  In this paper, we present a new algorithm for generic combinatorial
optimization, which we term quantum dueling. Traditionally, potential solutions
to the given optimization problems were encoded in a ``register'' of qubits.
Various techniques are used to increase the probability of finding the best
solution upon measurement. Quantum dueling innovates by integrating an
additional qubit register, effectively creating a ``dueling'' scenario where
two sets of solutions compete. This dual-register setup allows for a dynamic
amplification process: in each iteration, one register is designated as the
'opponent', against which the other register's more favorable solutions are
enhanced through a controlled quantum search. This iterative process gradually
steers the quantum state within both registers toward the optimal solution.
With a quantitative contraction for the evolution of the state vector,
classical simulation under a broad range of scenarios and hyper-parameter
selection schemes shows that a quadratic speedup is achieved, which is further
tested in more real-world situations. In addition, quantum dueling can be
generalized to incorporate arbitrary quantum search techniques and as a quantum
subroutine within a higher-level algorithm. Our work demonstrates that
increasing the number of qubits allows the development of previously
unthought-of algorithms, paving the way for advancement of efficient quantum
algorithm design.
