Speaker
Description
Fermionic linear optics (FLO) with Gaussian resources is efficiently classically simulable, with many-body amplitudes reducing to determinants and Pfaffians. We show this is no longer true when fermions carry internal $d$-level labels and undergo mid-circuit number monitoring with classical feedforward. The measurement record routes the selected blocks into a fixed Bell-fusion geometry that ties teleportation byproducts to specific measured blocks: the antisymmetrized permutation sum no longer collapses to a single determinant or Pfaffian, and each post-selected branch amplitude becomes a matrix element of an ordered non-commutative trace polynomial in outcome-dressed FLO blocks. We prove an exponential sequential-memory barrier for any fusion-order-respecting MPO contraction in the free trace-polynomial benchmark, and numerically observe Porter-Thomas statistics together with rapid bond-dimension growth. Under a standard average-case hardness conjecture, this establishes mid-circuit measurement-induced non-commutativity as a route to sampling hardness for noninteracting fermions, without coherent two-body interactions in the FLO evolution.
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