广义三正交码到长度54的分类

中文摘要 AI 辅助

魔法态蒸馏是容错量子计算中广泛考虑的用于制备高保真非克利福德资源的原始方法。这类协议中最常考虑的一类是广义三正交码；这些码将$n$个有噪声的输入$\mathrm{T}$态蒸馏为纯化的第三级对角魔法态。大量先前的工作已经搜索了这个协议空间，通常使用不保证最优性的启发式方法。现有的分类工作仅限于蒸馏$k$个输出$\mathrm{T}$态且$n+k\leq 38$的协议（Nezami和Haah，Phys. Rev. A 106, 012437 (2022)）。在本工作中，我们将此分类显著扩展到所有长度$n\leq 54$的协议。我们将距离限制为$d\geq 3$以保持分类的合理规模，并且因为在距离$2$时高效搜索已被充分理解（Singh等人，arXiv:2606.28518）。此外，我们的结果与synthillation（Campbell和Howard，Phys. Rev. A 95, 022316 (2017)）互补，后者可以从$\mathrm{T}$态蒸馏第三级态，但仅在距离$2$时。在给定输出下，根据输入计数、空间占用和距离的最优性，我们在我们的范围内找到$74$个最优广义三正交协议，其中$65$个是文献中新的。为了实现我们的分类，我们使用方向导数方法将Nezami和Haah的单位三正交空间的分类从长度$38$扩展到$54$。使用这些作为稳定子空间，我们添加满足三正交约束的逻辑行以创建完整协议。

英文摘要

Magic state distillation is a widely considered primitive in fault-tolerant quantum computation for the preparation of high-fidelity non-Clifford resources. The most commonly considered class of such protocols are generalised triorthogonal codes; these distil $n$ noisy input $\mathrm{T}$ states into purified third-level diagonal magic states. Extensive prior work has searched this space of protocols, often using heuristic methods that do not guarantee optimality. Existing classification work is limited to protocols distilling $k$ output $\mathrm{T}$ states with $n+k\leq 38$ (Nezami and Haah, Phys. Rev. A 106, 012437 (2022)). In this work, we significantly expand this classification to all protocols with lengths $n\leq 54$. We restrict to distance $d\geq 3$ to keep the classification to a sensible size, and because efficient searches are well understood at distance $2$ (Singh et al., arXiv:2606.28518). Moreover, our results are complementary to synthillation (Campbell and Howard, Phys. Rev. A 95, 022316 (2017)), which can distil third-level states from $\mathrm{T}$ states, but only at distance $2$. Under optimality in terms of input count, space footprint, and distance for a given output, we find $74$ optimal generalised triorthogonal protocols in our range, $65$ of which are new to the literature. To achieve our classification, we extend the classification of unital triorthogonal spaces of Nezami and Haah from length $38$ to $54$ using a directional derivative method. Using these as the stabiliser spaces, we add logical rows that satisfy the triorthogonality constraints to create full protocols.

发表机构

• Massachusetts Institute of Technology（麻省理工学院）
• IBM Research, IBM T. J. Watson Research Center（IBM 研究院，IBM T.J. 沃森研究中心）
• Joint Center for Quantum Information and Computer Science, NIST, University of Maryland（量子信息与计算科学联合中心，美国国家标准与技术研究院，马里兰大学）

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