  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.
