Quantum advantage under spectral discretization for analytic Young-measure densities

Let $d\geq 2$, let $\nu_{x,y}(\xi)$ be a Young-measure density analytic in $x$ and $y$, discretize $\xi$ algebraically at $N_\xi\sim\varepsilon^{-1}$, and suppose $R_1$ is polynomially bounded in $|\log\varepsilon|$. At fine-scale accuracy $\delta=\varepsilon$, write $n_{\rm \det}$ for the resulting deterministic LP size and $\mathrm{QCP}_{\rm \det}$ for the quantum central path cost. Quantum advantage under spectral discretization. One should have

$$n_{\rm \det}\sim\varepsilon^{-d}|\log\varepsilon|^{2d},\qquad \mathrm{QCP}_{\rm \det}\sim\widetilde O\!\bigl(R_1\,\varepsilon^{-(1+d/2)}|\log\varepsilon|^{d}\bigr),$$

with this improving over direct classical solvers at cost $\widetilde O(\varepsilon^{-d})$ for all $d\geq2$. In the stochastic case with $N_\omega^r$ free, the advantage condition should be $N_\omega^r\gtrsim\varepsilon^{-(d-2)}$, and is therefore satisfied for all $d\geq2$ and $r\geq1$. This proposes a quantum advantage when the density is analytic in the macroscopic and microscale spatial variables, replacing algebraic spatial resolution by spectral resolution; the source gives no proof or status evidence for these complexity estimates.