From simulatability to universality of continuous-variable quantum computers

(Abridged.) Quantum computers promise to solve some problems exponentially faster than traditional computers, but we still do not fully understand why this is the case. While the most studied model of quantum computation uses qubits, which are the quantum equivalent of a classical bit, an alternative method for building quantum computers is gaining traction. Continuous-variable devices, with their infinite range of measurement outcomes, use systems such as electromagnetic fields. Given this infinite-dimensional structure, combined with the complexities of quantum physics, we are left with a natural question: when are continuous-variable quantum computers more powerful than classical devices? This thesis investigates this question by exploring the boundary of which circuits are classically simulatable and which unlock a quantum advantage over classical computers. A series of proofs are presented demonstrating the efficient simulatability of progressively more complex circuits, even those with high amounts of Wigner negativity. Specifically, circuits initiated with highly Wigner-negative Gottesman-Kitaev-Preskill states, which form a grid-like structure in phase space, can be simulated in polynomial time.