Low‐Measurement‐Complexity Variational Quantum Poisson Equation Solver and its Application in Heat Conduction Problems

ABSTRACT

Heat conduction problems in chips, engines, and batteries are often formulated as Poisson equations and solved using classical algorithms such as Cholesky decomposition and SuperLU. However, large‐scale systems require significant resources and computation time. We enhance a Variational Quantum Linear Solver‐based solver by introducing a hybrid Pauli–computational basis for matrix expansion, which requires only m+1 circuit measurement without additional ansatz circuits or ancilla qubits, reducing complexity and improving suitability for noisy intermediate‐scale quantum devices. A quantum–classical hybrid framework based on domain decomposition is also proposed, assigning quantum solvers to speed‐critical subdomains and classical methods to high‐accuracy regions. Simulations on Qiskit for a Chip‐on‐Board Light‐Emitting Diode heat conduction problem achieve 99.9197% fidelity and 0.09764 overall error. Finally, this paper compares the proposed method with classical solvers, particularly sparse LU factorization, in terms of algorithmic complexity, memory consumption, and related computational costs. In addition, the impact of noise on solution fidelity is analyzed, together with corresponding noise mitigation strategies.