE® algorithm, it is necessary to perform HS. When attempting to apply first quantization HS to multiparticle system problems, one can represent the inter-particle potential using a soft-Coulomb approximation and employ the methods described above. However, in general, potential functions depend not only on distance but also on the positional coordinates of each particle. Furthermore, in cases of interaction potentials that are not analytically known, such as machine learning potentials, the degree of quantum speedup remains unclear and requires further investigation.

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