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This research establishes a rigorous bridge between non-commutative polynomial optimization and quantum dynamics by proving that differential constraints in operator variables can be mapped to standard semidefinite programming hierarchies. The method effectively bounds observables in large-scale quantum spin systems undergoing Hamiltonian evolution, demonstrating high accuracy even at low levels of the relaxation hierarchy for long-duration quench processes. This advancement provides a scalable mathematical framework for analyzing complex time-dependent phenomena in many-body quantum physics without requiring full state tomography.
Non-commutative optimization problems with differential constraints
arXiv:2408.02572v5 Announce Type: replace 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.
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.
Quantum Physics (quant-ph); Optimization and Control (math.OC)
arXiv:2408.02572 [quant-ph]
(or arXiv:2408.02572v5 [quant-ph] for this version)
https://doi.org/10.22331/q-2026-07-22-2169 DOI(s) linking to related resources
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