Path integral and qubit encoding techniques for quantum simulations of discrete planar rotor lattices

Abstract

Typical path integral Monte Carlo approaches use the primitive approximation to compute the probability density for a given path. In this thesis, we investigate the utility of pair approximating the action in path integral ground state simulations targeting planar rotations. The pair propagator, which was initially introduced to study superfluidity in condensed Helium, is naturally well-suited for systems interacting with a pair-wise potential. Consequently, paths sampled using the pair action tend to be closer to the exact paths (compared to primitive Trotter paths) for such systems leading to convergence with less imaginary time steps. Our approach relies on using the pair factorization in conjunction with a rejection-free path integral ground state paradigm to study a chain of planar rotors interacting with a pair-wise dipole-dipole interaction. We first use a heat kernel expansion to analyze the asymptotics of the pair propagator in imaginary time. Then, we exhibit the utility of the pair factorization scheme via convergence studies comparing the pair and primitive propagators. Finally, we compute energetic and structural properties of this system including the orientational correlation and Binder ratio as functions of the coupling strength to examine the behavior of the pair-DVR method near criticality. Density matrix renormalization group calculations are used for benchmarking throughout. Near term quantum devices have recently garnered significant interest as promising candidates for investigating difficult-to-probe regimes in many-body physics. To this end, various qubit encoding schemes targeting second quantized Hamiltonians have been proposed and optimized. In this thesis, we also investigate two qubit representations of the planar rotor lattice Hamiltonian. The first representation is realized by decomposing the rotor Hamiltonian projectors in binary and mapping them to spin-1/2 projectors. The second approach relies on embedding the planar rotor lattice Hilbert space in a larger space and recovering the relevant qubit encoded system as a quotient space projecting down to the physical degrees of freedom. This is typically called the unary mapping and is used for bosonic systems. We establish the veracity of the two encoding approaches using sparse diagonalization on small chains and discuss quantum phase estimation resource requirements to simulate small planar rotor lattices on near-term quantum devices.