# An Efficient Quantum Circuit for Flow Model Execution Using Quantum Neural Networks

arXiv: https://arxiv.org/abs/2610.11537 · HTML: https://mlsanity.com/paper/2610.11537/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 49 references that are in the map.
- First posted on arXiv 8 Oct 2026.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [Discrete Flow Matching](https://mlsanity.com/paper/2407.15595/) (2024)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Spectral Norm Regularization for Improving the Generalizability of Deep Learning](https://mlsanity.com/paper/1705.10941/) (2017)
- [Quantum Graph Neural Networks](https://mlsanity.com/paper/1909.12264/) (2019)
- [On the Guidance of Flow Matching](https://mlsanity.com/paper/2502.02150/) (2025)
- [Quantum Machine Learning in Drug Discovery: Applications in Academia and Pharmaceutical Industries](https://mlsanity.com/paper/2409.15645/) (2024)
- [Wavefunction Flows: Efficient Quantum Simulation of Continuous Flow Models](https://mlsanity.com/paper/2510.08462/) (2025)

## Abstract (arXiv)

Flow models generate trajectories from an initial distribution to a target distribution by solving an ordinary differential equation defined by a velocity field. Flow matching learns this velocity field by modeling the transport dynamics between the two distributions. Wavefunction flow establishes a formal connection between flow models and quantum dynamics by introducing a continuity Hamiltonian, which drives the Schr"odinger evolution of quantum states. In this paper, we investigate accurate and efficient quantum simulation of the wavefunction flow, thereby realizing the efficient implementation of flow models on quantum computers. We first leverage a quantum read-only memory (QROM)-based phase kickback framework for the wavefunction flow simulation, generating probability densities that closely match those produced by the corresponding conventional flow model. To address the high circuit-resource cost, we further incorporate a trained quantum neural network (QNN) into the phase kickback framework, replacing QROM for data encoding. Numerical experiments demonstrate that our proposed method implements flow models on quantum computers more efficiently, since it maintains the accuracy of wavefunction flow simulation compared with the QROM-based framework, and significantly reduces the circuit resources.