Large integer factorization is a prominent research challenge, particularly in the context of quantum computing. The classical computation of prime factors for an integer entails exponential time complexity. Quantum computing offers the potential for significantly faster computational processes compared to classical processors. We proposed a new quantum algorithm, Shallow Depth Factoring (SDF), to factor an integer. SDF consists of three steps. First, it converts a factoring problem to an optimization problem without an objective function. Then, we use a Quantum Feasibility Labeling (QFL) to label every possible solution according to whether it is feasible or infeasible for the optimization problem. Finally, the Variational Quantum Search (VQS) is used to find all feasible solutions. The SDF algorithm 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：大整数分解是一个突出的研究挑战，特别是在量子计算领域。经典计算机计算一个整数的素因子需要指数时间复杂度，而量子计算相比经典处理器具有实现计算过程显著加速的潜力。我们提出了一种新的量子算法——浅层分解算法（SDF）——用于整数分解。SDF包含三个步骤：首先，它将分解问题转化为无目标函数的优化问题；其次，我们使用量子可行性标记（QFL）对每个可能解进行标记，判断其在优化问题中是否可行；最后，利用变分量子搜索（VQS）找出所有可行解。SDF算法采用浅层量子电路实现高效分解，电路深度随待分解整数规模线性增长。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误差的敏感度。