Generating Dicke State Graphs
Abstract
Graph theory is a powerful tool in quantum computing, with applications ranging from quantum circuit synthesis and optimization to entanglement mapping. Recent work has shown how one can use edge-colored graphs to model photonic experiments that generate GHZ and W states. However, the latter work also proved that verifying that a graph models a Dicke state experiment is coNP-complete. In this work, we provide families of graphs that generate |Dka⟩ ⊗ |0⟩⊗ b , where b = |a−2k| is the number of spectator modes. The graph setup consists of a doubled complete subgraph on a vertices and a collection of auxiliary vertices. We prove that every coincidence carries exactly k excitations, every weight-k computational basis state on bitstrings of length a is realized, and each of those bitstrings is realized exactly (n/2)! times, where n = a+b. Since verifying the Dicke FORALL condition is coNP-complete in general, constructing explicit families that provably generate Dicke states is of interest.
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BibTeX
@misc{herrman2026generating,
title = {{Generating Dicke State Graphs}},
author = {Rebekah Herrman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22564},
eprint = {2609.22564},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22564},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Generating Dicke State Graphs
AU - Rebekah Herrman
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22564
UR - https://arxiv.org/abs/2609.22564
AB - Graph theory is a powerful tool in quantum computing, with applications ranging from quantum circuit synthesis and optimization to entanglement mapping. Recent work has shown how one can use edge-colored graphs to model photonic experiments that generate GHZ and W states. However, the latter work also proved that verifying that a graph models a Dicke state experiment is coNP-complete. In this work, we provide families of graphs that generate |D_k^a⟩ ⊗ |0⟩^⊗ b , where b = |a−2k| is the number of spectator modes. The graph setup consists of a doubled complete subgraph on a vertices and a collection of auxiliary vertices. We prove that every coincidence carries exactly k excitations, every weight-k computational basis state on bitstrings of length a is realized, and each of those bitstrings is realized exactly (n/2)! times, where n = a+b. Since verifying the Dicke FORALL condition is coNP-complete in general, constructing explicit families that provably generate Dicke states is of interest.
ER -