Mixed Quantum-Semiclassical Simulation

We study the quantum simulation of mixed quantum-semiclassical (MQS) systems, of fundamental interest in many areas of physics, such as molecular scattering and gravitational backreaction. A basic question for these systems is whether quantum algorithms of MQS systems would be valuable at all, when one could instead study the full quantum-quantum system. We study MQS simulations in the context where a semiclassical system is encoded in a Koopman-von Neumann (KvN) Hamiltonian and a standard quantum Hamiltonian describes the quantum system. In this case, because KvN and quantum Hamiltonians are constructed with the same operators on a Hilbert space, standard theorems guaranteeing simulation efficiency apply. We show that, in this context, $\textit{many-body}$ MQS particle simulations give only nominal improvements in qubit resources over quantum-quantum simulations due to logarithmic scaling in the ratio, $S_q/S_c$, of actions between quantum and semiclassical systems. However, $\textit{field}$ simulations can give improvements proportional to the ratio of quantum to semiclassical actions, $S_q/S_c$. Of particular note, due to the ratio $S_q/S_c \sim 10^{-18}$ of particle and gravitational fields, this approach could be important for semiclassical gravity. We demonstrate our approach in a model of gravitational interaction, where a harmonic oscillator mediates the interaction between two spins. In particular, we demonstrate a lack of distillable entanglement generation between spins due to classical mediators, a distinct difference in dynamics relative to the fully quantum case.