Variational Quantum Algorithms for Differential Equations on a Noisy Quantum Computer

The role of differential equations (DEs) in science and engineering is of paramount importance, as they provide the mathematical framework for a multitude of natural phenomena. Since quantum computers promise significant advantages over classical computers, quantum algorithms for the solution of DEs have received a lot of attention. Particularly interesting are algorithms that offer advantages in the current noisy intermediate-scale quantum (NISQ) era, characterized by small and error-prone systems. We consider a framework of variational quantum algorithms, quantum circuit learning (QCL), in conjunction with derivation methods, in particular the parameter shift rule, to solve DEs. As these algorithms were specifically designed for NISQ computers, we analyze their applicability on NISQ devices by implementing QCL on an IBM quantum computer. Our analysis of QCL without the parameter shift rule shows that we can successfully learn different functions with three-qubit circuits. However, the hardware errors accumulate with increasing number of qubits, and thus, only a fraction of the qubits available on the current quantum systems can be effectively used. We further show that it is possible to determine derivatives of the learned functions using the parameter shift rule on the IBM hardware. The parameter shift rule results in higher errors, which limits its execution to low-order derivatives. Despite these limitations, we solve a first-order DE on the IBM quantum computer. We further explore the advantages of using multiple qubits in QCL by learning different functions simultaneously and demonstrate the solution of a coupled DE on a simulator.