Average-case hardness of squared hafnian estimation

Establish that the problem $|\mathrm{GHE}|_\pm^2$, which asks for additive approximation of $|\operatorname{Haf}(X)|^2$ for a random symmetric complex Gaussian matrix $X$, is $\#P$-hard in the stated oracle sense: every oracle solving the problem should imply ${P}^{\#P}\subseteq\mathrm{BPP}^{\mathcal O}$.

Background

The paper defines ∣GHE∣±2|\mathrm{GHE}|_\pm^2 as the task of estimating the squared hafnian of a random symmetric complex Gaussian matrix to additive error proportional to its expected value. The authors prove a reduction showing that an approximately sampling classical algorithm for the relevant Gaussian boson sampling instances would place this estimation problem in FBPPNP\mathrm{FBPP}^{\mathrm{NP}}. The remaining average-case hardness assertion is explicitly posed as a conjecture; if true, it would yield the usual polynomial-hierarchy collapse consequence for approximate Gaussian boson sampling.

References

$|\mathrm{GHE}|\pm2$ is -hard, in the sense that if $\mathcal O$ is any oracle that solves $|\mathrm{GHE}|\pm2$, then ${P}{#{P}\subseteqBPP{\mathcal O}$. Conjecture~\ref{conj:ghe} is analogous to the conjecture for additive approximation of squared permanents of complex matrices, $|\mathrm{GPE}|_\pm2$, being -hard .

— Proof of the hiding conjecture for Gaussian boson sampling with an arbitrary number of squeezed input modes  (2608.19314 - Shou et al., 19 Aug 2026) in Conjecture labeled \ref{conj:ghe}, Section 1, after Theorem 3 (Hardness result)

A crucial open problem is to improve the additive imprecision level for #P-hardness from $\exp(-O(n\log n))$ to $\poly(n){-1}$, which remains open even for the standard BosonSampling setting.

— Threshold and Parity BosonSampling in the Linear-Mode Regime  (2608.24008 - Go et al., 25 Aug 2026) in Section Conclusion