# Lie Group Diffusion Models for Hardware-Aware Quantum Circuit Synthesis

arXiv: https://arxiv.org/abs/2606.29636 · HTML: https://mlsanity.com/paper/2606.29636/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 37 references that are in the map.
- First posted on arXiv 28 Jun 2026.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [SDXL: Improving Latent Diffusion Models for High-Resolution Image Synthesis](https://mlsanity.com/paper/2307.01952/) (2023)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Quantum Circuit Optimization with AlphaTensor](https://mlsanity.com/paper/2402.14396/) (2024)
- [Quantum circuit synthesis with diffusion models](https://mlsanity.com/paper/2311.02041/) (2023)
- [Topological Quantum Compiling with Reinforcement Learning](https://mlsanity.com/paper/2004.04743/) (2020)
- [Generative quantum advantage for classical and quantum problems](https://mlsanity.com/paper/2509.09033/) (2025)
- [Exponential quantum advantage in processing massive classical data](https://mlsanity.com/paper/2604.07639/) (2026)

## Built on by (most cited first, 1 of 1)

- [Circuit Hypernetworks for Quantum-Augmented Diffusion Language Models](https://mlsanity.com/paper/2609.24657/) (2026)

## Abstract (arXiv)

An important task in quantum computing is unitary circuit synthesis compatible with physical hardware constraints. This problem has a natural hybrid structure as local single-qubit gates are continuous variables on the Lie group SU(2) while the entangling circuit structure is discrete and hardware-dependent. In this work, we use generative models to perform quantum circuit synthesis incorporating both the natural SU(2) manifold geometry of quantum gates and hardware constraints that determine the overall circuit structure. Our model comprises two components: a circuit skeleton selector that chooses an entangling circuit and a diffusion model that generates quantum gates on the given circuit template by performing diffusion on the curved manifold SU(2) simeq S3 itself. We demonstrate this approach with unitary compilation of physically motivated three-qubit Hamiltonian simulation targets such as the Transverse Field Ising Model and the Heisenberg-XXZ Model and show that Lie group diffusion outperforms comparable baselines. The synthesised circuits can also be customised subject to constraints, which we demonstrate by producing circuits with large and small gate rotation angles for the same target unitary evolution. We also investigate the fidelity-complexity frontier of the synthesised gates to demonstrate that the circuit selector learns to select circuits that balance fidelity with complexity rather than collapsing onto the most expansive entangling template. These results demonstrate that Lie group diffusion provides a natural generative framework for hardware-aware quantum circuit synthesis.