Large integer factorization is a prominent research challenge, particularly in the context of quantum computing. This holds significant importance, especially in information security that relies on public key cryptosystems. The classical computation of prime factors for an integer has exponential time complexity. Quantum computing offers the potential for significantly faster computational processes compared to classical processors. In this paper, we propose a new quantum algorithm, Shallow Depth Factoring (SDF), to factor a biprime integer. SDF consists of three steps. First, it converts a factoring problem to an optimization problem without an objective function. Then, it uses a Quantum Feasibility Labeling (QFL) method to label every possible solution according to whether it is feasible or infeasible for the optimization problem. Finally, it employs the Variational Quantum Search (VQS) to find all feasible solutions. The SDF utilizes shallow-depth quantum circuits for efficient factorization, with the circuit depth scaling linearly as the integer to be factorized increases. Through minimizing the number of gates in the circuit, the algorithm enhances feasibility and reduces vulnerability to errors.

翻译：大整数因式分解是一个突出的研究挑战，特别是在量子计算领域。这对依赖公钥密码系统的信息安全具有重要意义。经典计算对一个整数进行质因数分解具有指数级的时间复杂度。相比之下，量子计算有望实现比经典处理器快得多的计算速度。本文提出一种新的量子算法——浅层深度因式分解（SDF），用于分解双素数整数。SDF包含三个步骤：首先，将因式分解问题转化为无目标函数的优化问题；其次，使用量子可行性标注（QFL）方法，根据每个可行解对优化问题的可行性进行标注；最后，采用变分量子搜索（VQS）找出所有可行解。SDF利用浅层量子电路实现高效因式分解，电路深度随待分解整数增加而线性扩展。通过最小化电路中的门数量，该算法增强了可行性并降低了对错误的敏感性。