Explorations in Quantum Error Correction and Simulation of Topological Phases

Abstract

Much excitement has arisen at the intersection between the field of quantum information theory with more traditional disciplines in physics such as high-energy and quantum condensed matter. This thesis consists of two independent explorations that are unified by the common language of quantum information: quantum error correction in holography and entanglement in topological phases of matter. The first part develops a fully algebraic formulation of quantum error correction tailored to the AdS/CFT correspondence. Building upon earlier work, it removes the assumption of a factorizable boundary Hilbert space, making the framework applicable to gauge theories such as $N=4$ super-symmetric Yang–Mills. This generalized framework yields a fully algebraic interpretation of the entanglement wedge reconstruction property in AdS/CFT as well as the Ryu–Takayanagi formula, and establishes an equivalence between the two. The second part of this thesis focuses on the exploration of interacting topological phases in strongly correlated lattice models. We begin by considering the setting of hardcore bosonic particles, and demonstrate that periodic driving naturally enables the realization of correlated hopping interactions that emulate flux attachment. Using large-scale density matrix renormalization group computations, we characterize the ground state of the resulting correlated hopping models on both the square and honeycomb lattice, finding bosonic integer and fractional quantum Hall states, respectively. Motivated by the possibility of adiabatically preparing such topological phases in cold atomic experiments, we map out the nearby phase diagram surrounding the bosonic integer quantum Hall state and propose an experimental implementation based upon laser-assisted tunneling of neutral atoms in a two-dimensional optical lattice. Finally, we turn to a classic, spin-model setting for exploring topological order: the nearest-neighbor spin-1/2 Heisenberg antiferromagnet on the Kagome lattice. We present a reexamination of the low-lying energy spectrum of this model using neural quantum states—a recently developed numerical method leveraging neural networks as a variational ansatz. While this method holds promise for simulating complex quantum systems, we highlight its limitations and potential pitfalls, emphasizing the importance of careful implementation and interpretation.