来自克莱默斯 - 万尼尔对偶性和模关系的量子元胞自动机
Quantum Cellular Automata from Kramers-Wannier Dualities and Modular Relations
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中文总结 AI 辅助
研究从克莱默斯 - 万尼尔对偶性等出发，探讨与引力拓扑响应相关的量子元胞自动机。通过多种情形构建其关系，推导新射影关系，给出制备纠缠态协议，统一了场论、高维对偶性和量子元胞自动机中引力拓扑响应的研究。
中文摘要 AI 辅助
近期工作构建了类似于1 + 1维克莱默斯 - 万尼尔对偶性的高维非可逆对称性类似物。其连续统描述常将纯引力拓扑项视为无关的抵消项，但这些项在晶格中有重要表现，能区分有限深度量子电路制备的态与非平凡量子元胞自动机纠缠的态。受此不匹配启发，我们表明与引力拓扑响应相关的量子元胞自动机出现在多种相关情形中，包括对称性拓扑操作生成的射影$\mathrm{SL}(2,\mathbb{Z}_N)$关系的晶格实现等。我们推导了新的射影$\mathrm{SL}(2,\mathbb{Z}_N)$关系，其射影相位是由施蒂费尔 - 惠特尼类构建的引力拓扑响应。还给出了用有限深度酉电路、测量和纠错制备相关量子元胞自动机纠缠态的通用协议。这些结果统一了场论、高维对偶性和量子元胞自动机中引力拓扑响应的研究。
英文摘要
Recent work has constructed higher-dimensional analogs of non-invertible symmetries similar to 1+1d Kramers-Wannier duality. Although their continuum descriptions often treat purely gravitational topological terms as inessential counterterms, these terms can have an essential lattice manifestation: they distinguish states prepared by finite-depth quantum circuits (FDQCs) from those entangled by nontrivial quantum cellular automata (QCAs). Motivated by this mismatch, we show that QCAs associated with gravitational topological responses arise in several related settings: (1) lattice realizations of projective $\mathrm{SL}(2,\mathbb{Z}_N)$ relations generated by topological operations on symmetries; (2) squares of dualities that generalize the relation between fermionization and Kramers-Wannier duality; (3) lattice implementations of QCAs through higher-form gauging; and (4) invertible phases protected by generalized time-reversal symmetries. We derive new projective $\mathrm{SL}(2,\mathbb{Z}_N)$ relations whose projective phases are gravitational topological responses constructed from Stiefel-Whitney classes. We furthermore give a general protocol for preparing the associated QCA-entangled states using finite-depth unitary circuits, measurements, and error correction. These results unify the study of gravitational topological responses in field theories, higher dimensional dualities, and quantum cellular automata.
