Quantum computing platforms are expected to outperform their classical counterparts in solving certain technical problems such as factorization and searching process. With remarkable progresses in the field, a lot of efforts and attentions are being made to experimentally demonstrate such supremacy of quantum computation. As a primitive attempt, we have implemented simple and well-known two-qubit algorithms in our superconducting qubit system. Our system consists of two transmon qubits, one of which is a frequency-tunable qubit, embedded in a single copper cavity forming a circuit QED system. The entangling gate that is essential for implementing quantum algorithms has been realized by utilizing MAP (microwaveactivated phase) gate. We evaluate the performance of two-qubit processor by estimating the fidelity of the entangling gate with two-qubit quantum process tomography (QPT). Then we demonstrate the implementation of two-qubit algorithms such as Deutsch-Josza algorithm and Grover search.