TY - JOUR

T1 - Bounding the Joint Numerical Range of Pauli Strings by Graph Parameters

AU - Xu, Zhen Peng

AU - Schwonnek, René

AU - Winter, Andreas

N1 - Funding Information: Z.-P.X. acknowledges support from the National Natural Science Foundation of China (Grant No. 12305007), Anhui Provincial Natural Science Foundation (Grant No. 2308085QA29), the Deutsche Forschungsgemeinschaft (projects 447948357 and 440958198), the Sino-German Center for Research Promotion (project M-0294), the European Research Council (Consolidator Grant No. 683107/TempoQ), and the Alexander von Humboldt Foundation. R.S. acknowledges financial support from Quantum Valley Lower Saxony, Quantum Frontiers, and the BMBF projects ATIQ, QuBRA, and CBQD. A.W. was supported by the European Commission QuantERA grant ExTRaQT (Spanish MICIN project PCI2022-132965), by the Spanish MINECO (projects PID2019-107609GB-I00 and PID2022-141283NB-I00) with the support of European Regional Development Fund funds, by the Spanish MICIN with funding from European Union NextGenerationEU (Grant No. PRTR-C17.I1) and the Generalitat de Catalunya, by the Spanish MTDFP through the QUANTUM ENIA project Quantum Spain, funded by the European Union NextGenerationEU program within the framework of the \u201CDigital Spain 2026 Agenda,\u201D by the Alexander von Humboldt Foundation, and by the Institute for Advanced Study of the Technical University Munich.

PY - 2024/4/22

Y1 - 2024/4/22

N2 - The relations among a given set of observables on a quantum system are effectively captured by their so-called joint numerical range, which is the set of tuples of jointly attainable expectation values. Here we explore geometric properties of this construct for Pauli strings, whose pairwise commutation and anticommutation relations determine a graph G. We investigate the connection between the parameters of this graph and the structure of minimal ellipsoids encompassing the joint numerical range, and we develop this approach in different directions. As a consequence, we find counterexamples to a conjecture by de Gois et al. [Phys. Rev. A 107, 062211 (2023)], and answer an open question raised by Hastings and O'Donnell [STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pp. 776-789], which implies a new graph parameter that we call "β(G)."Furthermore, we provide new insights into the perennial problem of estimating the ground-state energy of a many-body Hamiltonian. Our methods give lower bounds on the ground-state energy, which are typically hard to come by, and might therefore be useful in a variety of related fields.

AB - The relations among a given set of observables on a quantum system are effectively captured by their so-called joint numerical range, which is the set of tuples of jointly attainable expectation values. Here we explore geometric properties of this construct for Pauli strings, whose pairwise commutation and anticommutation relations determine a graph G. We investigate the connection between the parameters of this graph and the structure of minimal ellipsoids encompassing the joint numerical range, and we develop this approach in different directions. As a consequence, we find counterexamples to a conjecture by de Gois et al. [Phys. Rev. A 107, 062211 (2023)], and answer an open question raised by Hastings and O'Donnell [STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pp. 776-789], which implies a new graph parameter that we call "β(G)."Furthermore, we provide new insights into the perennial problem of estimating the ground-state energy of a many-body Hamiltonian. Our methods give lower bounds on the ground-state energy, which are typically hard to come by, and might therefore be useful in a variety of related fields.

UR - http://www.scopus.com/inward/record.url?scp=85191154831&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2308.00753

DO - 10.48550/arXiv.2308.00753

M3 - Article

AN - SCOPUS:85191154831

VL - 5

JO - PRX Quantum

JF - PRX Quantum

IS - 2

M1 - 020318

ER -