来自可逆拓扑量子场论的非克利福德量子元胞自动机
Non-Clifford quantum cellular automata from invertible topological quantum field theories
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中文总结 AI 辅助
研究从可逆拓扑量子场论构建非克利福德量子元胞自动机的方法，通过统一代数构造，在多维度产生新的QCA无限族，重新表述特定QCA并开发一般构造，还证明了部分5维QCA平凡，为高维构造及相关分类提供途径。
中文摘要 AI 辅助
量子元胞自动机（QCA）描述保持局域性的量子动力学，并连接量子信息、多体物理和拓扑量子场论（TQFT）。然而，从TQFT构建QCA具有挑战性。虽然拓扑作用可产生实现所需基态的对易哈密顿量，但它本身并未指定全局部算子代数的自同构。在这项工作中，我们开发了一种统一的代数构造，将哈密顿量的对易生成元扩展到全张量积希尔伯特空间上的完整分隔翻转代数，给出了相应QCA的微观定义。在三个空间维度上，我们的形式主义统一了所有先前已知的与维特群的\(\mathbb Z_8\times\mathbb Z_2\)子群相关的QCA构造，包括\(U(1)_2\)和\(U(1)_4\) QCA。相同的代数结构直接在\(d = 4k - 1\)维中产生新的广义\(U(1)_2\)和\(U(1)_4\)非克利福德QCA的无限族。我们还重新表述了4维\(w_2w_3\) QCA，并用于从与吴类的任意乘积相关的TQFT开发QCA的一般构造。此构造包括两个无限族。第一个由\(d = 2n + 3m - 1\)维中的\(w_2^nw_3^m\) QCA组成，而第二个由\(d = 4k\)维中的\(w_2w_{4k - 1}\) QCA组成。作为对比结果，我们明确构造了5维\(w_3^2\)和\(w_2^3\) QCA的有限深度量子电路，从而证明它们是平凡的，这与配边分类一致。总体而言，这些结果将可逆TQFT转换为微观QCA，提供了超越克利福德设置的高维构造的可扩展途径，并开辟了一种系统方法来分类它们的稳定结构和边界异常。
英文摘要
Quantum cellular automata (QCAs) describe locality-preserving quantum dynamics and connect quantum information, many-body physics, and topological quantum field theory (TQFT). Constructing a QCA from a TQFT, however, is challenging. Although a topological action can produce a commuting Hamiltonian realizing the desired ground state, it does not by itself specify an automorphism of the full local operator algebra. In this work, we develop a unified algebraic construction that extends the commuting generators of the Hamiltonian to a complete separator-flipper algebra on the full tensor-product Hilbert space, providing a microscopic definition of the corresponding QCA. In three spatial dimensions, our formalism unifies all previously known QCA constructions associated with the $\mathbb Z_8\times\mathbb Z_2$ subgroup of the Witt group, including the $U(1)_2$ and $U(1)_4$ QCAs. The same algebraic structure directly yields new infinite families of generalized $U(1)_2$ and $U(1)_4$ non-Clifford QCAs in dimensions $d=4k-1$. We also reformulate the 4-dimensional $w_2w_3$ QCA and use it to develop a general construction of QCAs from TQFTs associated with arbitrary products of Wu classes. This construction includes two infinite families. The first consists of $w_2^nw_3^m$ QCAs in dimension $d=2n+3m-1$, while the second consists of $w_2w_{4k-1}$ QCAs in dimension $d=4k$. As a contrasting result, we explicitly construct finite-depth quantum circuits for the 5-dimensional $w_3^2$ and $w_2^3$ QCAs, thereby proving that they are trivial, in agreement with the cobordism classification. Overall, these results convert invertible TQFTs into microscopic QCAs, provide a scalable route to higher-dimensional constructions beyond the Clifford setting, and open a systematic approach to classifying their stable structures and boundary anomalies.
