Discoveries / quantum-information-computing

The Fractional Colouring Conjecture for Triply Efficient Pauli Shadow Tomography

Conjecture 13 of King, Gosset, Kothari and Babbush asserts that for the set $B_\varepsilon(\rho)$ of Pauli observables with expectation value at least $\varepsilon$ in magnitude, the fractional chromatic number of the induced anticommutation graph is $O(\varepsilon^{-2})$; it would give a triply efficient Pauli shadow tomography algorithm. False: there are states and observables for which no finite constant bounds $\chi_f \varepsilon^2$.