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

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 n/ (9/4) about0.855 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 # -hardness of estimating output probabilities to additive error 2-O(n squared 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 n) gates, replacing the global-Haar construction's quadratic budget. The accuracy proved is finer than the 1/N scale, where N= n/2, required by sampling-to-counting; hardness of sampling within constant total-variation distance remains open.