Measurement-induced noncommutativity in adaptive fermionic linear optics

Fermionic linear optics (FLO) with Gaussian resources is known to be efficiently classically simulable. We show that this changes for fermions with internal degrees of freedom once the circuit is equipped with mid-circuit number monitoring and classical feedforward. In our architecture, the measurement record routes the selected blocks into a fixed-order Bell-fusion pairing geometry. On the level of the classical description, the permutation sum then no longer collapses to a single determinant or Pfaffian. Each accepted branch expands as a signed sum of path-ordered products of typically non-commuting dressed blocks, and branch amplitudes are matrix elements of the resulting non-commutative trace polynomials. Numerically, we observe Porter-Thomas statistics for the conditional branch distribution and rapid growth of the minimal order-respecting matrix product operator bond dimension. Under reasonable complexity assumptions, these results identify mid-circuit measurement-induced non-commutativity as a route to sampling hardness for noninteracting fermions, with the non-Gaussianity confined to a fixed preparation-and-readout interface rather than to the coherent FLO evolution.