  Quantum search algorithms, such as Grover's algorithm, are expected to
efficiently solve constrained combinatorial optimization problems. However,
implementing a quantum search algorithm for solving the traveling salesman
problem (TSP) on a circuit poses a potential challenge because current quantum
search algorithms for TSP assume that an initial state of equal superposition
of feasible solution states satisfying the constraint is already prepared a
priori. The time complexity of brute-force preparation of the initial state
increases exponentially with the factorial growth of feasible solutions, posing
a considerable obstacle in designing quantum circuits for large-scale TSP. To
overcome this problem, we propose a two-step quantum search algorithm with two
distinct operators for preparing the initial state and solving TSP. The
algorithm first amplifies an equal superposition state of all feasible
solutions of TSP and subsequently amplifies the optimal solution states among
these feasible solution states. Our algorithm, encoded in the higher-order
unconstrained binary optimization (HOBO) representation, notably reduces the
required number of qubits, enabling efficient preparation of the initial state
with a unified circuit design and solving TSP with a quadratic speedup in the
absence of prior knowledge of feasible solutions.
