Universal Fault-Tolerant Quantum Computing with Stabiliser Code Families

Scalable universal quantum computing requires fault-tolerant implementations of a universal set of logical operators. Several important results constraining this goal exist in specific contexts, along with particular methods for overcoming these constraints, but no broadly applicable framework has been developed. We address this by defining a general notion of fault tolerance of quantum channels on scalable quantum error-correcting code families. With this definition, we present a no-go theorem that precludes a universal set of unitary fault-tolerant logical operator implementations for a wide range of stabiliser code families, including concatenated codes and conventional topological stabiliser codes such as surface and colour codes. Deriving this theorem also illuminates a general approach for how non-unitary channels can circumvent its constraints, which we show is manifested in a range of apparently distinct universal, fault-tolerant schemes.