[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-450316-105":59,"doc-detail-450316-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","quantum-computing-algorithms-for-inverse-problems-on-graphs-an-np-complete-inverse-problem","QUANTUM COMPUTING ALGORITHMS FOR INVERSE PROBLEMS ON GRAPHS - An NP-Complete Inverse Problem","","Study inverse problems on finite graphs from partial vertex distance data: given a subset of vertices B and all pairwise graph distances d(b1,b2), the goal is to reconstruct the underlying graph structure. The work develops quantum computing methods combining a qubit graph representation with Grover’s search. Uniqueness is proved for trees when B is the set of leaves, and an algorithm outputs the required connected graph(s) with matching distances. The approach uses O(|X|^2) qubits and yields a quadratic speedup, while a slight modification becomes NP-complete via reduction from all NP problems.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/quantum-computing-algorithms-for-inverse-problems-on-graphs-an-np-complete-inverse-problem/450316/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/quantum-computing-algorithms-for-inverse-problems-on-graphs-an-np-complete-inverse-problem/450316.png","ImageObject",300,407,{"name":92,"@type":93},"Rowan","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-06","2026-09-30",true,{"@type":102,"interactionType":103,"userInteractionCount":81},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What inverse problem is studied for finite graphs?","Question",{"text":112,"@type":113},"The paper studies reconstructing a finite graph from distances between a specified subset of vertices B, where distances are defined as the minimal number of edges along paths connecting vertices.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"When is the graph uniquely determined in the proposed framework?",{"text":117,"@type":113},"A uniqueness result is proved: if the graph is a tree and B is the set of leaves, then the graph is uniquely determined among connected graphs with the same number of vertices.",{"name":119,"@type":110,"acceptedAnswer":120},"How does the quantum algorithm work and what is its resource usage?",{"text":121,"@type":113},"The algorithm combines a qubit representation of a graph with Grover’s search to produce a graph matching the required vertex distances. It can be implemented using only O(|X|^2) qubits, with quadratic improvement over standard classical methods.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},450316,1790909995,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":81,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},1099514067415,"https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502","[https://helda.helsinki.fi](https://helda.helsinki.fi)  \nQuantum computing algorithms for inverse problems on graphs and an NP-complete inverse problem Ilmavirta, Joonas; Lassas, Matti; Lu, Jinpeng; Oksanen, Lauri; Ylinen, Lauri 2025-08  \nAmerican Institute of Mathematical Sciences  \n[http://hdl.handle.net/10138/590602](http://hdl.handle.net/10138/590602)  \nIlmavirta, J, Lassas, M, Lu, J, Oksanen, L & Ylinen, L 2025, 'Quantum computing algorithms for inverse problems on graphs and an NP-complete inverse problem', Inverse problems and imaging, vol. 19, no. 4, 2024049, pp. 660-692. [https://doi.org/10.3934/ipi](https://doi.org/10.3934/ipi). 2024049  \nDownloaded from Helda, University of Helsinki institutional repository. [https://helda.helsinki.fi](https://helda.helsinki.fi)[ ](https://helda.helsinki.fi)[This is an electronic reprint of the original article.](This is an electronic reprint of the original article.)  \nThis reprint may differ from the original in pagination and typographic detail.  \nPlease cite the original version.  \nQUANTUM COMPUTING ALGORITHMS  \nFOR INVERSE PROBLEMS ON GRAPHSAND AN NP-COMPLETE INVERSE PROBLEM  \nJoonas Ilmavirta􀀀1, Matti Lassas􀀀2, Jinpeng Lu􀀀∗2 , Lauri Oksanen􀀀2 and Lauri Ylinen􀀀3  \n1 Department of Mathematics and Statistics, University of Jyv¨askyl¨a, Finland  \n2 Department of Mathematics and Statistics, University of Helsinki, Finland  \n3 Department of Electronics and Nanoengineering, Aalto University, Finland (Communicated by Hongyu Liu)  \nAbstract. We consider an inverse problem for a finite graph (X, E) where we are given a subset of vertices B ⊂ X and the distances d (X, E)(b1, b2) between all pairs of vertices b1, b2 ∈ B . The distance between vertices x1, x2 ∈ X is defined as the minimal number of edges in paths connecting the vertices.  \nThis problem can be regarded as a discrete version of the boundary rigidity problem in Riemannian geometry or the inverse travel time problem in geophysics. We develop quantum computing methods to find a solution to the problem, and show that the solution is unique under certain conditions. We prove the following uniqueness result: when (X, E) is a tree and B is the set of leaves of the tree, the graph (X, E) is uniquely determined in the class of all connected graphs having a fixed number of vertices. We present a quantum algorithm which, under arbitrary conditions, produces the graph (X, E), or oneof those, which has the given number of vertices and the required distances between vertices in B . To this end we develop an algorithm that takes in aqubit representation of a graph and combine it with Grover’s search algorithm.  \nThe algorithm can be implemented using only O(|X|2 ) qubits, the same order as the number of elements in the adjacency matrix of (X, E), and it has a quadratic improvement in computational cost compared to standard classical algorithms. Finally, we consider applications in the theory of computation, and show that a slight modification of the inverse problem above is NP-complete:  \nall NP-problems can be reduced to a discrete inverse problem that we consider.  \n1. Introduction. We consider inverse travel time problems for graphs and prove a uniqueness result: a tree with n vertices can be uniquely reconstructed from the leafto-leaf distances in the class of all connected graphs with n vertices. Furthermore, we demonstrate how quantum computing algorithms can be used to solve discrete inverse problems, in particular, inverse travel time problems for finite graphs. We also show that a certain generalized inverse travel time problem for graphs is an NP-complete problem.  \n2020 Mathematics Subject Classification. Primary: 68Q12, 52C25; Secondary: 68Q17 .  \nKey words and phrases. Inverse travel time problem, boundary rigidity, graph, quantum algorithm, NP-completeness.  \n∗ Corresponding author: Jinpeng Lu.  \nQUANTUM COMPUTING ALGORITHMS FOR INVERSE PROBLEMS 661  \nThe inverse travel time problem for graphs can be seen as a discrete analog","cbCaiscI8362MmU2","https://ap.wps.com/l/cbCaiscI8362MmU2","pdf",599165,34,"English","# Abstract\n# Introduction\n## Inverse travel time and boundary rigidity\n## Applications and related discrete inverse problems\n## NP-completeness of the generalized problem","[{\"question\":\"What inverse problem is studied for finite graphs?\",\"answer\":\"The paper studies reconstructing a finite graph from distances between a specified subset of vertices B, where distances are defined as the minimal number of edges along paths connecting vertices.\"},{\"question\":\"When is the graph uniquely determined in the proposed framework?\",\"answer\":\"A uniqueness result is proved: if the graph is a tree and B is the set of leaves, then the graph is uniquely determined among connected graphs with the same number of vertices.\"},{\"question\":\"How does the quantum algorithm work and what is its resource usage?\",\"answer\":\"The algorithm combines a qubit representation of a graph with Grover’s search to produce a graph matching the required vertex distances. It can be implemented using only O(|X|^2) qubits, with quadratic improvement over standard classical methods.\"}]","QUANTUM COMPUTING ALGORITHMS FOR INVERSE PROBLEMS ON GRAPHS - An NP-Complete Inverse Problem | PDF",1790732857,86]