广义可分量子码之塔
Towers of generalized divisible quantum codes
摘要：
可分二进制经典码是其中每个码字具有可被固定整数整除的权重的码。如果除数是$2^ν$，对于正整数$ν$，那么可以构造Calderbank-Shor-Steane（CSS）码，其中$X$稳定器空间是可分的经典码，其在Clifford层次的$ν$层中允许横向门。我们考虑一个推广的整除性，允许一个系数向量的奇整数，每个码字有零点积模的除数。在这种广义上，我们构造了一个CSS码的除数$2^{ν+1}$和代码距离$d$从任何CSS码的代码距离$d$和除数$2^ν$，其中的横向$X$是一个非平凡的逻辑运算符。新码的编码速率比旧码的编码速率大约小$d$倍。特别是，对于大的$d$和$ν\ge 2$，我们的建设产生了一个CSS代码的参数$[[O(d^{ν-1}), Ω(d),d]]$承认一个横向门在$ν$ -th级的Clifford层次结构。对于我们的建设，我们介绍了转换魔术状态蒸馏协议的基础上Clifford测量的基础上的代码与横向$T$门。我们的塔包含，作为一个子类，广义三重甚至CSS代码，出现在所谓的规范固定或代码切换方法。
英文摘要：
A divisible binary classical code is one in which every code word has weight divisible by a fixed integer. If the divisor is $2^ν$ for a positive integer $ν$, then one can construct a Calderbank-Shor-Steane (CSS) code, where $X$-stabilizer space is the divisible classical code, that admits a transversal gate in the $ν$-th level of Clifford hierarchy. We consider a generalization of the divisibility by allowing a coefficient vector of odd integers with which every code word has zero dot product modulo the divisor. In this generalized sense, we construct a CSS code with divisor $2^{ν+1}$ and code distance $d$ from any CSS code of code distance $d$ and divisor $2^ν$ where the transversal $X$ is a nontrivial logical operator. The encoding rate of the new code is approximately $d$ times smaller than that of the old code. In particular, for large $d$ and $ν\ge 2$, our construction yields a CSS code of parameters $[[O(d^{ν-1}), Ω(d),d]]$ admitting a transversal gate at the $ν$-th level of Clifford hierarchy. For our construction we introduce a conversion from magic state distillation protocols based on Clifford measurements to those based on codes with transversal $T$-gates. Our tower contains, as a subclass, generalized triply even CSS codes that have appeared in so-called gauge fixing or code switching methods.
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