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.

翻译：量子计算模拟器在经典硬件上模拟量子算法，是可扩展量子计算机广泛应用前测试量子算法的关键平台。量子计算模拟面临的核心挑战在于管理有限算术精度带来的数值误差，特别是资源高效的定点算术中的截断误差。尽管这一问题至关重要，目前量化截断误差如何影响量子算法精度的系统性研究仍较为有限。本文提出一个严格的定量分析框架，研究定点量子计算模拟中的截断误差传播问题，重点关注Grover量子搜索算法。首先，我们引入Grover迭代过程中量子态的简化双值振幅表示方法，并证明其理论有效性。基于该表示，我们推导出刻画量子门操作间截断误差累积的显式数学表达式。通过理想概率分布与模拟概率分布之间的$\ell_2$距离量化总体模拟误差，获得按$O(2^{n-f})$比例变化的渐近误差界，其中$n$为量子比特数，$f$为分数比特精度。在实际定点量子计算模拟器上进行的大量数值模拟与实证实验证实，观测误差与我们的理论预测精确吻合。最后，我们给出确定实现特定误差阈值所需最小分数比特精度的闭式公式，为平衡精度与资源利用的模拟器设计者提供明确指导。