Local minima in quantum systems

[2309.16596] Local minima in quantum systems Skip to main content Search Submit Donate Log in Quantum Physics arXiv:2309.16596 (quant-ph) [Submitted on 28 Sep 2023] Title: Local minima in quantum systems Authors: Chi-Fang Chen , Hsin-Yuan Huang , John Preskill , Leo Zhou View PDF HTML (experimental) Abstract: Finding ground states of quantum many-body systems is known to be hard for both classical and quantum computers. As a result, when Nature cools a quantum system in a low-temperature thermal bath, the ground state cannot always be found efficiently. Instead, Nature finds a local minimum of the energy. In this work, we study the problem of finding local minima in quantum systems under thermal perturbations. While local minima are much easier to find than ground states, we show that finding a local minimum is computationally hard for classical computers, even when the task is to output a single-qubit observable at any local minimum. In contrast, we prove that a quantum computer can always find a local minimum efficiently using a thermal gradient descent algorithm that mimics the cooling process in Nature. To establish the classical hardness of finding local minima, we consider a family of two-dimensional Hamiltonians such that any problem solvable by polynomial-time quantum algorithms can be reduced to finding ground states of these Hamiltonians. We prove that for such Hamiltonians, all local minima are global minima. Therefore, assuming quantum computation is more powerful than classical computation, finding local minima is classically hard and quantumly easy. Comments: 9+80 pages, 4 figures Subjects: Quantum Physics (quant-ph) ; Disordered Systems and Neural Networks (cond-mat.dis-nn); Computational Complexity (cs.CC); Mathematical Physics (math-ph); Optimization and Control (math.OC) Cite as: arXiv:2309.16596 [quant-ph] (or arXiv:2309.16596v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2309.16596 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Nature Physics 21, 654--660 (2025) Related DOI : https://doi.org/10.1038/s41567-025-02781-4 Focus to learn more DOI(s) linking to related resources Submission history From: Leo Zhou [ view email ] [v1] Thu, 28 Sep 2023 16:59:05 UTC (1,537 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: quant-ph < prev | next > new | recent | 2023-09 Change to browse by: cond-mat cond-mat.dis-nn cs cs.CC math math-ph math.MP math.OC References & Citations INSPIRE HEP SciX Google Scholar Semantic Scholar 2 blog links ( what is this? ) export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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