Polynomial Approximations for Quantum Hamiltonian Complexity

Abstract

Quantum Hamiltonian Complexity (QHC) sits at the intersection of quantum complexity theory and quantum many-body physics. However, many results in QHC still remain disconnected across domains. The technique of polynomial approximations, derived from classical approximation theory methods, has recently seen remarkable success at proving results in QHC, and may change this detached paradigm. This thesis aims to better unify results in QHC by applying polynomial approximations to provide an alternative proof of the Lieb-Schultz-Mattis (LSM) theorem, a seminal result in condensed matter physics. The LSM theorem provides an upper bound on the spectral gap of quantum many-body systems under certain symmetry conditions, and is deeply related to topological quantum computing. We hope to provide a more elementary proof of LSM based in classical techniques, such as concentration bounds, and demonstrate new applications of the polynomial approximation technique. In this work, we outline the general approach for using polynomial approximations to prove the LSM theorem for a 1D periodic spin-1/2 chain, and characterize the valid ground states for this system. We also provide progress towards this end in the form of examples of unconcentrated LSM ground states, as well as discussing the uniform superposition ground state and alternate perspectives on a full proof or counterexample.