[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-121800-en":3,"doc-seo-121800-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},121800,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Improved machine learning algorithm for predicting ground state properties - Supplementary Information","Supplementary material to the study on predicting ground-state properties using a machine learning model trained on data whose scaling is only logarithmic in system size. It derives a simpler representation for Tr(Ôρ_g.s.(x)) by approximating it with sums of smooth and then simple local functions, supported by spectral flow and Lieb–Robinson bounds. The document establishes a norm inequality for observables decomposed into geometrically local terms, enabling an ML design that achieves rigorous sample-complexity guarantees, along with bounds for training and prediction errors and notes on computational time.","Improved machine learning algorithm for predicting ground state properties  \nLaura Lewis, 1, * Hsin-Yuan Huang, 1 Viet T. Tran,2 Sebastian Lehner,2 Richard Kueng,2 and John Preskill 1, 3  \n1 California Institute of Technology, Pasadena, CA, USA  \n2 Johannes Kepler University, Linz, Austria  \n3 AWS Center for Quantum Computing, Pasadena, CA, USA (Dated: December 7, 2023)  \nSupplementary Information  \nContents  \nI. Simple form for ground state property 2  \nA. Approximation by a sum of smooth functions 2  \nB. Simplification using discretization 10  \nC. Simple form for ground state property 12  \nD. Technical lemmas for finding constants and bounding integrals 13  \nII. Norm inequality for observables 18  \nA. Facts and lemmas 18  \nB. Proof of Theorem 4 19  \nIII. ML algorithm and sample complexity 23  \nA. ML algorithm 23  \nB. Rigorous guarantee 25  \nC. ℓ 1-Norm bound on coefficients of linear hypothesis 26  \nD. Training error bound 28  \nE. Prediction error bound 29  \nF. Computational time for training and prediction 30  \nIV. Details of numerical experiments 30  \nThese appendices provide detailed proofs of the statements in the main text. We discuss our main contribution that Tr(􀁏􀀚) can be approximated by a machine learning model given training data scaling logarithmically in system size, where 􀁏 is an unknown observable and 􀀚 is the ground state of a Hamiltonian. The proof of this result has three main parts. The first two parts yield important results necessary for the design of the ML algorithm and its sample complexity.  \nWe recommend that readers start with Section I, which derives a simpler form for the ground state property Tr(􀁏􀀚(􀁸)) that we wish to predict. In Section II, we give a norm inequality characterizing the Pauli coefficients of any observable that can be written as a sum of geometrically local observables. The norm inequality reveals a structure of the ground state property Tr(􀁏􀀚(􀁸)) that we can use to design an ML algorithm that uses very few training data. In Section III, we present our ML algorithm and prove its sample complexity using standard tools in ML theory, including known guarantees on the LASSO (least absolute shrinkage and selection operator) algorithm’s performance. Finally, in Section IV, we describe numerical experiments performed to assess the performance of the algorithm in practice.  \n* Electronic address: [llewis@alumni.caltech.edu](llewis@alumni.caltech.edu)  \n2  \nI. SIMPLE FORM FOR GROUND STATE PROPERTY  \nThis section is dedicated to deriving a simpler form for the ground state property Tr(􀁏􀀚(􀁸)) as a function of 􀁸 . Note that throughout this section, log denotes the natural logarithm. We consider the assumptions (a)-(d) from Appendix F.5 of [1], with (b) and (d) adjusted for our setting, which wereproduce here for convenience:  \n(a) Physical system: We consider 􀁮 finite-dimensional quantum systems that are arranged at locations, or sites, in a 􀁤-dimensional space, e.g., a spin chain (􀁤 = 1), a square lattice (􀁤 = 2), or a cubic lattice (􀁤 = 3) . Unless specified otherwise, our big-􀁏 , Ω , Θ notation is with respect to the thermodynamic limit 􀁮 → ∞ .  \n(b) Hamiltonian: 􀁈 (􀁸) decomposes into a sum of geometrically local terms 􀁈 (􀁸) = ∑︀􀁌􀁪=1 ℎ􀁪 (⃗􀁸􀁪), each of which only acts on an 􀁏(1) number of sites in a ball of 􀁏(1) radius. Here, ⃗􀁸􀁪 ∈ R􀁱 ,􀁱 = 􀁏(1) and 􀁸 is the concatenation of 􀁌 vectors ⃗􀁸1 , . . . ,⃗􀁸 􀁌 with dimension 􀁭 = 􀁌􀁱 = 􀁏 (􀁮) . Individual terms ℎ􀁪 (⃗􀁸􀁪) obey ‖ℎ􀁪 (⃗􀁸􀁪)‖∞ ≤ 1 and also have bounded directional derivative: ‖􀁀ℎ􀁪 /􀁀􀁵^‖∞ ≤ 1 , where 􀁵^ is a unit vector in parameter space.  \n(c) Ground-state subspace: We consider the ground state 􀀚(􀁸) for the Hamiltonian 􀁈(􀁸) to be defined as 􀀚 (􀁸) = lim􀀌 →∞ 􀁥 −􀀌􀁈(􀁸)/Tr(︀􀁥−􀀌􀁈(􀁸))︀ . This is equivalent to a uniform mixture over the eigenspace of 􀁈(􀁸) with the minimum eigenvalue.  \n(d) Observable: 􀁏 can be written as a sum of few-body observables 􀁏 = ∑︀􀁪 􀁏􀁪 , where each 􀁏􀁪 only acts on an 􀁏(1) number of sites. Hence, we can also write 􀁏 = ∑︀􀁐∈􀁓(","cbCaic8zZppivQei","https://ap.wps.com/l/cbCaic8zZppivQei","pdf",3490731,1,33,"English","en",105,"# Contents\n## Simple form for ground state property\n## Norm inequality for observables\n## ML algorithm and sample complexity\n## Details of numerical experiments","[{\"question\":\"What quantity does the document focus on predicting?\",\"answer\":\"It targets the ground-state property Tr(Ôρ_g.s.(x)), where Ô is an unknown observable and ρ_g.s.(x) is the ground state of a Hamiltonian.\"},{\"question\":\"How is Tr(Ôρ_g.s.(x)) simplified for the ML approach?\",\"answer\":\"The material derives a simpler form by approximating Tr(Ôρ_g.s.(x)) with a sum of smooth local functions and then approximating that sum with simple functions, with technical lemmas to bound integrals.\"},{\"question\":\"What guarantees are provided for the machine learning method?\",\"answer\":\"It presents an ML algorithm and proves rigorous sample complexity and error bounds for training and prediction, using standard ML theory tools including performance guarantees for LASSO.\"}]","Improved machine learning algorithm for predicting ground state properties - 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