Abstract
Graph states form a broad class of multipartite entangled states underlying measurement-based quantum computation, quantum networks, and stabilizer codes. However, systematic entanglement distillation for arbitrary graph states remains challenging because the circuit design space grows rapidly with the number of parties. We introduce a group of Clifford operations that we call "factorized graph-preserving". It enables us to efficiently enumerate and optimize graph-state purification circuits at finite size for realistic noisy hardware. These operations map products of graph-basis states to products of graph-basis states, so their action can be represented as permutations of graph-basis labels. Moreover, this useful gate set admits a compact factorized description determined by simple graph-theoretic features. This structure also allows, after some initial cached precomputation, drastically lower computational complexity for simulating a gate. We further organize these operations over local-complementation (LC) orbits using minimum-edge representatives (MERs), which let us design purification circuits that apply to all locally equivalent graph states (up to a basis change). Using this framework, we optimize noisy finite-size multipartite distillation circuits for several graph-state families. Numerical results show that the resulting graph-preserving circuits can outperform standard recurrence-based purification protocols under realistic gate and measurement noise. Our results establish LC-orbit structure and factorized graph-preserving operations as practical tools for scalable, topology-aware and hardware-constrained graph-state distillation protocol design. Our work can also be interpreted as a graph-based heuristic for finding transversal gates.
My contributions
- Characterized factorized graph-preserving Clifford operations through bipartiteness and leaf structure, obtaining compact homogeneous and bilocal gate families for graph-basis simulation.
- Organized gate families across local-complementation orbits using minimum-edge representatives, allowing purification circuits to be transferred between locally Clifford-equivalent graph states.
Research overview
This work represents a restricted family of Clifford operations as permutations of graph-basis labels. Factoring these operations and organizing them by local-complementation orbits makes it possible to search for purification circuits across related graph states. Numerical studies evaluate finite-size circuits with gate and measurement noise.
Cite this work
Mingyuan Wang, Guus Avis, Kenneth Goodenough, Stefan Krastanov. Efficient Graph State Purification with Factorized Graph-Preserving Operations across Local Clifford Orbits. arXiv:2606.23809 (2026). 10.48550/arXiv.2606.23809.
BibTeX citation
@misc{wang2026graphstatepurification,
title = {{Efficient Graph State Purification with Factorized Graph-Preserving Operations across Local Clifford Orbits}},
author = {Mingyuan Wang and Guus Avis and Kenneth Goodenough and Stefan Krastanov},
year = {2026},
eprint = {2606.23809},
archivePrefix = {arXiv},
primaryClass = {quant-ph},
doi = {10.48550/arXiv.2606.23809},
url = {https://doi.org/10.48550/arXiv.2606.23809}
}