Generic and scalable differential-equation solver for quantum scientific computing

One of the most important topics in quantum scientific computing is solving differential equations. In this paper, a generalized quantum functional expansion (QFE) framework is proposed. In the QFE framework, a functional expansion of the solution is encoded into a quantum state and the time evolution of the quantum state is solved with variational quantum simulation (VQS). The quantum functional encoding supports different numerical schemes of functional expansions. If the solution space is infinite-order differentiable, the lower bound of the required number of qubits under the quantum function encoding scheme is the double logarithm of the inverse error bound. Furthermore, a parallel Pauli operation strategy is proposed to significantly improve the scalability of VQS. The number of circuits in VQS is exponentially reduced to only the quadratic order of the number of ansatz parameters. Four example differential equations are solved to demonstrate the generic QFE framework.