Quantum Simulation of Noncausal Kinematic Transformations

We propose the implementation of Galileo group symmetry operations or, in general, linear coordinate transformations in a quantum simulator. With an appropriate encoding, unitary gates applied to our quantum system give rise to Galilean boosts or spatial and time parity operations in the simulated dynamics. This framework provides us with a flexible toolbox that enhances the versatility of quantum simulation theory, allowing the direct access to dynamical quantities that would otherwise require full tomography. Furthermore, this method enables the study of noncausal kinematics and phenomena beyond special relativity in a quantum controllable system.

Figure 1

Scheme of gates acting on the active (colored circles) and control (white circles) qubits, implementable in different quantum platforms. In this example, we depict a quantum simulation process in which two or more (spacetime nonlocal) kinematic transformations ${\Pi }_x,{\Pi }_y,\dots$, have been performed at an intermediate time of the simulated dynamics. The first and fourth boxes (starting from the right) are used to generate the dynamics given by an evolution operator $U_t$. The small boxes represent local gates $U_{c1,c2}$ acting on the control qubits.