Explorations of the Similarity Renormalization Group on Quantum Computers

Renormalization Group (RG) methods have served as a powerful tool for increasing the tractability of the Nuclear-Many Body Problem. By applying continuous changes in resolution to our nuclear interactions, RG methods decouple low-energy and high-energy degrees of freedom while preserving low-energy observables. This allows for an effective low-energy description. Here we adapt the Similarity Renormalization Group (SRG), which applies continuous unitary transformations that band-diagonalize a nuclear many-body hamiltonian, to quantum computers. As proof of concept we demonstrate the SRG evolution of a nuclear hamiltonian with combined one-body, two-body, and three-body terms using the Double Bracket Quantum Algorithm. Furthermore, we explore SRG flows in systems that undergo quantum phase transitions, like the Lipkin-Meshov-Glick Model. We aim to reformulate the SRG to an RG evolution in phase space, via the Moyal double bracket equation. We seek to adapt the Double-Bracket Quantum Algorithm to accommodate the Moyal Double Bracket, to induce non-perturbative SRG flows of the Hamiltonian in phase space that expands operators to all orders in 1/ hbar. The goal is to fully capture the spectrum of the Hamiltonian, even at values of the potential that indicate a transition from single particle to collective degrees of freedom.