AI 中文总结
本文针对三重高效影子层析的分数着色猜想，构造反例证明其不成立，还给出依赖图交换指数或β数的通用构造，关键利用词典积图放大相关参数。
AI 中文摘要
分数图着色对泡利可观测量的影子层析有用。实际中，希望任何实验感兴趣的泡利算子集，其反交换图的分数色数χ_f较小。King、Gosset、Kothari和Babbush[PRX Quantum 6, 010336 (2025)]的猜想13指出，若B_ε(ρ)是某个量子态ρ中期望值绝对值至少为ε的泡利可观测量集合，则B_ε(ρ)诱导的反交换图G的分数色数为O(ε^{-2})，即存在常数C，使得对所有态和图都有χ_f·ε² ≤ C。若该猜想成立，将意味着：只要B_ε集合存在高效分数着色算法，就存在对泡利可观测量任意子集S的三重高效泡利影子层析算法。本文通过构造一类态和可观测量，证明该猜想不成立——不存在满足该界的有限C；还给出了更通用的构造，依赖图的交换指数或β数。证明的关键要素可视为放大技巧的实例：分数色数、β数和期望值通过词典积图被放大。
英文摘要
Fractional graph colorings are useful for the Shadow tomography of Pauli observables. In practice, it is desirable that any experimentally interesting set of Pauli operators has a small fractional chromatic number $χ_{f}$ for its anticommutation graph. Conjecture 13 in King, Gosset, Kothari, and Babbush [PRX Quantum 6, 010336 (2025)] states that if $B_ε(\varrho)$ is the set of Pauli observables having expectation value magnitude at least $ε$ in some given quantum state $\varrho$, then the fractional chromatic number of the anticommutation graph $G$ induced by $B_ε(\varrho)$ is $O(ε^{-2})$. In other words, it asserts that there exists a constant $C$ such that $χ_{f} \cdot ε^2 \leq C$ on all states and graphs. If the conjecture were true, it would imply that there exists a triply efficient Pauli shadow tomography algorithm for {\it any} subset $S$ of Pauli observables, provided that there is also an efficient fractional coloring algorithm for the set $B_ε$. Here we show that the conjecture is false by constructing a family of states and observables for which no finite $C$ satisfying the bound exists. We also give a more general construction relying on the commutation index or $β$ number of a graph. The key ingredient in the proofs can be seen as an instance of the amplification trick, where fractional chromatic numbers, $β$ numbers, and expectation values are amplified through lexicographic graph products.
Comments14 pages. Found with GPT Sol 5.6. Comments welcome!
