在CSS码与电路中寻找对角逻辑门
Finding diagonal logical gates in CSS codes and circuits
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中文总结 AI 辅助
本文提出高效算法，可从给定ansatz门中找出CSS码与电路的所有对角逻辑门，其运行时间为O(n³)，还讨论了该方法向其他门及量子位的推广，为容错量子计算提供支持。
中文摘要 AI 辅助
寻找高效的非Clifford逻辑或魔术态制备方案，是容错量子计算领域的核心挑战之一。诸多已提出的方案依赖于作用于空间中CSS码的对角非Clifford逻辑门，或时空里装饰的CSS型错码提取电路。本文提出并实现了高效算法，用于从给定的一组 ansatz 门中，找出特定CSS码（电路）的所有（时空）逻辑门。根据ansatz门的选择，这意味着寻找横向门、更通用的保局域性逻辑电路、折叠门或类似结构。本文聚焦于Clifford层级中的量子比特对角门，同时也讨论了向任意对角非层级门、特定非对角门，以及素数和复合维量子位的推广。该方法通过将码空间保形门重新表述为X校验矩阵在相位函数上的“拉回”的核来实现，该映射在有限阿贝尔2-群之间进行。本文实现了一种快速“过滤”方法来寻找该核。对于含O(n)个量子比特的qLDPC码，或含O(n)个门的电路，在朴素稠密实现下，寻找容错逻辑门的运行时间为O(n³)，且存在利用稀疏性进行改进的潜力。
英文摘要
Finding efficient schemes for non-Clifford logic or magic state preparation is one of the central challenges on the way to fault-tolerant quantum computation. Many of the proposed schemes rely on diagonal non-Clifford logical gates acting on CSS codes in space or decorating CSS-type syndrome-extraction circuits in spacetime. Here we propose and implement efficient algorithms to find all (spacetime) logical gates of a given CSS code (circuit) composed from a prescribed set of ansatz gates. Depending on the choice of ansatz gates, this means finding transversal gates, more general locality-preserving logical circuits, folding gates, or similar. While we focus on qubit diagonal gates in the Clifford hierarchy, we also discuss the generalization to arbitrary diagonal non-hierarchy gates, certain non-diagonal gates, as well as prime and composite-dimensional qudits. Our method works by rephrasing code-space preserving gates as the kernel of the ``pullback'' of the $X$ check matrix onto phase functions, which maps between finite abelian 2-groups. We implement a fast ``filtration'' method to find this kernel. The runtime for finding fault-tolerant logical gates in a qLDPC code with $O(n)$ qubits or a circuit with $O(n)$ gates in a naive dense implementation is $O(n^3)$, with potential for improvement making use of sparsity.
