标题： 固定点 Grover 量子算法仿真的渐近误差界和分数位设计 显示英文标题

标题： Asymptotic Error Bounds and Fractional-Bit Design for Fixed-Point Grover's Quantum Algorithm Emulation

摘要： 量子计算（QC）仿真器是在经典硬件上模拟量子算法的不可或缺的平台，在可扩展的量子计算机变得广泛可用之前尤其如此。 量子计算仿真中的一个关键挑战是管理数值误差，特别是有限算术精度导致的截断误差，尤其是在资源高效的定点算术中。 尽管其重要性，系统地量化截断误差如何影响量子算法准确性的研究却很有限。 本文提出了一种严格的定量框架，用于分析定点量子计算仿真中的截断误差传播，重点研究了Grover量子搜索算法。 首先，我们在Grover迭代过程中引入量子态的简化二值振幅表示，并证明其理论有效性。 利用这种表示方法，我们推导出明确的数学表达式，用以刻画截断误差在量子门操作中的累积情况。 我们通过理想概率分布与仿真概率分布之间的 $\ell_2$ 距离来量化整体仿真误差，得到的渐近界限随 $O(2^{n-f})$ 缩放，其中 $n$ 是量子比特的数量， $f$ 是分数位精度。 大量的数值模拟和在实际定点量子计算仿真器上的实验验证表明，观察到的误差与我们的理论预测完全一致。 最后，我们提供了一个闭式公式，用于确定实现指定误差阈值所需的最小分数位精度，为仿真器设计者在准确性与资源利用率之间取得平衡提供了明确指导。 显示英文摘要

摘要： Quantum computing (QC) emulators, which simulate quantum algorithms on classical hardware, are indispensable platforms for testing quantum algorithms before scalable quantum computers become widely available. A critical challenge in QC emulation is managing numerical errors from finite arithmetic precision, especially truncation errors in resource-efficient fixed-point arithmetic. Despite its importance, systematic studies quantifying how truncation errors impact quantum algorithm accuracy are limited. In this paper, we propose a rigorous quantitative framework analyzing truncation error propagation in fixed-point QC emulation, focusing on Grover's quantum search algorithm. First, we introduce a simplified two-value amplitude representation of quantum states during Grover's iterations and prove its theoretical validity. Using this representation, we derive explicit mathematical expressions characterizing truncation error accumulation across quantum gate operations. We quantify the overall emulation error by the $\ell_2$ distance between ideal and emulated probability distributions, obtaining asymptotic bounds scaling as $O(2^{n-f})$, where $n$ is the number of qubits and $f$ is fractional-bit precision. Extensive numerical simulations and empirical experiments on a practical fixed-point QC emulator confirm that observed errors precisely match our theoretical predictions. Finally, we provide a closed-form formula to determine the minimal fractional-bit precision required to achieve a specified error threshold, offering clear guidelines for emulator designers balancing accuracy and resource utilization.