# Bridge of Psi's: Quantum Circuit Optimization with Schr"odinger Bridges

arXiv: https://arxiv.org/abs/2609.25947 · HTML: https://mlsanity.com/paper/2609.25947/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 54 references that are in the map.
- First posted on arXiv 22 Sep 2026.

## Builds on (most cited first, 25 of 25)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Swin Transformer V2: Scaling Up Capacity and Resolution](https://mlsanity.com/paper/2111.09883/) (2021)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Simulation-free Schr"odinger bridges via score and flow matching](https://mlsanity.com/paper/2307.03672/) (2023)
- [Isambard-AI: a leadership class supercomputer optimised specifically for Artificial Intelligence](https://mlsanity.com/paper/2410.11199/) (2024)
- [U-DiTs: Downsample Tokens in U-Shaped Diffusion Transformers](https://mlsanity.com/paper/2405.02730/) (2024)
- [Inference-time Scaling of Diffusion Models through Classical Search](https://mlsanity.com/paper/2505.23614/) (2025)
- [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)
- [Quarl: A Learning-Based Quantum Circuit Optimizer](https://mlsanity.com/paper/2307.10120/) (2023)
- [Phase Gadget Synthesis for Shallow Circuits](https://mlsanity.com/paper/1906.01734/) (2019)
- [A Verified Optimizer for Quantum Circuits](https://mlsanity.com/paper/1912.02250/) (2019)
- [Synthesis of discrete-continuous quantum circuits with multimodal diffusion models](https://mlsanity.com/paper/2506.01666/) (2025)
- [Synthesizing Quantum-Circuit Optimizers](https://mlsanity.com/paper/2211.09691/) (2022)
- [Quanto: Optimizing Quantum Circuits with Automatic Generation of Circuit Identities](https://mlsanity.com/paper/2111.11387/) (2021)
- [Scalable Neural Decoders for Practical Fault-Tolerant Quantum Computation](https://mlsanity.com/paper/2604.08358/) (2026)
- [AltGraph: Redesigning Quantum Circuits Using Generative Graph Models for Efficient Optimization](https://mlsanity.com/paper/2403.12979/) (2024)
- [Equivariant Reinforcement Learning for Clifford Quantum Circuit Synthesis](https://mlsanity.com/paper/2605.10910/) (2026)
- [Linear-Time T-Gate Optimization via Random Abstraction](https://mlsanity.com/paper/2605.13929/) (2026)
- [When Close Enough Is Not Enough: Autoregressive Drift in Quantum Circuit Synthesis](https://mlsanity.com/paper/2607.12780/) (2026)

## Abstract (arXiv)

Quantum circuit optimization replaces a circuit with an equivalent one of fewer gates and lower depth, reducing execution cost and error rate. We ask whether a generative model can learn this transformation directly from examples, rather than selecting from a fixed rewrite library or rigid algebraic routines. We present Bridge of Psi's (BOPS), a generative model based on Schr"odinger bridges, using a custom denoiser architecture, that learns a transformation from a source circuit into an equivalent optimized circuit. We train it on data constructed to be hard for existing optimizers, by applying rewrite rules backwards so that each input has a known lower-cost target. On held-out 8 qubits × 64 depth Clifford+T circuits, BOPS reduces gate count by 2.46× and depth by 2.45× in geometric mean, outperforming all nine baseline optimizers. This constitutes the first generative model bridging quantum circuits and frontier machine learning methods, opening up the quantum compilation stack to learned optimization along multiple axes.