A Quantum Approach for Exploring the Numerical Results of the Heat Equation

Abstract

This paper presents a quantum algorithm for solving the one-dimensional heat equation with Dirichlet boundary conditions. The algorithm utilizes discretization techniques and employs quantum gates to emulate the heat propagation operator. Central to the algorithm is the Trotter–Suzuki decomposition, enabling the simulation of the time evolution of the temperature distribution. The initial temperature distribution is encoded into quantum states, and the evolution of these states is driven by quantum gates tailored to mimic the heat propagation process. As per the literature, quantum algorithms exhibit an exponential computational speedup with increasing qubit counts, albeit facing challenges such as exponential growth in relative error and cost functions. This study addresses these challenges by assessing the potential impact of quantum simulations on heat conduction modeling. Simulation outcomes across various quantum devices, including simulators and real quantum computers, demonstrate a decrease in the relative error with an increasing number of qubits. Notably, simulators like the simulator_statevector exhibit lower relative errors compared to the ibmq_qasm_simulator and ibm_osaka. The proposed approach underscores the broader applicability of quantum computing in physical systems modeling, particularly in advancing heat conductivity analysis methods. Through its innovative approach, this study contributes to enhancing modeling accuracy and efficiency in heat conduction simulations across diverse domains.

Keywords

• quantum computing
• partial differential equations
• quantum heat equations
• Trotter–Suzuki decomposition
• quantum circuits