Quantum algorithm for partial differential equations of non-conservative systems with spatially varying parameters

Abstract

Partial differential equations (PDEs) are crucial for modeling various physical phenomena such as heat transfer, fluid flow, and electromagnetic waves. In computer-aided engineering (CAE), the ability to handle fine resolutions and large computational models is essential for improving product performance and reducing development costs. However, solving large-scale PDEs, particularly for systems with spatially varying material properties, poses significant computational challenges. In this paper, we propose a quantum algorithm for solving second-order linear PDEs of non-conservative systems with spatially varying parameters, using the linear combination of Hamiltonian simulation (LCHS) method. Our approach transforms those PDEs into ordinary differential equations represented by qubit operators, through spatial discretization using the finite difference method. Then, we provide an algorithm that efficiently constructs the operator corresponding to the spatially varying parameters of PDEs via a logic minimization technique, which reduces the number of terms and subsequently the circuit depth. We also develop a scalable method for realizing a quantum circuit for LCHS, using a tensor-network-based technique, specifically a matrix product state (MPS). We validate our method with applications to the acoustic equation with spatially varying parameters and the dissipative heat equation. Our approach includes a detailed recipe for constructing quantum circuits for PDEs, leveraging efficient encoding of spatially varying parameters of PDEs and scalable implementation of LCHS, which we believe marks a significant step towards advancing quantum computing's role in solving practical engineering problems.

21 pages, 5 figures

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• Quantum circuits for partial differential equations in Fourier space
• Quantum algorithms for solving a drift-diffusion equation: A complexity analysis
• A Quantum-Inspired Algorithm for Wave Simulation Using Tensor Networks
• Simulating Quantum Turbulence with Matrix Product States
• Sublinear Classical-to-Quantum Data Encoding using $n$-Toffoli Gates
• Orthogonal frequency-division multiplexing for simultaneous gate operations on multiple qubits via a shared control line
• An efficient explicit implementation of a near-optimal quantum algorithm for simulating linear dissipative differential equations