发表机构
Indian Institute of Science（印度科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究刻画了可通过横向物理Z旋转实现逻辑对角门的CSS码嵌套对，提出附加构造方法扩展CSS码以实现多个容错逻辑Z旋转，需付出物理量子比特开销增加的代价。
AI 中文摘要
由嵌套经典码$C_2 \subseteq C_1$构造的Calderbank-Shor-Steane（CSS）码，通常针对优良的码参数进行优化，但实用量子计算同样需要容错逻辑门。本研究刻画了一类嵌套对$(C_1, C_2)$，其生成的CSS码可通过横向物理Z旋转实现目标逻辑对角门。在此过程中，我们重现了Camps-Moreno等人的结论：CSS码仅能通过横向物理Z旋转实现逻辑单量子比特Z旋转和多量子比特受控Z旋转。基于该刻画，我们提出了“附加构造”方法：以一个$[[n',k']]$ CSS码$Q'$和目标逻辑Z旋转（单量子比特或多受控）$U_L$为输入，通过系统地附加$n''$个物理量子比特扩展$Q'$，得到一个$[[n,k]]$ CSS码$Q$，其中$n = n'+n''$且$k=k'$；目标逻辑门$U_L$通过对$n''$个附加物理量子比特应用精心选择的物理横向Z旋转在$Q$中实现。该CSS码$Q$可能存在最小距离的损失，但该损失可通过构造中的参数选择加以控制。通过重复应用附加构造，我们可将任意CSS码$Q'$扩展为支持多个期望逻辑Z旋转容错实现的CSS码$Q$，代价是随目标逻辑门数量增加而上升的物理量子比特开销。
英文摘要
Calderbank-Shor-Steane (CSS) codes, constructed from nested classical codes $C_2 \subseteq C_1$, are typically optimized for good code parameters. However, practical quantum computing equally demands fault-tolerant logical gates. In this work, we characterize nested pairs $(C_1, C_2)$ whose resulting CSS codes realize a target logical diagonal gate via transversal physical $Z$-rotations. In doing so, we recover a result of Camps-Moreno et al. that CSS codes can realize only logical single-qubit $Z$-rotations and multi-qubit controlled $Z$-rotations via transversal physical $Z$-rotations. Building on our characterization, we develop the ''appending construction'', that takes as input an $[[n',k']]$ CSS code $Q'$ and a target logical $Z$-rotation (single-qubit or multi-controlled) $U_L$, and extends $Q'$ by systematically appending $n''$ physical qubits to obtain an $[[n,k]]$ CSS code $Q$ with $n = n'+n''$ and $k=k'$. The target logical gate $U_L$ is realized in $Q$ by applying a well-chosen physical transversal $Z$-rotation to the $n''$ appended physical qubits. Moreover, any logical gate realized via transversal physical $Z$-rotations in the input code $Q'$ remains transversally realizable in the extended code $Q$. The CSS code $Q$ may incur a loss in minimum distance, but the loss can be controlled through the parameter choices made in the construction. By repeatedly applying the appending construction, we can extend any CSS code $Q'$ to obtain a CSS code $Q$ that supports fault-tolerant implementations of multiple desired logical $Z$-rotations. The cost to be paid for this is the increased physical qubit overhead as the number of target logical gates grows. To illustrate our methodology, we construct CSS code families that transversally realize addressable logical single-qubit and multi-controlled $Z$-rotations.
Comments68 pages. Improves exposition
