The XP Stabilizer Formalism

Abstract

Quantum computers are expected to have advantages over classical computers in solving a range of high impact problems, but they are highly susceptible to errors due to environmental noise. The Pauli Stabiliser formalism generalises classical error-correction methods and makes use of quantum error correction codes to protect quantum information. In this thesis, we introduce the XP stabiliser formalism, which is a generalisation of the Pauli stabiliser formalism with a number of useful applications. Quantum algorithms are typically written in terms of quantum circuits which involve a series of unitary gates followed by measurements which form the output of the computation. To implement quantum algorithms reliably, we need to perform unitary gates fault-tolerantly so that errors do not propagate in an uncontrolled way. Transversal logical operators are one way of applying unitary gates fault-tolerantly on Pauli stabiliser codes. Identifying transversal logical operators for a given Pauli stabiliser code is challenging, and existing methods have exponential complexity in one or more of the parameters of the code. Making use of the XP formalism, we present efficient algorithms which identify all transversal logical operators that are diagonal in the computational basis for any Pauli stabiliser code. We also show how to construct codes with a transversal implementation of any desired diagonal logical operator. The Pauli stabiliser formalism can also be used to efficiently represent certain quantum states, but many states of interest lie outside the formalism. In the XP formalism, a wider range of states can be represented than in the Pauli stabiliser formalism, including hypergraph states which have interesting non-local properties. The braiding of non-Abelian anyons is a proposed pathway to universal fault-tolerant quantum computation. Certain XP stabiliser codes are known to harbour non-Abelian anyons, and can be studied within the new formalism.