Average-case hardness of monitored-FLO branches

Establish that for fixed local dimension d ≥ 2 and constant κ > 0 with m = κ n^2, in Haar-random number-conserving fermionic linear optics (FLO) instances conditioned on a collision-free monitoring record c in the monitored adaptive FLO architecture, approximating the conditional branch probability p(β | c) within relative error 1/poly(n) is #P-hard on average over instances and for a constant fraction of outcomes β among the d^{2n} possible Bell-fusion result strings.

Background

The paper introduces a monitored adaptive FLO architecture in which mid-circuit block-occupation measurements and feedforward route selected blocks into a fixed-order Bell-fusion geometry. Conditioned on a collision-free monitoring record in the dilute regime m = Θ(n^2), the FLO layer induces an operator kernel on an n×n block submatrix S of the single-particle propagator, and each branch amplitude is a matrix element of a non-commutative trace polynomial in dressed blocks.

Numerical evidence shows Porter–Thomas-like anti-concentration and rapid growth in the minimal order-respecting matrix-product-operator bond dimension, suggesting classical intractability analogous to other quantum sampling proposals. Motivated by this, the authors posit an average-case hardness conjecture asserting that estimating the conditional branch probabilities p(β | c) to relative error 1/poly(n) is #P-hard on average over Haar-random monitored-FLO instances.