Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Programming and Computer Software

Evaluation of Differential Equation Solutions through Traditional Techniques and Neural Networks in Quantum Simulations

Romik Sargsyan, Roman Sahakyan, E. A. Gazazyan, Sh. Sargsyan, Sevak Sargsyan, Hrachya V. Astsatryan

Published Oct 1, 2024• 2 citations
Publisher

Abstract

Abstract The study presents a comparative analysis of algorithms of traditional numerical methods and methods using neural networks for solving ordinary differential equations. The evaluation includes both mathematical aspects and aspects of machine learning. Using the Julia programming language, neural networks and the necessary algorithms for conducting practical experiments are implemented. These experiments demonstrate outstanding achievements in the field of quantum computing. Moreover, comparing the efficiency metrics of solving differential equations using numerical methods and neural networks reveals interesting results, depending on the chosen architecture and network parameters.

DOI
10.1134/s0361768824700452

Related papers

Neural Networks vs. Traditional Techniques: Differential Equation Approaches in Quantum Gate Analysis

Romik Sargsyan, Ruben O. Sahakyan, E. A. Gazazyan +3 more

The research evaluates the traditional numerical algorithms and neural network-based methods for solving ordinary differential equations. The evaluation covers both mathematical and machine learning perspectives. Practical experiments are carried out by implementing neural networks and the necessary algorithms using the Julia programming language. These experiments highlight significant advancements in the realm of quantum computing. Additionally, analyzing the efficiency metrics of solving differential equations with numerical methods versus neural networks uncovers intriguing results, which vary based on the chosen architecture and network parameters.

Workshop on Data Analytics and Mathematical Modeling•Jun 26, 2024•0 citations

Experimental Evaluation of Quantum Machine Learning Algorithms

Ricardo Daniel Monteiro Simoes, Patrick Huber, Nicola Meier +3 more

Machine learning and quantum computing are both areas with considerable progress in recent years. The combination of these disciplines holds great promise for both research and practical applications. Recently there have also been many theoretical contributions of quantum machine learning algorithms with experiments performed on quantum simulators. However, most questions concerning the potential of machine learning on quantum computers are still unanswered such as How well do current quantum machine learning algorithms work in practice? How do they compare with classical approaches? Moreover, most experiments use different datasets and hence it is currently not possible to systematically compare different approaches. In this paper we analyze how quantum machine learning can be used for solving small, yet practical problems. In particular, we perform an experimental analysis of kernel-based quantum support vector machines and quantum neural networks. We evaluate these algorithm on 5 different datasets using different combinations of quantum feature maps. Our experimental results show that quantum support vector machines outperform their classical counterparts on average by 3 to 4% in accuracy both on a quantum simulator as well as on a real quantum computer. Moreover, quantum neural networks executed on a quantum computer further outperform quantum support vector machines on average by up to 5% and classical neural networks by 7%.

IEEE Access•Jan 1, 2023•66 citations

Evaluating the performance of classical, quantum, and hybrid quantum neural networks in solving differential equations

Navid Markazi, Behrouz Mirza

We investigate the performance of classical, quantum, and hybrid quantum neural networks in solving partial differential equations through both supervised and unsupervised learning. The performance of each type of network is evaluated in solving the damped harmonic oscillator, the Einstein field equations, and the time-independent Schrödinger equation. We employed physics-informed supervised neural networks to solve the damped harmonic oscillator and the Einstein field equations, and developed an unsupervised neural network algorithm based on the shooting method for the time-independent Schrödinger equation. For the quantum neural networks, we introduced three variational parameterized quantum feature maps and two quantum circuits for the hybrid quantum neural networks. Each network was examined in three different configurations and evaluated across multiple random seeds. Under favorable parameter initializations, the hybrid quantum neural networks achieve higher accuracy than the classical neural networks in most cases across all tested problems. In contrast, the quantum neural networks attain the best accuracy for the damped harmonic oscillator and also perform well in the Schrödinger equation problem. Both the quantum and the hybrid quantum neural networks require fewer parameters than the classical neural networks and converge faster during optimization. All three models exhibit sensitivity to parameter initialization, with the quantum neural networks displaying the highest variability across different problems.

Scientific Reports•Nov 10, 2025•2 citations

Neural Network Continuous-Variable Quantum Computing

Martin Knudsen, Christian B. Mendl

We explore how a continuous-variable (CV) quantum computer could solve a classic differential equation, making use of its innate capability to represent real numbers in qumodes. Specifically, we construct variational CV quantum circuits [Killoran et al., Phys.~Rev.~Research 1, 033063 (2019)] to approximate the solution of one-dimensional ordinary differential equations (ODEs), with input encoding based on displacement gates and output via measurement averages. Our simulations and parameter optimization using the PennyLane / Strawberry Fields framework demonstrate good convergence for both linear and non-linear ODEs.

•Jan 1, 2020•0 citations

Quantum Algorithms for Solving Ordinary Differential Equations

Benjamin Zanger, Christian B. Mendl, Martin Schulz +1 more

Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing quantum computers for the purpose of solving differential equations. We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital quantum computer, and (ii) representing and solving high-order Runge-Kutta methods as optimization problems on quantum annealers. As realizations applied to two-dimensional linear ordinary differential equations, we devise and simulate corresponding digital quantum circuits, and implement and run a 6$^{\mathrm{th}}$ order Gauss-Legendre collocation method on a D-Wave 2000Q system, showing good agreement with the reference solution. We find that the quantum annealing approach exhibits the largest potential for high-order implicit integration methods. As promising future scenario, the digital arithmetic method could be employed as an oracle within quantum search algorithms for inverse problems.

•Dec 17, 2020•1 citation

Dynamic analysis of shortcut to adiabaticity based on physical information neural network

Liu Ming, Siqi Zhang, Hong Li

A quantum shortcut to adiabaticity scheme based on physics-informed neural networks is proposed in this work. Compared with traditional shortcut to adiabaticity technology, our method innovatively integrates machine learning methods by employing parameterized physics-informed neural networks to solve parameterized differential equations. The neural networks serve as an approximating function for quantum adiabatic evolution processes, while incorporating parameter-dependent differential equations and various physical constraints as components of the loss function. Through networks training, we effectively simulate quantum system dynamics and derive the driving control field for population inversion. Numerical simulations show that the quantum system can achieve rapid population inversion within significantly reduced time while maintaining high fidelity and exceptional robustness against parameter fluctuations. The neural networks exhibit remarkable computational capabilities, particularly suitable for generating control functions in complex quantum systems. Compared with conventional counter-diabatic driving and transitionless quantum driving methods, this PINN-based framework not only achieves better control performance but also provides the improved practicality for experimental implementations. The success of this method demonstrates its promising applications in quantum control tasks, including but not limited to quantum state preparation, quantum gate optimization, and adiabatic quantum computing acceleration.

Acta Physica Sinica•Jan 1, 2025•0 citations
Abstract The study presents a comparative analysis of algorithms of traditional numerical methods and methods using neural networks for solving ordinary differential equations. The evaluation includes both mathematical aspects and aspects of machine learning. Using the Julia programming language, neural networks and the necessary algorithms for conducting practical experiments are implemented. These experiments demonstrate outstanding achievements in the field of quantum computing. Moreover, comparing the efficiency metrics of solving differential equations using numerical methods and neural networks reveals interesting results, depending on the chosen architecture and network parameters.
The research evaluates the traditional numerical algorithms and neural network-based methods for solving ordinary differential equations. The evaluation covers both mathematical and machine learning perspectives. Practical experiments are carried out by implementing neural networks and the necessary algorithms using the Julia programming language. These experiments highlight significant advancements in the realm of quantum computing. Additionally, analyzing the efficiency metrics of solving differential equations with numerical methods versus neural networks uncovers intriguing results, which vary based on the chosen architecture and network parameters.
Machine learning and quantum computing are both areas with considerable progress in recent years. The combination of these disciplines holds great promise for both research and practical applications. Recently there have also been many theoretical contributions of quantum machine learning algorithms with experiments performed on quantum simulators. However, most questions concerning the potential of machine learning on quantum computers are still unanswered such as How well do current quantum machine learning algorithms work in practice? How do they compare with classical approaches? Moreover, most experiments use different datasets and hence it is currently not possible to systematically compare different approaches. In this paper we analyze how quantum machine learning can be used for solving small, yet practical problems. In particular, we perform an experimental analysis of kernel-based quantum support vector machines and quantum neural networks. We evaluate these algorithm on 5 different datasets using different combinations of quantum feature maps. Our experimental results show that quantum support vector machines outperform their classical counterparts on average by 3 to 4% in accuracy both on a quantum simulator as well as on a real quantum computer. Moreover, quantum neural networks executed on a quantum computer further outperform quantum support vector machines on average by up to 5% and classical neural networks by 7%.
We investigate the performance of classical, quantum, and hybrid quantum neural networks in solving partial differential equations through both supervised and unsupervised learning. The performance of each type of network is evaluated in solving the damped harmonic oscillator, the Einstein field equations, and the time-independent Schrödinger equation. We employed physics-informed supervised neural networks to solve the damped harmonic oscillator and the Einstein field equations, and developed an unsupervised neural network algorithm based on the shooting method for the time-independent Schrödinger equation. For the quantum neural networks, we introduced three variational parameterized quantum feature maps and two quantum circuits for the hybrid quantum neural networks. Each network was examined in three different configurations and evaluated across multiple random seeds. Under favorable parameter initializations, the hybrid quantum neural networks achieve higher accuracy than the classical neural networks in most cases across all tested problems. In contrast, the quantum neural networks attain the best accuracy for the damped harmonic oscillator and also perform well in the Schrödinger equation problem. Both the quantum and the hybrid quantum neural networks require fewer parameters than the classical neural networks and converge faster during optimization. All three models exhibit sensitivity to parameter initialization, with the quantum neural networks displaying the highest variability across different problems.
We explore how a continuous-variable (CV) quantum computer could solve a classic differential equation, making use of its innate capability to represent real numbers in qumodes. Specifically, we construct variational CV quantum circuits [Killoran et al., Phys.~Rev.~Research 1, 033063 (2019)] to approximate the solution of one-dimensional ordinary differential equations (ODEs), with input encoding based on displacement gates and output via measurement averages. Our simulations and parameter optimization using the PennyLane / Strawberry Fields framework demonstrate good convergence for both linear and non-linear ODEs.
Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing quantum computers for the purpose of solving differential equations. We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital quantum computer, and (ii) representing and solving high-order Runge-Kutta methods as optimization problems on quantum annealers. As realizations applied to two-dimensional linear ordinary differential equations, we devise and simulate corresponding digital quantum circuits, and implement and run a 6th^{\mathrm{th}}th order Gauss-Legendre collocation method on a D-Wave 2000Q system, showing good agreement with the reference solution. We find that the quantum annealing approach exhibits the largest potential for high-order implicit integration methods. As promising future scenario, the digital arithmetic method could be employed as an oracle within quantum search algorithms for inverse problems.
A quantum shortcut to adiabaticity scheme based on physics-informed neural networks is proposed in this work. Compared with traditional shortcut to adiabaticity technology, our method innovatively integrates machine learning methods by employing parameterized physics-informed neural networks to solve parameterized differential equations. The neural networks serve as an approximating function for quantum adiabatic evolution processes, while incorporating parameter-dependent differential equations and various physical constraints as components of the loss function. Through networks training, we effectively simulate quantum system dynamics and derive the driving control field for population inversion. Numerical simulations show that the quantum system can achieve rapid population inversion within significantly reduced time while maintaining high fidelity and exceptional robustness against parameter fluctuations. The neural networks exhibit remarkable computational capabilities, particularly suitable for generating control functions in complex quantum systems. Compared with conventional counter-diabatic driving and transitionless quantum driving methods, this PINN-based framework not only achieves better control performance but also provides the improved practicality for experimental implementations. The success of this method demonstrates its promising applications in quantum control tasks, including but not limited to quantum state preparation, quantum gate optimization, and adiabatic quantum computing acceleration.