Negative Quasi-Probability in the Context of Quantum Computation

This thesis deals with the question of what resources are necessary and sufficient for quantum computational speedup. In particular, we study what resources are required to promote fault tolerant stabilizer computation to universal quantum computation. In this context we discover a remarkable connection between the possibility of quantum computational speedup and negativity in the discrete Wigner function, which is a particular distinguished quasi-probability representation for quantum theory. This connection allows us to establish a number of important results related to magic state computation, an important model for fault tolerant quantum computation using stabilizer operations supplemented by the ability to prepare noisy non-stabilizer ancilla states. In particular, we resolve in the negative the open problem of whether every non-stabilizer resource suffices to promote computation with stabilizer operations to universal quantum computation. Moreover, by casting magic state computation as resource theory we are able to quantify how useful ancilla resource states are for quantum computation, which allows us to give bounds on the required resources. In this context we discover that the sum of the negative entries of the discrete Wigner representation of a state is a measure of its usefulness for quantum computation. This gives a precise, quantitative meaning to the negativity of a quasi-probability representation, thereby resolving the 80 year debate as to whether this quantity is a meaningful indicator of quantum behaviour. We believe that the techniques we develop here will be widely applicable in quantum theory, particularly in the context of resource theories.