Climbing the Diagonal Clifford Hierarchy
Climbing the Diagonal Clifford Hierarchy
复制标题
攀登对角克利福德层次结构
DOI：
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发表时间：
2021
期刊：
影响因子：
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通讯作者：
Robert Calderbank
中科院分区：
文献类型：
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作者：
Jingzhen Hu;Q. Liang;Robert Calderbank
幻态蒸馏和Shor因式分解算法本质上利用了逻辑对角门。我们介绍了一种在Clifford层次结构中实现某个级别的目标逻辑对角门的CSS码的综合方法。该方法结合了三个基本操作：连接、移除$Z$稳定器和添加$X$稳定器。它显式地跟踪由保留了css代码的对角物理门引起的逻辑门。第一步是连接，输入是一个css代码和一个位于$L$级别的物理对角门，在同一级别诱导一个逻辑对角门。输出是一个新的代码，对于该代码，位于$L+1$电平的物理对角门诱导出原始的逻辑门。下一步是明智地删除$Z$稳定器，以提高诱导逻辑运算符的水平。我们确定了三种将逻辑Clifford层次从$L$攀升到$L+1$的方法，每种方法都建立在诱导逻辑算子的泡利系数上的递归关系上。移除$Z$稳定器可能会缩短距离，而第三个基本操作的目的是添加$X$稳定器，以补偿此类损失。对于相干噪声模型，我们描述了如何通过简单地应用Pauli$X$矩阵在无消相干的子空间中在计算和存储中间结果之间切换。以往的逻辑门综合方法主要集中在码的状态，并给出了一个CSS码被横切$Z$-旋转固定的充分条件。相反，我们通过分析横向对角门对决定编码的稳定子群的作用，得到了充要条件。两个概念证明证明了该方法的有效性：$[[2^{L+1}-2，2，2]]$三正交码族和$[[2^m，inom{m}{r}，2^{min{r，m-r}}]]$量子Reed-Muller码族。
Magic state distillation and the Shor factoring algorithm make essential use of logical diagonal gates. We introduce a method of synthesizing CSS codes that realize a target logical diagonal gate at some level $l$ in the Clifford hierarchy. The method combines three basic operations: concatenation, removal of $Z$-stabilizers, and addition of $X$-stabilizers. It explicitly tracks the logical gate induced by a diagonal physical gate that preserves a CSS code. The first step is concatenation, where the input is a CSS code and a physical diagonal gate at level $l$ inducing a logical diagonal gate at the same level. The output is a new code for which a physical diagonal gate at level $l+1$ induces the original logical gate. The next step is judicious removal of $Z$-stabilizers to increase the level of the induced logical operator. We identify three ways of climbing the logical Clifford hierarchy from level $l$ to level $l+1$, each built on a recursive relation on the Pauli coefficients of the induced logical operators. Removal of $Z$-stabilizers may reduce distance, and the purpose of the third basic operation, addition of $X$-stabilizers, is to compensate for such losses. For the coherent noise model, we describe how to switch between computation and storage of intermediate results in a decoherence-free subspace by simply applying Pauli $X$ matrices. The approach to logical gate synthesis taken in prior work focuses on the code states, and results in sufficient conditions for a CSS code to be fixed by a transversal $Z$-rotation. In contrast, we derive necessary and sufficient conditions by analyzing the action of a transversal diagonal gate on the stabilizer group that determines the code. The power of our approach is demonstrated by two proofs of concept: the $[[2^{l+1}-2,2,2]]$ triorthogonal code family, and the $[[2^m,inom{m}{r},2^{min{r,m-r}}]]$ quantum Reed-Muller code family.
影响因子：
6.4
作者：
Hu, Jingzhen;Liang, Qingzhong;Calderbank, Robert
通讯作者：
Calderbank, Robert
影响因子：
2.9
作者：
Narayanan Rengaswamy;Robert Calderbank;H. Pfister
通讯作者：
Narayanan Rengaswamy;Robert Calderbank;H. Pfister