Distinguishing unknown non-orthogonal qubits is an essential requirement for addressing various challenges in quantum computation, including quantum machine learning, quantum communications, and quantum technologies. For instance, while quantum teleportation enables the transfer of unknown individual qubits between distant parties, an algorithm is necessary to define the associated state of a teleported qubit at the receiving end. In this paper, we propose a novel quantum algorithm designed to effectively determine the state of a given unknown qubit and distinguish a subset of non-orthogonal qubits. The proposed algorithm can efficiently identify the state of an unknown qubit in the form cosθ2|0〉+sinθ2|1〉 using the Mz operator. By estimating the angle θ through the measurement of entanglement degree, the proposed algorithm can identify the state of an unknown qubit. Experimental validation of the proposed algorithm is conducted using the IBM quantum computer simulator chip ibmqx2. Furthermore, a t-test is conducted to compare the proposed algorithm with the direct measurement approach. The results indicate a significant difference between the two methods, demonstrating the superior performance of the proposed algorithm.