发表机构
Department of Physics, Center for Integrated Plasma Studies, University of Colorado Boulder（科罗拉多大学博尔德分校物理系，综合等离子体研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对线性平流算子，提出一种基于迎风格式与LCUs嵌入的量子算法，实现指数级加速、抑制伪振荡，并设计高效失败恢复机制，为流体模拟提供子程序。
AI 中文摘要
线性平流算子是流体和等离子体问题中普遍存在的构建模块。我们开发了一种用于实现该算子的量子算法。当平流速度在空间上恒定时，我们的算法在每个时间步上比经典算法指数级更高效，并且避免了陡峭梯度附近的伪振荡。该算法最清晰地通过一维平流方程在具有周期性边界条件的均匀空间网格上加以说明，并可扩展到更高维度。该算法采用一阶迎风格式，能够捕获波包络中的不连续性，但该格式非酉。我们使用酉算子线性组合（LCUs）嵌入非酉迎风格式，并开发了迎风酉算子的高效量子门分解，该分解使用$O(n^2)$个双量子比特门执行一步平流，其中$N=2^n$是空间网格点数，而经典计算机的成本为$O(N)$。尽管LCUs在每个时间步引入小的有界失败概率，但我们表明失败的累积不会导致如天真预期那样的指数时间复杂性。此外，当LCUs失败时，我们开发了一种概率方案来从失败状态恢复，避免了模拟的完全重启。该失败恢复方案使用量子傅里叶变换（QFT），并有效实现了未知量子态的量子不定积分。如果波包络被良好分辨以仅包含低傅里叶模式，则恢复（其本身可能失败）比完全重启更高效。我们在经典上模拟了我们的方案，并在Quantinuum的离子阱量子比特上演示了小规模问题。我们的量子算法为涉及线性平流的物理模拟提供了子程序。
英文摘要
The linear advection operator is an ubiquitous building block in fluid and plasma problems. We develop a quantum algorithm for enacting the operator. When the advection velocity is constant in space, our algorithm is exponentially more efficient per time step than classical and avoids spurious oscillations near steep gradients. The algorithm is most cleanly illustrated using the one-dimensional advection equation on a uniform spatial grid with periodic boundary conditions, which can be extended to higher dimensions. The algorithm uses a first-order upwind scheme, which captures discontinuities in the wave envelope but is not unitary. We embed the non-unitary upwind scheme using Linear Combinations of Unitaries (LCUs), and develop an efficient quantum gate decomposition of the upwind unitary, which performs one step of advection using $O(n^2)$ two-qubit gates, where $N=2^n$ is the number of spatial grid points, as opposed to a classical computer which costs $O(N)$. Although LCUs introduces a small bounded probability of failure per time step, we show that the accumulation of failures does not lead to exponential-in-time complexity as one would naively expect. Moreover, when LCUs fails, we develop a probabilistic scheme to recover from the failure state, which avoids a full restart of the simulation. The failure recovery scheme uses quantum Fourier transform (QFT) and effectively achieves quantum indefinite integration of an unknown quantum state. The recovery, which can itself fail, is more efficient than a full restart if the wave envelop is well-resolved to include only low Fourier modes. We emulate our scheme classically and demonstrate small problems on Quantinuum's trapped-ion qubits. Our quantum algorithm provides a subroutine for physics simulations that involve linear advection.
Comments24 pages, 8 figures, work presented at the 2025 APS Four Corners Meeting
