Recently, several powerful machines dedicated to solving combinatorial optimization problems through the Ising-model formulation have appeared. The trigger for the paradigm shift to a specialized machine for solving optimization problems was the D-Wave machine, which implements quantum annealing. Quantum annealing employs quantum fluctuations to find an optimal solution to an optimization problem with discrete variables. In particular, we input the optimization problem in the form of the Ising Hamiltonian, which is a specialized form of the quadratic unconstrained binary optimization problem. However, when we employ quantum annealing for a practical optimization problem, there are several issues. One typical issue is absence of the detailed form of the cost function, which characterizes the optimization problem. To input problems into specialized machines for solving an optimization problem, it is necessary to determine the unknown parameters within the Ising model. We propose a method to estimate the unknown parameters in the Ising Hamiltonian using compressed sensing. Furthermore, we analyze the theoretical limitations of our proposed method by employing the replica method, which is a sophisticated tool in statistical mechanics.