Counterexamples to the fractional coloring conjecture for triply efficient shadow tomography

Abstract: 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 $χ<em>{f}$ for its anticommutation graph. Conjecture 13 in King, Gosset, Kothari, and Babbush [PRX Quantum 6, 010336 (2025)] states that if $B</em>ε(\varrho)$ 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 $O(ε<sup>{-2})$. In other words, it asserts that there exists a constant $C$ such that $χ<em>{f} \cdot ε² \leq C$ 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 $B</em>ε$. 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.