General quantum algorithm for solving integer programming using qudits.

Integer programming (IP) is an optimization problem that involves discrete variables and constraints. It is commonly used to formulate real-world problems but can be challenging to solve using classical methods. Most existing quantum algorithms for solving IP rely on qubits, mapping the problem into a quadratic unconstrained binary optimization form. In our work, we develop a measurement-based quantum algorithm inspired by Simon's period-finding algorithm and our atom-based approach to solving the IP problem. This algorithm uses a discrete multi-level quantum system, known as a qudit, to encode the values of the variables. To implement the constraints, we apply a series of entangling unitary operators that flip ancillary qubits when a constraint is satisfied. After this process, we measure the qubits, which collapse the state of the system into a superposition of the basis states that satisfy the constraints. Furthermore, the cost function for the problem is encoded in the phase of the quantum state. By maximizing this phase, we can find the solution to the optimization problem.