基于卷积定理的量子乘法算法
Quantum Multiplication Algorithm Based on the Convolution Theorem
AI总结：
针对经典大数乘法算法接近理论极限的问题，受卷积定理和量子振幅放大启发，提出时间复杂度为$O(\sqrt{n}\log^2 n)$的量子整数乘法算法，性能优于Harvey算法且适用范围更广，为该基础问题提供量子视角。
摘要：
大数高效乘法问题一直是经典计算领域的长期挑战，数百年来已得到广泛研究。现有经典算法似乎已接近其理论极限，几乎没有进一步提升的空间。然而，随着量子计算机的出现，以及对可在量子硬件上执行乘法运算的量子算法的需求，一种新的范式应运而生。本文受卷积定理和量子振幅放大范式的启发，提出了一种用于整数乘法的量子算法，其时间复杂度为$O(\sqrt{n}\log^2 n)$，性能优于目前已知的最优经典算法——时间复杂度为$O(n \log n)$的Harvey算法。与Harvey算法不同，我们的算法不存在仅适用于极大数的限制，因此可灵活应用于各类整数乘法任务。本文还回顾了经典乘法算法的历史与发展，促使我们探索量子资源如何为这一基础问题提供新的视角与可能性。
英文摘要：
The problem of efficient multiplication of large numbers has been a long-standing challenge in classical computation and has been extensively studied for centuries. It appears that the existing classical algorithms are close to their theoretical limit and offer little room for further enhancement. However, with the advent of quantum computers and the need for quantum algorithms that can perform multiplication on quantum hardware, a new paradigm emerges. In this paper, inspired by convolution theorem and quantum amplitude amplification paradigm we propose a quantum algorithms for integer multiplication with time complexity $O(\sqrt{n}\log^2 n)$ which outperforms the best-known classical algorithm, the Harvey algorithm with time complexity of $O(n \log n)$. Unlike the Harvey algorithm, our algorithm does not have the restriction of being applicable solely to extremely large numbers, making it a versatile choice for a wide range of integer multiplication tasks. The paper also reviews the history and development of classical multiplication algorithms and motivates us to explore how quantum resources can provide new perspectives and possibilities for this fundamental problem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
