# LLM-Driven Algorithm Design for Quantum Circuit Synthesis based on Binary Decision Diagrams

arXiv: https://arxiv.org/abs/2609.05327 · HTML: https://mlsanity.com/paper/2609.05327/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 54 references that are in the map.
- First posted on arXiv 4 Sep 2026.

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

- [AlphaEvolve: A coding agent for scientific and algorithmic discovery](https://mlsanity.com/paper/2506.13131/) (2025)
- [ReEvo: Large Language Models as Hyper-Heuristics with Reflective Evolution](https://mlsanity.com/paper/2402.01145/) (2024)
- [HSEvo: Elevating Automatic Heuristic Design with Diversity-Driven Harmony Search and Genetic Algorithm Using LLMs](https://mlsanity.com/paper/2412.14995/) (2024)
- [Multi-objective Evolution of Heuristic Using Large Language Model](https://mlsanity.com/paper/2409.16867/) (2024)
- [EoH-S: Evolution of Heuristic Set using LLMs for Automated Heuristic Design](https://mlsanity.com/paper/2508.03082/) (2025)
- [Qiskit HumanEval: An Evaluation Benchmark For Quantum Code Generative Models](https://mlsanity.com/paper/2406.14712/) (2024)
- [CALM: Co-evolution of Algorithms and Language Model for Automatic Heuristic Design](https://mlsanity.com/paper/2505.12285/) (2025)
- [Agent-Q: Fine-Tuning Large Language Models for Quantum Circuit Generation and Optimization](https://mlsanity.com/paper/2504.11109/) (2025)
- [QCircuitBench: A Large-Scale Dataset for Benchmarking Quantum Algorithm Design](https://mlsanity.com/paper/2410.07961/) (2024)
- [QuanBench: Benchmarking Quantum Code Generation with Large Language Models](https://mlsanity.com/paper/2510.16779/) (2025)
- [TrajEvo: Trajectory Prediction Heuristics Design via LLM-driven Evolution](https://mlsanity.com/paper/2508.05616/) (2025)
- [BDD2Seq: Enabling Scalable Reversible-Circuit Synthesis via Graph-to-Sequence Learning](https://mlsanity.com/paper/2511.08315/) (2025)

## Abstract (arXiv)

Quantum circuits are central to implementing quantum algorithms on quantum devices, where quantum gates must be reversible. Many quantum algorithms rely on Boolean functions, which must therefore be implemented reversibly within quantum circuits. Reversible circuit synthesis provides a way to translate such Boolean functions into reversible circuits. Binary decision diagrams (BDDs) offer a scalable approach to this task, but the resulting BDDs and circuits depend heavily on variable ordering. Existing ordering heuristics commonly minimize BDD size because it is closely tied to the circuit size. However, BDD size is an imperfect proxy for the quantum cost of the synthesized circuit (QCC). We propose textttQuantumEvo, an evolutionary framework that uses an LLM as a heuristic generator for QCC-aware BDD variable ordering. Instead of predicting orderings directly, textttQuantumEvo searches over ordering heuristics initialized from multiple heuristic families. Candidate heuristics directly manipulate variable orderings using standard BDD operations and are selected by downstream QCC. The discovered heuristic, HGA-QE, modifies the sifting step inside a genetic algorithm so that the procedure is better aligned with QCC. Across the benchmark set, HGA-QE achieves a 70.9% tie-or-win rate against the per-function best baseline and is strictly best on 13.5% of the functions. The results demonstrate broadly competitive QCC performance, with HGA-QE showing a clearer relative advantage in strict wins on the two benchmark suites drawn from sources different from the data used for heuristic discovery.