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The overhead cost of performing universal fault-tolerant quantum computationfor large scale quantum algorithms is very high. Despite several attempts atalternative schemes, magic state distillation remains one of the most efficientschemes for simulating non-Clifford gates in a fault-tolerant way. However,since magic state distillation circuits are not fault-tolerant, all Cliffordoperations must be encoded in a large distance code in order to have comparablefailure rates with the magic states being distilled. In this work, we introducea new concept which we call redundant ancilla encoding. The latter combinedwith flag qubits allows for circuits to both measure stabilizer generators ofsome code, while also being able to measure global operators tofault-tolerantly prepare magic states, all using nearest neighbor interactions.In particular, we apply such schemes to a planar architecture of the triangularcolor code family. In addition to our scheme being suitable for experimentalimplementations, we show that for physical error rates near $10^{-4}$ and undera full circuit-level noise model, our scheme can produce magic states using anorder of magnitude fewer qubits and space-time overhead compared to the mostcompetitive magic state distillation schemes. Further, we can take advantage ofthe fault-tolerance of our circuits to produce magic states with very lowlogical failure rates using encoded Clifford gates with noise rates comparableto the magic states being injected. Thus, stabilizer operations are notrequired to be encoded in a very large distance code. Consequently, we believeour scheme to be suitable for implementing fault-tolerant universal quantumcomputation with hardware currently under development.

Very low overhead fault-tolerant magic state preparation using redundant ancilla encoding and flag qubits