Going beyond energy: ground-state properties unlocked in a certifiable and scalable way
A team of researchers theoretically obtain, for the first time, certifiable bounds on many-body ground-state properties beyond the energy with competitive performance. The resulting method is general and scalable
A set of particles that interact with each other tends to minimize its energy. When the whole system actually reaches the minimum energy, its description is given by the so-called ‘ground-state’, or state of minimal energy. Scientists are thus very interested in discovering and analyzing its properties, which of course include -but are not limited to- the energy. For instance, magnetic and conduction properties are other worthwhile knowing features, to name just a few.
However, discovering the exact value of these ground-state properties becomes harder as the number of particles increases, reaching a point where even a supercomputer would not be able to find the solution. To circumvent this obstacle, there are two numerical methods that bound the energy of the ground state: the variational and the relaxation methods. They provide an upper and a lower bound, respectively. This means that, even though the exact ground-state energy is, strictly speaking, still uncertain, it will be for sure contained between those two values. The closer the upper and the lower bound are, the less uncertainty there will be on the energy.
Variational and relaxation methods have proven to be really effective, providing tight enough bounds to be able to infer the ground-state energy with the desired accuracy in a wide variety of physical problems. Nevertheless, when these techniques are applied to other different properties, it is not possible to know whether the obtained quantities are close to the real value, as they are no longer ensured to be upper or lower bounds. Finding non-trivial bounds would not only allow scientists to check if the information provided by variational methods was on the right path, but would also lead by itself to a certified estimation on the real value of these ground-state properties.
This issue has now been tackled by an international team in a Physical Review X article, with Dr. Jie Wang from the Chinese Academy of Sciences as its leading author and the participation of ICFO researchers Dr. Jacopo Surace and ICREA Prof. Antonio Acín, as well as the Perimeter Institute for Theoretical Physics, Université Grenoble Alpes, Sobbornne Université, Collège de France, ESAT, Inria Paris-Saclay, Institute Polytechnique of Paris, LAAS-CNRS and the Institute of Mathematics from Toulouse. They show how, by taking into account the variational and relaxation results for the energy, one can derive certifiable bounds on other ground-state properties in a scalable way.
Towards the certification of any ground-state property
With their approach, consisting in a numerical method called semi-definite programming (SDP) relaxation, one can be sure that the actual value of a given ground-state property lays within the obtained range. Again, just as what happened with the energy, the ability to get the limiting bounds closer will increase the accuracy of the predictions. The novelty of the article lies in the fact that the energy bounds given by the variational and relaxation methods are now considered. The strategy leads to a significant improvement (by an order of magnitude) with respect to previous attempts, being the first time that a method exhibits competitive performance when certifying ground-state properties beyond the energy.
The team benchmarked their method with several models (Heisenberg models) that described an array of interacting particles with spin ½ (for instance, electrons). The addressed properties were the spin-spin correlation functions, which give direct information about whether the system behaves ferromagnetically or antiferromagnetically. In all the cases their SDP relaxation provided bounds in agreement with the expectations. They also successfully bounded the spin-spin correlation function of a particular system (J1 – J2 square lattice Heisenberg model for 100 spins) whose exact calculation is currently out of reach, showcasing the potential of the proposed technique.
Although SDP relaxations are just in their infancy and many improvements can be made to bring upper and lower bounds closer together, the results obtained show great promise. Moreover, their tool is completely general, so in principle it could be applied to any other observable of interest. “Our method offers deep insights into the understanding of phases of matter, how particles arrange to minimize their energy, which is essential to understand many phenomena, from chemical processes to material design”, claims ICREA Prof. at ICFO Antonio Acín. “And we are confident that our approach will become a central tool in the study of ground-state problems, a ubiquitous issue in science”.
Funding Acknowledgements:
This work is supported by the ERC AdG CERQUTE, the Government of Spain (NextGenerationEU PRTR-C17.I1 and Quantum in Spain, Severo Ochoa CEX2019-000910-S), Fundació Cellex, Fundació Mir-Puig, Generalitat de Catalunya (CERCA programme), the AXA Chair in Quantum Information Science, EU projects PASQUANS2, NEQST, and Quantera Veriqtas, the National Natural Science Foundation of China under Grant No. 12201618, the European Union Horizon’s 2020 research and innovation programme under the Marie Sklodowska Curie Grant Agreement No. 101031549 (QuoMoDys), the NSF Grant No. OAC-1835443 and the ERC Adv. Grant Grant Agreement No. 885682. Research at the Perimeter Institute for Theoretical Physics is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science. V. M. was supported by the FastQI grant funded by the Institut Quantique Occitan, the PHC Proteus Grant No. 46195TA, the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Actions, Grant Agreement No. 813211 (POEMA), by the AI Interdisciplinary Institute ANITI funding, through the French “Investing for the Future PIA3” program under the Grant Agreement No. ANR-19-PI3A-0004 as well as by the National Research Foundation, Prime Minister’s Office, Singapore under its Campus for Research Excellence and Technological Enterprise (CREATE) program. This work was partially performed using HPC resources from CALMIP (Grant No. 2023-P23035). M. O. R. acknowledges funding by the ANR for the JCJC grant LINKS (ANR-23-CE47-0003)