老虎码的代数结构
Algebraic structure of Tiger codes
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中文总结 AI 辅助
研究老虎码的代数结构，通过核定义等方法证明相关性质、构造基并进行傅里叶变换等，还扩展到非线性约束及研究逻辑操作实现，确立其为描述玻色子编码的稳健数学框架。
中文摘要 AI 辅助
老虎码是一族多模玻色子量子码，统一了包括猫码、对猫码和双模二项式码等几种先前已知的构造。本文对这些码进行了严格的代数处理。从码空间的核定义出发，证明湮灭型约束有有限生成集，构造显式正交基，表明码的逻辑结构由基础链复形的同调决定。还开发了码空间上的傅里叶变换，扩展框架到非线性数约束，最后研究逻辑操作的实现，包括推广逻辑泡利算子构造和构造非克利福德门等，这些结果确立了老虎码作为描述广泛玻色子编码的数学稳健框架。
英文摘要
Tiger codes form a family of multimode bosonic quantum codes that unify several previously known constructions, including cat, paircat, and the two-mode binomial code. In this work, we give a rigorous algebraic treatment of these codes. Starting from a kernel definition of the codespace, we prove that the annihilation-type constraints admit a finite generating set, construct an explicit orthonormal basis, and show that the logical structure of the code is governed by the homology of an underlying chain complex, as expected in the original work on Tiger codes of Xu et al. We then develop a Fourier transform over the codespace to prove that the span of phase-rotated projected coherent states is dense therein, and to yield dual $X$- and $Z$-type descriptions of the code. We further extend the framework to non-linear number constraints, encompassing codes such as the four-legged cat or the repetition cat code. Finally, we investigate the implementation of logical operations. We first generalise the construction of logical Pauli operators proposed by Xu et al. to arbitrary logical spaces, and then construct non-Clifford gates using physical polynomial phase rotations of the form $e^{iP(\hat{\boldsymbol{n}})}$. We derive criteria on the real polynomial $P$ which, for positive single-logical-qubit Tiger codes satisfying an additional sign assumption, such as the paircat code, characterise the polynomials $P$ that preserve the codespace by decomposing them into a family of univariate polynomials. Through this decomposition, we relate the degrees of the resulting components to the induced logical action in the Clifford hierarchy. These results establish Tiger codes as a mathematically robust framework for describing a broad class of bosonic encodings.
