Non-local fault-tolerance and quasiprobabilty models in quantum computing

Abstract

In this thesis we explore two topics related to quantum computing. In the first part of this thesis we explore how geometrically non-local quantum low-density parity-check (LDPC) codes can be utilised to enable fault-tolerant quantum computing with reduced physical overheads. We design a scheme to implement any Clifford gate via measurements of logical Pauli operators on quantum LDPC codes. When supplemented with magic states this is sufficient for performing universal quantum computing. We prove that our scheme preserves the distance of the code and show that low overheads are maintained. We then show how we can construct quantum LDPC codes that are tailored to account for biased depolarising noise. We conduct numerical simulations using the BP-OSD decoder, and showing favorable scaling compared to non-tailored quantum LDPC codes and tailored surface codes. Overall, we provide evidence that fault-tolerant quantum computing with LDPC codes has the potential to reduce overheads significantly when compared to conventional surface code schemes. In the second part of this thesis we explore quasiprobability representations for quantum computation in the magic state model. We define a new model based on generalized Jordan-Wigner transformations that has a close connection to the probability representation of universal quantum computation based on the $\Lambda$ polytopes. It leads to an efficient classical simulation algorithm for magic state quantum circuits for which the input state is positively represented, and it outperforms previous representations in terms of the states that can be positively represented. For each number of qubits the model defines a polytope contained in the $\Lambda$ polytope with some shared vertices. We characterise these vertices, which by polar duality also allows us to characterise new facets of the stabiliser polytope.