Polynomial fractional-coloring bound for large-expectation Pauli observables

Determine whether there exists a constant exponent \(\kappa>0\) such that, for every \(n\)-qubit state \(\varrho\), every \(\epsilon\in(0,1)\), and the set \(B_\epsilon=\{P\in\mathcal P_n:|\operatorname{tr}(\varrho P)|\geq\epsilon\}\) of Pauli observables with expectation magnitude at least \(\epsilon\), the anticommutation graph \(G(B_\epsilon)\) has fractional chromatic number \(O(\epsilon^{-\kappa})\), thereby establishing a polynomial fractional-coloring bound sufficient for sample-efficient two-copy Clifford-measurement shadow tomography.

Background

The paper disproves the cited fractional coloring conjecture asserting the stronger universal bound $\chi_f(G(B_\epsilon))=O(\epsilon^{-2})$, by constructing states and Pauli-observable families for which $\chi_f(G(B_\epsilon))\epsilon^2$ is unbounded. The counterexamples achieve growth at least on the order of $\epsilon^{-2.07598}$, so they rule out the conjectured exponent 2 but do not exclude all polynomial dependence on $1/\epsilon$.

The authors explain that any bound of the form $\chi_f(G(B_\epsilon))=O(\epsilon^{-\kappa})$, for some fixed $\kappa>0$, would still imply a sample-efficient two-copy Clifford-measurement shadow-tomography protocol for arbitrary Pauli sets. The question is therefore explicitly left unresolved after the stronger conjecture is shown to be false.