Hastings Matthew B, Haah Jeongwan
Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA.
Station Q Quantum Architectures and Computation, Microsoft Research, Redmond, Washington 98052, USA.
Phys Rev Lett. 2018 Feb 2;120(5):050504. doi: 10.1103/PhysRevLett.120.050504.
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 ε is Ω[log(1/ε)]. We show that this conjecture is false. We find a family of quantum error correcting codes of parameters ⟦∑[under i=w+1]over m,∑[under i=0]over w,∑[under i=w+1]over r+1⟧ for any integers m>2r, r>w≥0, by puncturing quantum Reed-Muller codes. When m>νr, our code admits a transversal logical gate at the νth level 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(log^{γ}(1/ε)), where γ=log(n/k)/log(d)<0.678 for some m, r, w. The smallest code in our family for which γ<1 is on ≈2^{58} qubits.
据推测，对于用于T门的任意魔态蒸馏协议，在输出错误率为ε时，每个输出魔态所需的有噪声输入魔态数量为Ω[log(1/ε)]。我们证明了这个推测是错误的。通过打孔量子里德 - 穆勒码，我们找到了一族参数为⟦∑[下标i = w + 1][上标m](m / i), ∑[下标i = 0][上标w](m / i), ∑[下标i = w + 1][上标r + 1](r + 1 / i)⟧的量子纠错码，其中m > 2r，r > w ≥ 0为任意整数。当m > νr时，我们的码在克利福德层级的第ν级允许一个横向逻辑门。在ν = 3（T门）级别的魔态蒸馏协议中，输入魔态与输出魔态的比例为O(log^{γ}(1/ε))，其中对于某些m、r、w，γ = log(n / k) / log(d) < 0.678。我们族中γ < 1的最小码约有2^{58}个量子比特。
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