In recent years, topological phases of matter have presented exciting new avenues to achieve scalable quantum computation. In this thesis, we investigate a class of quantum many-body spin models known as symmetry-protected topological (SPT) phases for use in quantum information processing and storage. We explore the fault-tolerant properties of SPT phases, and how they can be utilized in the design of a quantum computer. Of central importance in this thesis is the concept of quantum error-correction, which in addition to its importance in fault-tolerant quantum computation, is used to characterise the stability of topological phases at finite temperature.
We begin with an introduction to quantum computation, quantum error correction, and topological phases of matter. We then focus on the fundamental question of whether symmetry-protected topological phases of matter can exist in thermal equilibrium; we prove that systems protected by global onsite symmetries cannot be ordered at nonzero temperature. Subsequently, we show that certain three-dimensional models with generalised higher-form symmetries can be thermally SPT ordered, and we relate this order to the ability to perform fault-tolerant measurement-based quantum computation. Following this, we assess feasibility of these phases as quantum memories, motivated by the fact that SPT phases in three dimensions can possess protected topological degrees of freedom on their boundary. We find that certain SPT ordered systems can be self-correcting, allowing quantum information to be stored for arbitrarily long times without requiring active error correction. Finally, we develop a framework to construct new schemes of fault-tolerant measurement-based quantum computation. As a notable example, we develop a cluster-state scheme that simulates the braiding and fusion of surface-code defects, offering novel alternative methods to achieve fault-tolerant universal quantum computation.