Quantum algorithms and quantum machine learning for differential equations

Abstract

The fast and accurate solution of differential equations is a highly researched topic. Classical methods are able to solve very large systems, however, this can require highperformance computers and very long computational times. Since quantum computers promise significant advantages over classical computers, quantum algorithms for the solution of differential equations have received a lot of attention. Particularly interesting are algorithms that are relevant in the current Noisy-Intermediate-Scale-Quantum (NISQ) era, characterized by small and error-prone systems. In this context, promising candidates are variational quantum algorithms which are hybrid quantumclassical algorithms where only a part of the algorithm is executed on the quantum computer. Thus, they typically require fewer qubits and qubit gates and can tolerate the errors stemming from an imperfect quantum computer. One important example of variational quantum algorithms is the so-called quantum circuit learning (QCL) algorithm, which can be used to approximate functions. Here, an ansatz function is formed with a data encoding layer, subsequently transformed by a shallow parameterized circuit and finally the measurement of an expectation value defines the function value. In this thesis, this method is investigated in great detail, developing new and improved circuit designs and comparing their usefulness in approximating different functions. The method is also tested on a real quantum computer, which has not been reported in literature yet. For this purpose, the algorithm is executed on the superconducting quantum computer IBM Quantum System One in Ehningen to investigate its applicability in the NISQ era. The concept of QCL can be combined with the parameter shift rule to determine derivatives. This enables the solution of nonlinear differential equations. This procedure is subjected to thorough testing across a multitude of differential equations while being compared to other quantum algorithms for solving differential equations. The strengths of this algorithm are shown but also the weaknesses are analyzed. Going beyond the current state of research, the method is extended to solve coupled differential equations with a single circuit, significantly reducing the computational effort. Lastly, a differential equation is successfully solved on the quantum computer IBM Quantum System One in Ehningen.