  This thesis is split into two parts, which are united in the sense that they
involve applying ideas from quantum information to fundamental physics.
  The first part is focused on examining discrete-time models in quantum
computation (discrete-time quantum walks and quantum cellular automata) as
discretized models of relativistic systems. One of the results here is a
theorem demonstrating that a large class of discrete-time quantum walks have
relativistic dynamics in the continuum limit. Additionally, the problem of
fermion doubling for these models is investigated, and it is seen that the
problem can be circumvented in two dimensional space. This was already known
for one dimensional systems.
  Another result involves taking the limits of causal free field theories in
discrete spacetime to recover continuum field theories, something that is not
straightforward because of the nontrivial nature of the vacuum in quantum field
theory. Additionally, it is shown that general systems of fermions evolving
causally in discrete spacetime can be represented by quantum cellular automata,
which makes them efficiently simulable by quantum computers. A related result
is that quantum cellular automata composed of fermions are equivalent to
regular quantum cellular automata.
  In the second part of this thesis, the focus is on the foundations of
statistical physics. The main result of this part is a general bound on the
time it takes a quantum system to effectively reach equilibrium. The discussion
also includes a practical definition of equilibration that takes our
measurement capabilities into account. Finally, the nature of the equilibrium
state is also discussed, with a focus on initial state independence, which
relates to the important question of when the equilibrium state is a Gibbs
state.
