The Topological Properties of Fault-Tolerant Circuits

The design of time-independent local Hamiltonians that realise quantum algorithms is derived from the study of perfect state transfer. The novel features of this evolution are the perfect realisation of the computation, and the ability to implement operations in parallel rather than sequentially. This is applied to a quantum circuit for concatenated error correcting codes, and the properties of the eigenstates are compared to those of topological memories. An erroneous implementation of the algorithm (one where a single error could lead to a fault in the output) is found not to be topological, whereas a true fault-tolerant algorithm possesses self-correcting properties, suggesting the existence of a unifying structure for fault-tolerant memories.