Measurement-induced non-commutativity in adaptive fermionic linear optics

TL;DR

This paper demonstrates that mid-circuit measurements and classical feedforward in fermionic linear optics circuits induce non-commutativity, leading to classical intractability and sampling hardness without adding interactions.

Contribution

It introduces measurement-induced non-commutativity in fermionic linear optics, showing how it causes classical simulation to become hard, unlike standard FLO.

Findings

Output distribution follows Porter-Thomas statistics

Rapid growth of matrix product operator bond dimension

Sampling becomes computationally hard due to non-commutativity

Abstract

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 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 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 sampling hardness for noninteracting fermions under reasonable complexity assumptions, without introducing coherent two-body interactions into the FLO evolution.