A robust phase of continuous transversal gates in quantum stabilizer codes

A quantum error correcting code protects encoded logical information against errors. Transversal gates are a naturally fault-tolerant way to manipulate logical qubits but cannot be universal themselves. Protocols such as magic state distillation are needed to achieve universality via measurements and postselection. A phase is a region of parameter space with smoothly varying large-scale statistical properties except at its boundaries. Here, we find a phase of continuously tunable logical unitaries for the surface code implemented by transversal operations and decoding that is robust against dephasing errors. The logical unitaries in this phase have an infidelity that is exponentially suppressed in the code distance compared to their rotation angles. We exploit this to design a simple fault-tolerant protocol for continuous-angle logical rotations. This lowers the overhead for applications requiring many small-angle rotations such as quantum simulation.