基于置换矩阵表示的微分方程量子算法及其在伯格斯方程中的应用
Quantum algorithm for differential equations via permutation matrix representation with application to the Burgers equation
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中文总结 AI 辅助
该研究开发了一种基于置换矩阵表示（PMR）的量子算法，用于求解非线性粘性伯格斯方程，通过分解生成器适配哈密顿模拟线性组合（LCHS）算法，量化并缓解稳定偏移带来的指数级后选择开销，还将PMR扩展至一般流体方程。
中文摘要 AI 辅助
我们开发了一种用于求解非线性粘性伯格斯方程动力学的量子算法。我们对空间离散化后的方程应用Carleman线性化程序，随后采用填充方案以实现量子比特寄存器上的执行。现有的基于Carleman的量子算法通常在预言机模型中构建提升后的线性微分方程，而此处我们使用置换矩阵表示（PMR）将填充后的生成器分解为对角掩码和可逆算术置换，证明其与哈密顿模拟线性组合（LCHS）算法天然兼容。在LCHS所需的假设下——最重要的是线性生成器的厄米部分为半正定，可能需经稳定偏移处理——该算法制备出与截断提升系统解成比例的归一化量子态；稳定偏移会引入指数级后选择开销，我们对此进行量化并通过重标方案缓解。我们表明，该算法的缩放与Carleman生成器的非对角范数相关，而非矩阵范数，这对其他对角占优的生成器可能具有优势。我们还将PMR方案扩展至包含高阶导数、非线性项，或涉及多个流体变量、空间维度的一般流体方程。该构造展示了PMR如何作为便利的哈密顿模拟基元，服务于更广泛的基于LCU的算法。
英文摘要
We develop a quantum algorithm for solving the dynamics of the nonlinear viscous Burgers equation. We apply the Carleman linearization procedure on the spatially discretized equation, followed by a padding scheme that allows implementation on qubit registers. Existing Carleman-based quantum algorithms commonly formulate the lifted linear differential equation in an oracle model. Here we decompose the padded generator into diagonal masks and reversible arithmetic permutations using the Permutation Matrix Representation (PMR), which we show to be naturally compatible with the Linear Combination of Hamiltonian Simulations (LCHS) algorithm. Under the assumptions required by LCHS - most importantly positive semidefiniteness of the Hermitian part of the linear generator, possibly after a stabilizing shift - the algorithm prepares a normalized quantum state proportional to the solution of the truncated lifted system; the stabilizing shift introduces an exponential postselection overhead, which we quantify and mitigate through a rescaling scheme. We show that our algorithm scales with the off-diagonal norm of the Carleman generator instead of the matrix norm, which can be advantageous for other generators that are diagonally dominant. We also extend the PMR scheme to general fluid equations that may contain higher-order derivatives or nonlinear terms, or may involve multiple fluid variables or spatial dimensions. The construction illustrates how PMR can serve as a convenient Hamiltonian-simulation primitive for a broader class of LCU-based algorithms.
