Jens Eisert

Mid‑circuit Measurements Add Magic, Making Adaptive FLO Classically Hard

Measurement-induced non-commutativity in adaptive fermionic linear optics We show that mid-circuit measurements of fermions with internal degrees of freedom can induce the equivalent of "magic" to free fermionic circuits and render them classically hard to sample. https://t.co/r7rZBcyLHL Fermionic linear optics (FLO) with Gaussian resources is efficiently classically simulable. We show that this is no longer the case for such quantum circuits for fermions with internal degrees of freedom, equipped with #midcircuit 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 classical description, this implies realizing a situation in which the permutation sum no longer collapses to a single determinant or Pfaffian. Each post-selected 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 as the output distribution and a rapid growth of the minimal order-respecting matrix product operator bond dimension. These results thus establish mid-circuit measurement-induced non-commutativity as a route to #samplinghardness for noninteracting #fermions under reasonable complexity assumptions, without introducing coherent two-body interactions into the FLO evolution. Warm thanks to Chenfeng Cao and Yifan Tang for the wonderful collaboration.