具有空间变化参数的非保守系统偏微分方程的量子算法
Quantum algorithm for partial differential equations of non-conservative systems with spatially varying parameters
AI总结：
本文提出一种基于哈密顿量模拟线性组合（LCHS）的量子算法，用于求解具有空间变化参数的非保守系统二阶线性偏微分方程，通过逻辑最小化技术减少电路深度，并利用矩阵乘积态实现可扩展量子电路，为实际工程问题提供了高效量子计算方案。
摘要：
偏微分方程（PDEs）对于建模传热、流体流动和电磁波等各种物理现象至关重要。在计算机辅助工程（CAE）中，处理精细分辨率和大型计算模型的能力对于提高产品性能和降低开发成本至关重要。然而，求解大规模PDEs，特别是对于具有空间变化材料属性的系统，带来了重大的计算挑战。在本文中，我们提出了一种量子算法，使用哈密顿量模拟的线性组合（LCHS）方法，求解具有空间变化参数的非保守系统的二阶线性PDEs。我们的方法通过使用有限差分法进行空间离散化，将这些PDEs转化为由量子比特算子表示的常微分方程。然后，我们提供了一种算法，通过逻辑最小化技术高效地构建与PDEs空间变化参数相对应的算子，该技术减少了项数从而降低了电路深度。我们还开发了一种实现LCHS量子电路的可扩展方法，使用基于张量网络的技术，具体为矩阵乘积态（MPS）。我们通过应用于具有空间变化参数的声学方程和耗散热方程来验证我们的方法。我们的方法包括构建PDEs量子电路的详细方案，利用对PDEs空间变化参数的高效编码和LCHS的可扩展实现，我们相信这标志着推进量子计算在解决实际工程问题中的作用迈出了重要一步。
英文摘要：
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.
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