[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-121836-en":3,"doc-seo-121836-105":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":21,"is_downloadable":21,"audit_status":21,"page_count":11,"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},121836,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Improved Machine Learning Algorithm for Predicting Ground State Properties","A classical machine learning method is presented to predict ground state properties of n-qubit gapped geometrically local Hamiltonians. The approach introduces an inductive bias that encodes geometric locality through a nonlinear feature map, then learns a regression function via ℓ1-regularized LASSO. After training on only O(log n) data points from other Hamiltonians in the same quantum phase, the model achieves average prediction error at most ε with high probability. Training and prediction time scale as O(n log n), supported by experiments up to 45 qubits.","Article [https://doi.org/10.1038/s41467-024-45014-7](https://doi.org/10.1038/s41467-024-45014-7)  \nImproved machine learning algorithm for predicting ground state properties  \nReceived: 26 March 2023  \nAccepted: 8 January 2024  \n\n| Published online: 30 January 2024 |\n| --- |\n| Check for updates |\n\nLaura Lewis1,5, Hsin-Yuan Huang 1,2,6 , Viet T. Tran3, Sebastian Lehner3, Richard Kueng3 & John Preskill1,4  \nFinding the ground state of a quantum many-body system is a fundamental problem in quantum physics. In this work, we give a classical machine learning (ML) algorithm for predicting ground state properties with an inductive bias encoding geometric locality. The proposed ML model can efficiently predict ground state properties of an n-qubit gapped local Hamiltonian after learning from only OðlogðnÞÞ data about other Hamiltonians in the same quantum phase of matter. This improves substantially upon previous results that require OðncÞ data for a large constant c. Furthermore, the training and prediction time of the proposed ML model scale as Oðn log nÞ in the number of qubits n. Numerical experiments on physical systems with up to 45 qubits confirm the favorable scaling in predicting ground state properties using a small training dataset.  \nFinding the ground state of a quantum many-body system is a fundamental problem with far-reaching consequences for physics, materials science, and chemistry. Many powerful methods1–7 have been proposed, but classical computers still struggle to solve many general classes of the ground state problem. To extend the reach of classical computers, classical machine learning (ML) methods have recently been adapted to study this and related problems both empirically and theoretically8–35. A recent work36 proposes a polynomial-time classical ML algorithm that can efficiently predict ground state properties of gapped geometrically local Hamiltonians, after learning from data obtained by measuring other Hamiltonians in the same quantum phase of matter. Furthermore36, shows that under a widely accepted conjecture, no polynomial-time classical algorithm can achieve the same performance guarantee. However, although the ML algorithm given in36 uses a polynomial amount of training data and computational time, the polynomial scaling OðncÞ has a very large degree c. Here, f ðxÞ = OðgðxÞÞ denotes that f(x) is asymptotically upper bounded by g(x) up to constant factors with respect to the limit n → ∞. Moreover, when the prediction error ϵ is small, the amount of training data grows exponentially in 1/ϵ, indicating that a very small prediction error cannot be achieved efficiently.  \nIn this work, we present an improved ML algorithm for predicting ground state properties. We consider an m-dimensional vector x ∈ [−1, 1]m that parameterizes an n-qubit gapped geometrically local Hamiltonian given as  \nHðxÞ = X hj ð!xj Þ,  \nj  \nð1Þ  \nwhere x is the concatenation of constant-dimensional vectors!x1 , ... ,!xL parameterizing the few-body interaction hj ð!xj Þ . Let ρ(x) be the ground state of H(x) and O be a sum of geometrically local observables with ∥O ∥∞ ≤ 1. We assume that the geometry of then-qubit system is known, but we do not know how hj ð!xj Þ is parameterized or what the observable O is. The goal is to learn a function h* (x) that approximates the ground state property TrðOρðxÞÞ from a classical dataset,  \n􀀂x‘,y‘􀀃 , 8‘=1, ... ,N, ð2Þ  \nwherey‘ ≈ TrðOρðx‘ÞÞ records the ground state property forxℓ ∈ [−1, 1]m sampled from an arbitrary unknown distribution D. Here,  \n1California Institute of Technology, Pasadena, CA, USA. 2Massachusetts Institute of Technology, Cambridge, MA, USA. 3Johannes Kepler University, Linz, Austria. 4AWS Center for Quantum Computing, Pasadena, CA, USA. 5Present address: University of Cambridge, Cambridge, UK. 6Present address:  \nGoogle Quantum AI, Venice, CA, USA. e-mail: [hsinyuan@caltech.edu](hsinyuan@caltech.edu)  \ny‘ ≈ TrðOρðx‘ÞÞ means that yℓ has additive error at most ϵ. If y‘ = TrðOρðx‘Þ","cbCaiiUTQGHQvmWA","https://ap.wps.com/l/cbCaiiUTQGHQvmWA","pdf",1007463,2,1,"English","en",105,"# Introduction\n## Problem and motivation\n## Prior work and limitations\n# Improved ML algorithm\n## Hamiltonian setting and learning target\n## Feature map with geometric inductive bias\n## LASSO-based learning\n# Guarantees and scaling\n## Sample complexity and prediction error\n## Runtime analysis\n# Numerical validation\n## Experiments and observed performance","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The work targets the task of predicting ground state properties of quantum many-body systems using classical computation.\"},{\"question\":\"How does the proposed algorithm incorporate geometric locality?\",\"answer\":\"It uses a nonlinear feature map whose components reflect geometrically local subsets of coordinates, embedding the system geometry as an inductive bias.\"},{\"question\":\"What training data and runtime scaling does the method achieve?\",\"answer\":\"The model can learn from only O(log n) training samples for n-qubit systems, with training and prediction time scaling as O(n log n), and numerical results support favorable performance up to 45 qubits.\"}]","Improved Machine Learning Algorithm for Predicting Ground State Properties | 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problem does the paper address?","Question",{"text":75,"@type":76},"The work targets the task of predicting ground state properties of quantum many-body systems using classical computation.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed algorithm incorporate geometric locality?",{"text":80,"@type":76},"It uses a nonlinear feature map whose components reflect geometrically local subsets of coordinates, embedding the system geometry as an inductive bias.",{"name":82,"@type":73,"acceptedAnswer":83},"What training data and runtime scaling does the method achieve?",{"text":84,"@type":76},"The model can learn from only O(log n) training samples for n-qubit systems, with training and prediction time scaling as O(n log n), and numerical results support favorable performance up to 45 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