Distillation with Sublogarithmic Overhead

Matthew B. Hastings, Jeongwan Haah
2018 Physical Review Letters  
It has been conjectured [1] 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 ϵ is Ω((1/ϵ)). We show that this conjecture is false. We find a family of quantum error correcting codes of parameters [[∑_i=w+1^m mi, ∑_i=0^wmi, ∑_i=w+1^r+1r+1i]] for any integers m > 2r, r > w > 0, by puncturing quantum Reed-Muller codes. When m > ν r, our code admits a transversal logical gate at the ν-th
more » ... vel of Clifford hierarchy. In a distillation protocol for magic states at the level ν = 3 (T-gate), the ratio of input to output magic states is O(^γ (1/ϵ)) where γ = (n/k)/(d)< 0.678 for some m,r,w. The smallest code in our family for which γ < 1 is on ≈ 2^58 qubits.
doi:10.1103/physrevlett.120.050504 pmid:29481193 fatcat:czwlroh3wrgfvjfxqh5y2g7sxy