  In this work, we have introduced two innovative quantum algorithms: the
Direct Column Algorithm and the Quantum Backtracking Algorithm to solve
N-Queens problem, which involves the arrangement of $N$ queens on an $N \times
N$ chessboard such that they are not under attack from each other on the same
row, column and diagonal. These algorithms utilizes Controlled W-states and
dynamic circuits, to efficiently address this NP-Complete computational
problem. The Direct Column Algorithm strategically reduces the search space,
simplifying the solution process, even with exponential circuit complexity as
the problem size grows, while Quantum Backtracking Algorithm emulates classical
backtracking techniques within a quantum framework which allows the possibility
of solving complex problems like satellite communication, routing and VLSI
testing.
