Logarithmic-Depth Fermion Sampling: Anticoncentration and Average-Case Hardness

Abstract

Sampling problems give some of the clearest conditional separations between quantum and classical computation. In Fermion Sampling, passive linear optics acts on a non-Gaussian magic input. Globally Haar-random transformations give anticoncentration and high-precision average-case hardness, but require linear depth and quadratically many gates. A natural open question is whether linear depth is necessary? We show that logarithmic depth suffices for both guarantees in the same ensemble. For fresh uniform matchings with independent Haar two-mode gates, the collision reaches the passive-Haar scale at a sharp threshold of $\log n/\log(9/4)\approx0.855\log_2 n$ layers, with an explicit limiting transition profile and a matching lower bound from two-particle correlations. The mechanism is input-dependent: the magic input suppresses the extensive weight of the slowest relaxation mode, leaving the next to set the scale. An occupation-basis input retains large collision even under full Haar randomness. At a larger logarithmic depth, we prove, in the real-RAM model, average-case $\#\mathsf{P}$-hardness of estimating output probabilities to additive error $2^{-O(n\log^2 n)}$ on any fixed fraction of instances above $3/4$. We construct hard instances in four native layers, embed them in typical schedules, and interpolate along Cayley paths using a rational linear-program decoder. A finite $192$-gate alphabet preserves the collision law exactly. Both guarantees use logarithmic native fermionic depth between arbitrary mode pairs and $O(n\log n)$ gates, replacing the global-Haar construction's quadratic budget. The accuracy proved is finer than the $1/N$ scale, where $N=\binom{n}{n/2}$, required by sampling-to-counting; hardness of sampling within constant total-variation distance remains open.