Average-case hardness conjecture for approximating the Sherrington–Kirkpatrick partition function

Let ${\bf J}=(J_{ij}:1\leqslant i<j\leqslant n)\in\mathbb{R}^{n(n-1)/2}$ have independent and identically distributed entries $J_{ij}\stackrel{d}{=}\mathcal{N}(0,1)$. Let $\widehat{Z}({\bf J})$ denote the target approximation to the partition function, and let an oracle $\mathcal{O}(\cdot)$ operate in a real-valued computational model, such as a Blum–Shub–Smale machine.

Average-case hardness conjecture. (a) If

$$\mathbb{P}\left(\widehat{Z}({\bf J})=\mathcal{O}({\bf J})\right)\geqslant \frac12+\frac{1}{{\rm poly}(n)},$$

then $P=\#P$. (b) More challengingly, if

$$\mathbb{P}\left(\widehat{Z}({\bf J})=\mathcal{O}({\bf J})\right)\geqslant \frac{1}{{\rm poly}(n)},$$

then $P=\#P$.

These are proposed as open average-case hardness statements for computing the Sherrington–Kirkpatrick partition function. The second variant asks for the same consequence under only inverse-polynomial success probability, and both statements concern algorithms operating on real-valued inputs.