Low-overhead Code Concatenation Approaches for Universal Quantum Computation.

Fault-tolerant quantum circuit design can be done by using a set of transversal gates. However, as there is no quantum error correction code with a universal set of transversal gates, several approaches have been proposed which, in combination of transversal gates, make universal fault-tolerant quantum computation possible. Magic state distillation, gauge fixing, code switching, code concatenation and pieceable fault tolerance are well-known examples of such approaches. However, the overhead of these approaches is one of the main bottlenecks for large-scale quantum computation. In this paper, two approaches for universal fault-tolerant quantum computation, mainly based on code concatenation, are proposed. The first approach combines the code concatenation approach with code switching, pieceable fault tolerance or magic state distillation and the second approach extends the nonuniformity of the concatenated codes by allowing to apply CNOT gates between different codes. The proposed approaches outperform the code concatenation approach in terms of both number of qubits and code distance and have also significantly less resource overhead than magic state distillation. This is achieved at the cost of reducing the effective distance of the concatenated code for implementing non-transversal gates.