Distillation with Sublogarithmic Overhead

It has been conjectured that, for any distillation protocol for magic states for the T gate, the number of noisy input magic states required per output magic state at output error rate epsilon is Omega[log(1/epsilon)]. We show that this conjecture is false. We find a family of quantum error correcting codes of parameters [[Sigma(m)(i=w vertical bar 1) ((m)(i)), Sigma(w)(i=0)((m)(i)), Sigma(r+1)(i=w vertical bar 1) ((r vertical bar 1)(i))]] for any integers m > 2r, r > w >= 0, by puncturing quantum Reed-Muller codes. When m > nu r, our code admits a transversal logical gate at the upsilon th level of Clifford hierarchy. In a distillation protocol for magic states at the level nu = 3 ( T gate), the ratio of input to output magic states is O(log(gamma)(1/epsilon)), where gamma = log(n/k)/log(d) < 0.678 for some m, r, w. The smallest code in our family for which gamma < 1 is on approximate to 2(58) qubits.