Representational Separation Between Unitary and Channel Quantum Generative Models via Shared Classical Randomness at Shallow Depth

This paper proves a scalable representational separation between shallow unitary Born models and channel-based quantum generative models. Starting from bounded-connectivity shallow circuits with computational-basis measurements, the authors add spatially separated local Pauli operations controlled by one shared classical random bit. This weak stochastic resource creates long-range correlations in the classical output distribution that bounded-connectivity shallow unitary models cannot reproduce. For one-dimensional nearest-neighbor architectures, reproducing the resulting distributions with a purely unitary model may require worst-case depth Ω(N). The paper also shows how measurement-based quantum computation (MBQC) can implement the shared randomness through adaptive measurement outcomes, with numerical experiments supporting the analytical claims.