通过色散流形上的几何量子态制备求解波传播问题

中文摘要 AI 辅助

我们提出了一种通过线性系统公式求解偏微分方程的量子算法，重点研究频域中由离散化亥姆霍兹方程描述的波传播问题。尽管量子线性系统求解器能够对系统自由度进行指数级压缩，但其运行时间复杂度通常由离散化算子的条件数决定。利用微分算子的解析结构可以提供显式矩阵求逆的替代方案，正如通过显式量子电路对屏蔽泊松方程所展示的那样。然而，对于亥姆霍兹方程，逆算子在色散面 $k^2=\omega^2/c^2$ 上变得奇异，使得直接的傅里叶空间态制备方法在指数级上变得低效。我们通过直接制备支撑在共振流形上的量子态，并通过傅里叶相位编码源位置来解决这一挑战。所提出的算法消除了与共振流形上后选择相关的指数级巨大开销，使得成功概率仅依赖于源的数量，且与计算域大小无关。更一般地，我们的方法适用于傅里叶空间解在色散流形上具有奇异支撑的双曲型微分方程，将其求解重新表述为几何量子态制备问题。

英文摘要

We present a quantum algorithm for solving partial differential equations through a linear-system formulation, focusing on wave propagation problems described by the discretized Helmholtz equation in frequency domain. Although quantum linear-system solvers offer exponential compression of the system degrees of freedom, their runtime complexity is generally governed by the condition number of the discretized operator. Exploiting the analytic structure of the differential operator can provide an alternative to explicit matrix inversion, as illustrated for the screened Poisson equation through an explicit quantum circuit. For the Helmholtz equation, however, the inverse operator becomes singular on the dispersion surface $k^2=ω^2/c^2$, rendering direct Fourier-space state-preparation methods exponentially inefficient. We address this challenge by directly preparing quantum states supported on the resonant manifold and encoding source locations through Fourier phases. The resulting algorithm eliminates the exponentially large overhead associated with post-selection on the resonant manifold, yielding a success probability that depends linearly on the number of sources and is independent of the computational domain size. More generally, our approach applies to hyperbolic differential equations whose Fourier-space solutions possess singular support on dispersion manifolds, recasting their solution as a problem of geometric quantum state preparation.

发表机构

• Quantum Art（量子艺术）

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