Non-commutative optimization problems with differential constraints

Mateus Araújo; Andrew J. P. Garner; Miguel Navascués

doi:10.22331/q-2026-07-22-2169

Non-commutative optimization problems with differential constraints

Mateus Araújo¹, Andrew J. P. Garner², and Miguel Navascués²

¹Departamento de Física Teórica, Atómica y Óptica, and Laboratory for Disruptive Interdisciplinary Science (LaDIS), Universidad de Valladolid, 47011 Valladolid, Spain
²Institute for Quantum Optics and Quantum Information (IQOQI) Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, Wien 1090, Austria

Published:	2026-07-22, volume 10, page 2169
Editor:	Abhinav Deshpande
Eprint:	arXiv:2408.02572v4
Doi:	https://doi.org/10.22331/q-2026-07-22-2169
Citation:	Quantum 10, 2169 (2026).

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Abstract

Non-commutative polynomial optimization (NPO) problems seek to minimize the state average of a polynomial of some operator variables, subject to polynomial constraints, over all states and operators, as well as the Hilbert spaces where those might be defined. Many of these problems are known to admit a complete hierarchy of semidefinite programming (SDP) relaxations. In this work, we consider a variant of NPO problems where a subset of the operator variables satisfies a system of ordinary differential equations. We prove that, under mild conditions of operator boundedness, for every such problem one can construct a standard NPO problem with the same solution. This allows us to define a complete hierarchy of SDPs to tackle the original differential problem. We apply this method to bound averages of local observables in quantum spin systems subject to a Hamiltonian evolution (i.e., a quench). We find that, even in the thermodynamic limit of infinitely many sites, low levels of the hierarchy provide very good approximations for reasonably long evolution times.

Popular summary

Many problems in quantum physics require optimizing a figure of merit over operators following an equation of motion. Determining how the magnetization of a material changes with time when one switches on a magnetic field, estimating how random an experimental outcome can be on a rotating device or predicting the future behavior of low-energy quantum systems are examples of such problems. In this paper, we show a systematic way to solve them all, by mapping them to a well-known class of “static problems” that can be tackled through hierarchies of semidefinite programming relaxations. As a spin-off, we develop an algorithm to bound the local properties of quenched quantum spin systems, valid even in the thermodynamical limit of infinitely many sites.

► BibTeX data

@article{Araujo2026noncommutative,
  doi = {10.22331/q-2026-07-22-2169},
  url = {https://doi.org/10.22331/q-2026-07-22-2169},
  title = {Non-commutative optimization problems with differential constraints},
  author = {Ara{\'{u}}jo, Mateus and Garner, Andrew J. P. and Navascu{\'{e}}s, Miguel},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2169},
  month = jul,
  year = {2026}
}

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