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Workshop on Data Analytics and Mathematical Modeling

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

Romik Sargsyan, Ruben O. Sahakyan, E. A. Gazazyan, Shushanik A. Sargsyan, Sevak Sargsyan, Hrachya V. Astsatryan

Published Jun 26, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.62343/csit.2024.4

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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.
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%.
Machine learning has achieved dramatic success over the past decade, with applications ranging from face recognition to natural language processing. Meanwhile, rapid progress has been made in the field of quantum computation including developing both powerful quantum algorithms and advanced quantum devices. The interplay between machine learning and quantum physics holds the intriguing potential for bringing practical applications to the modern society. Here, we focus on quantum neural networks in the form of parameterized quantum circuits. We will mainly discuss different structures and encoding strategies of quantum neural networks for supervised learning tasks, and benchmark their performance utilizing Yao.jl, a quantum simulation package written in Julia Language. The codes are efficient, aiming to provide convenience for beginners in scientific works such as developing powerful variational quantum learning models and assisting the corresponding experimental demonstrations.
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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.
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