Dissertation: Cameron Calcuth

Abstract

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.

Prior to the work conducted in this thesis, theorems of classical simulatability of continuous-variable quantum computations relied on positive phase-space representations of all circuit components. Circuits confined to Gaussian elements or those preserving positive Wigner functions are efficiently simulatable, whereas introducing Wigner‐negative resources, which indicate non-classical behaviour, is necessary to achieve universality. Although necessary, Wigner negativity does not provide a sufficient condition to achieve universal quantum computation.

In this thesis, 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.

The implications of these results extend to a new fundamental understanding of the computational power of continuous-variable quantum computers. Specifically, we demonstrate the first sufficient condition for achieving universality using continuous-variable devices. These results shine a light on the limits of our current understanding while also paving the way for further exploration of fundamental topics in quantum computing.