Generic and Scalable Differential Equation Solver for Quantum Scientific Computing

TL;DR

This paper introduces a scalable quantum framework for solving differential equations in scientific computing, utilizing a functional expansion encoded into quantum states and a novel parallel operation strategy to enhance efficiency.

Contribution

It proposes the generalized quantum functional expansion (QFE) framework and a parallel Pauli operation strategy, improving scalability and reducing circuit complexity in quantum differential equation solving.

Findings

Successfully solved four example differential equations using QFE.

Achieved exponential reduction in circuit number with the new strategy.

Lower qubit requirement scales with double logarithm of inverse error.

Abstract

One of the most important topics in quantum scientific computing is solving differential equations. In this paper, generalized quantum functional expansion (QFE) framework is proposed. In the QFE framework, a functional expansion of 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. The lower bound of the required number of qubits is double logarithm of the inverse error bound in the QFE framework. Furthermore, a new 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.