Topological order and the low-energy subspace

Abstract

This thesis concerns itself with two fundamental tasks in the theory of quantum information: the preparation of ground and low-energy states of local Hamiltonians on a quantum computer and the secure transmission of classical information over a quantum channel. The first part of this thesis addresses the former task and is dedicated in its entirety to quantum many-body physics. We will be focusing on topologically ordered lattice models in two spatial dimensions. It is known that the presence of long-range entanglement in the ground states of these models necessitates local quantum circuits of depth linear in the diameter of the system for their preparation. We study the consequences of this hardness result for the preparation of low-energy states of these models. In particular we show that preparing such states requires local quantum circuits of polynomial depth in the inverse energy density of the states and comment on the relevance of these results to quantum computation in the near term. Our technical contribution is to give a circuit depth lower bound for low-energy states independent of the dimension of the ground space. We then experimentally demonstrate how currently existing ion trap hardware has reached a level of maturity where noisy adaptive circuits of constant depth outperform noiseless, non-adaptive circuits of the same depth for the task of approximating a ground state of the the toric code. In the last part of this thesis, we switch gears to quantum Shannon theory and study private communication over a classical-quantum wiretap channel. We study the problem of non-additivity of the private information for this model. Surprisingly, we find that the private information is non-additive when either of the outputs of the channel is allowed to be quantum, while the input is classical. However, there exists a single-letter formula when the input state is quantum and the outputs are classical due to the private information of the channel becoming additive.