Application of a variational hybrid quantum-classical algorithm to heat conduction equation

The prosperous development of both hardware and algorithms for quantum computing (QC) potentially prompts a paradigm shift in scientific computing in various fields. As an increasingly active topic in QC, the variational quantum algorithm (VQA) leads a promising direction for solving partial differential equations on Noisy Intermediate Scale Quantum (NISQ) devices. Although a clear perspective on the advantages of QC over classical computing techniques for specific mathematical and physical problems exists, applications of QC in computational fluid dynamics to solve practical flow problems, though promising, are still in an early stage of development. To explore QC in practical simulation of flow problems, this work applies a variational hybrid quantum-classical algorithm, namely the variational quantum linear solver (VQLS), to resolve the heat conduction equation through finite difference discretization of the Laplacian operator. Details of VQLS implementation are discussed by various test instances of linear systems. Finally, the successful statevector simulations of the heat conduction equation in one and two dimensions demonstrate the validity of the present algorithm by proof-of-concept results. In addition, the heuristic scaling for the heat conduction problem indicates that the time complexity of the present approach is logarithmically dependent on the precision {\epsilon} and linearly dependent on the number of qubits n.