A new method for simulation of a binary homogeneous Markov process using a quantum computer was proposed. This new method allows using the distinguished properties of the quantum mechanical systems -- superposition, entanglement and probability calculations. Implementation of an algorithm based on this method requires the creation of a new quantum logic gate, which creates entangled state between two qubits. This is a two-qubit logic gate and it must perform a predefined rotation over the X-axis for the qubit that acts as a target, where the rotation accurately represents the transient probabilities for a given Markov process. This gate fires only when the control qubit is in state |1>. It is necessary to develop an algorithm, which uses the distribution for the transient probabilities of the process in a simple and intuitive way and then transform those into X-axis offsets. The creation of a quantum controlled n-th root of X gate using only the existing basic quantum logic gates at the available cloud platforms is possible, although the hardware devices are still too noisy, which results in a significant measurement error increase. The IBM's Yorktown 'bow-tie' back-end performs quite better than the 5-qubit T-shaped and the 14-qubit Melbourne quantum processors in terms of quantum fidelity. The simulation of the binary homogeneous Markov process on a real quantum processor gives best results on the Vigo and Yorktown (both 5-qubit) back-ends with Hellinger fidelity of near 0.82. The choice of the right quantum circuit, based on the available hardware (topology, size, timing properties), would be the approach for maximizing the fidelity.