Moment Optimization in the Navascués-Pironio-Acín Hierarchy
Abstract
The Navascués-Pironio-Acín (NPA) hierarchy provides a convergent sequence of semidefinite programming (SDP) relaxations for noncommutative polynomial optimisation, ubiquitous in quantum physics. However, its practical applicability is limited by the combinatorial growth in operator moments required at each level. Since not all moments contribute equally to bound tightness, selecting moments within a fixed computational budget is a relevant problem. We reframe moment selection as combinatorial subset selection and show it is governed by strong higher-order synergistic interactions among moments, quantified through a marginal synergy diagnostic adapted from complex systems theory. We develop and compare three optimisation methods: Parallel Tempering (PT), an RBM-based reinforcement learning policy, and Bayesian Optimisation (BO). On the I3322 Bell inequality benchmark, all three substantially outperform greedy approaches at costs around two orders of magnitude below brute force, with the RBM achieving the closest approach to optimal throughout the hard transition regime. We apply the framework to the 174 Bell inequalities in the $(4,4,2,2)$ scenario, finding heterogeneous convergence behaviour across inequalities, and to the one-dimensional Heisenberg spin chain, demonstrating that physically motivated monomial bases are internally compressible and are not globally optimal in general. A budget-aware search over a broader pool improves certified bounds on long-range correlations by nearly two orders of magnitude. These results establish a scalable framework for moment selection in noncommutative polynomial optimisation, with broad applications across quantum physics and quantum information.
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BibTeX
@misc{flora2026moment,
title = {{Moment Optimization in the Navascués-Pironio-Acín Hierarchy}},
author = {Francesco Flora and Losel Matos and Tim Heightman and Tamás Kriváchy and Adan Garriga and Antonio Acín},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.14755},
eprint = {2607.14755},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.14755},
}
RIS
TY - UNPB
TI - Moment Optimization in the Navascués-Pironio-Acín Hierarchy
AU - Francesco Flora
AU - Losel Matos
AU - Tim Heightman
AU - Tamás Kriváchy
AU - Adan Garriga
AU - Antonio Acín
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2607.14755
UR - https://arxiv.org/abs/2607.14755
AB - The Navascués-Pironio-Acín (NPA) hierarchy provides a convergent sequence of semidefinite programming (SDP) relaxations for noncommutative polynomial optimisation, ubiquitous in quantum physics. However, its practical applicability is limited by the combinatorial growth in operator moments required at each level. Since not all moments contribute equally to bound tightness, selecting moments within a fixed computational budget is a relevant problem. We reframe moment selection as combinatorial subset selection and show it is governed by strong higher-order synergistic interactions among moments, quantified through a marginal synergy diagnostic adapted from complex systems theory. We develop and compare three optimisation methods: Parallel Tempering (PT), an RBM-based reinforcement learning policy, and Bayesian Optimisation (BO). On the I_3322 Bell inequality benchmark, all three substantially outperform greedy approaches at costs around two orders of magnitude below brute force, with the RBM achieving the closest approach to optimal throughout the hard transition regime. We apply the framework to the 174 Bell inequalities in the $(4,4,2,2)$ scenario, finding heterogeneous convergence behaviour across inequalities, and to the one-dimensional Heisenberg spin chain, demonstrating that physically motivated monomial bases are internally compressible and are not globally optimal in general. A budget-aware search over a broader pool improves certified bounds on long-range correlations by nearly two orders of magnitude. These results establish a scalable framework for moment selection in noncommutative polynomial optimisation, with broad applications across quantum physics and quantum information.
ER -