# Representational separation between unitary and channel quantum generative models via shared classical randomness at shallow depth

arXiv: https://arxiv.org/abs/2608.05110 · HTML: https://mlsanity.com/paper/2608.05110/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 46 references that are in the map.
- First posted on arXiv 5 Aug 2026.

## Builds on (most cited first, 13 of 13)

- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Learning in Implicit Generative Models](https://mlsanity.com/paper/1610.03483/) (2016)
- [Differentiable Learning of Quantum Circuit Born Machine](https://mlsanity.com/paper/1804.04168/) (2018)
- [Enhancing Generative Models via Quantum Correlations](https://mlsanity.com/paper/2101.08354/) (2021)
- [Certified Robustness of Quantum Classifiers against Adversarial Examples through Quantum Noise](https://mlsanity.com/paper/2211.00887/) (2022)
- [Stochastic noise can be helpful for variational quantum algorithms](https://mlsanity.com/paper/2210.06723/) (2022)
- [Training-efficient density quantum machine learning](https://mlsanity.com/paper/2405.20237/) (2024)
- [Variational measurement-based quantum computation for generative modeling](https://mlsanity.com/paper/2310.13524/) (2023)
- [Method for noise-induced regularization in quantum neural networks](https://mlsanity.com/paper/2410.19921/) (2024)
- [Constructing Optimal Noise Channels for Enhanced Robustness in Quantum Machine Learning](https://mlsanity.com/paper/2404.16417/) (2024)
- [Measurement-based quantum computation from Clifford quantum cellular automata](https://mlsanity.com/paper/2312.13185/) (2023)
- [Minimizing classical resources in variational measurement-based quantum computation for generative modeling](https://mlsanity.com/paper/2604.11578/) (2026)

## Abstract (arXiv)

Near-term quantum hardware limits circuit depth and often imposes geometrically local connectivity for quantum generative models, restricting the output distributions accessible to shallow unitary Born models. Introducing stochasticity into a unitary quantum Born model can improve the empirical generative performance of the resulting channel model and, for a restricted small-scale architecture, has been proven to represent a strictly larger family of distributions than its unitary counterpart. However, whether such randomness provides a provable separation at fixed shallow depth for arbitrarily large systems has remained open. Here, we show that shared classical randomness, a comparatively weak resource from entanglement theory, is sufficient to establish such a strict scalable representational separation over the corresponding shallow unitary Born model. More specifically, we augment bounded-connectivity shallow unitary circuits, followed by computational-basis measurements, with spatially separated local Pauli operations, whose joint application is controlled by a single classically sampled random bit. The resulting shallow-depth channel model generates long-range correlations in the classical output distribution that no purely unitary shallow-depth model with bounded connectivity can reproduce. For one-dimensional nearest-neighbour architectures, reproducing such distributions with a purely unitary model can require depth Omega(N) in the worst case. We further show that measurement-based quantum computation (MBQC) provides a natural implementation of the required shared classical randomness through suitable adaptation of the random measurement outcomes. Numerical experiments on MBQC-based generative models support the analytical results.