Continuously tunable surface code logical gates via syndrome-adaptive transversal operations

Coherent single-qubit Z rotations such as T-gates feature in prominent quantum algorithms and subroutines such as the quantum Fourier transform. We propose an approximate protocol for realizing coherent rotations on the logical qubits encoded in a surface code using only coherent rotations on the physical qubits and syndrome measurements. Our protocol reduces the need for ancilla qubits in implementing non-Clifford gates as compared to magic-state distillation protocols. As a result, it holds promise to dramatically reduce the qubit overhead in useful quantum algorithms such as Shor's algorithm.