We propose a variational algorithm to study the real time dynamics of quantum systems as a ground-state problem on quantum devices. The method is based on the original proposal of Feynman and Kitaev to encode time into a register of auxiliary qubits. We prepare the Feynman-Kitaev Hamiltonian acting on the composed system as a qubit operator and find an approximate ground state using the Variational Quantum Eigensolver. We apply the algorithm to the study of the dynamics of a transverse field Ising chain with an increasing number of spins and time steps, proving a favorable scaling in terms of the number of two qubit gates. Through numerical experiments, we investigate its robustness against hardware noise, showing that the method can be use to evaluate dynamical properties of quantum systems and detect the presence of dynamical quantum phase transitions by measuring Loschmidt echoes. The scripts provided implement the algorithm both in Python, using the Qiskit library, and in Julia, using the Yao.jl package. Moreover, we included the data to reproduce the figures of the manuscript without running the simulations.