  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.
