Quantum Simulation of Klein Gordon Equation and Observation of Klein Paradox in IBM Quantum Computer

The Klein Gordon equation was the first attempt at unifying special relativity and quantum mechanics. While initially discarded this equation of "many fathers" can be used in understanding spinless particles that consequently led to the discovery of pions and other subatomic particles. The equation leads to the development of Dirac equation and hence quantum field theory. It shows interesting quantum relativistic phenomena like Klein Paradox and "Zitterbewegung", a rapid vibrating movement of quantum relativistic particles. The simulation of such quantum equations initially motivated Feynman to propose the idea of quantum computation. While many such simulations have been done till date in various physical setups, this is the first time a digital quantum simulation of Klein Gordon equation is proposed on IBM's quantum computer. Here we simulate the time-dependent Klein Gordon equation in a barrier potential and clearly observe the tunnelling of the particle and anti-particle through a strong potential claiming Klein Paradox. The simulation technique used here inspires the quantum computing community for further studying Klein Gordon equation and applying it to more complicated quantum mechanical systems.