[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85244-en":3,"doc-seo-85244-105":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":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85244,1374391974564,"Clementine","https://ap-avatar.wpscdn.com/avatar/14000253aa45c000a9e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779874745381141002",8,"Research & Report","Spectral Gap of Lee-Yang Hamiltonians","Lee-Yang theorem and quantum extensions constrain the zeros of a graph Hamiltonian’s partition function in the complex magnetic-field plane to the imaginary axis. For Lee-Yang Hamiltonians subject to a uniform Z-field of strength h, the paper proves a nonzero uniform lower bound: the ground state has a spectral gap at least h/4, independent of system size and coupling strengths. The argument leverages partition-function zero-freeness (Asano and Suzuki-Fisher) to obtain exponential decay of imaginary-time Z-operator correlations, yielding a polynomial-time quantum algorithm for ground-state energy.","Spectral gap of Lee-Yang Hamiltonians  \nChaithanya Rayudu ∗ University of New Mexico  \nJun Takahashi † University of Tokyo  \narXiv :2607 . 10765v1 [ quant-ph] 12 Jul 2026  \nAbstract  \nThe Lee-Yang theorem and its quantum extensions state that, for a broad class of Hamiltonians on any graph, the partition function’s zeros in the complex magnetic field plane lie only on the imaginary axis. For these Hamiltonians, we prove that under a uniform Z-field of any strength h, the ground state has a spectral gap of at least h/4, independent of the system size and of the coupling strengths. The proof uses the zero-freeness of the partition function as given by Asano and Suzuki-Fisher to show exponential decay of the imaginary-time correlations for any product of Z-operators. Our result gives a polynomial-time quantum algorithm for computing the ground state energy of any Lee-Yang Hamiltonian.  \n1 Introduction  \nThe spectral gap of a quantum Hamiltonian plays a fundamental role in determining its physical properties, especially at low temperature. A many-body system with a constant energy gap shows exponentially decaying spatial correlations in the ground state [1 , 2], which contrasts sharply with gapless critical systems. Gappedness also implies stability of the phase against local perturbations [3, 4] and leads to the notion of topological phases of matter [5, 6] where the closing of a gap marks a quantum phase transition [7, 8] . While serving an essential role in understanding physical properties, rigorously proving the existence of a gap for a given Hamiltonian in general is notoriously difficult. For example, the Haldane conjecture [9] that the antiferromagnetic Heisenberg chain is gapped for integer spin remains open even after four decades, despite overwhelming numerical evidence [10 , 11 , 12] .  \nThe spectral gap also plays an important role in quantum computation. Adiabatic quantum computation [13 , 14 , 15], which is a universal model of quantum computation [16], works along paths of Hamiltonians where the run time scales with inverse powers of the minimum gap along a path [17] . An inverse polynomial gap along a path that begins at a Hamiltonian with an easily prepared ground state thus makes the ground state at its end efficiently preparable. The inverse gap also controls the cost of ground-state preparation by phase estimation [18], given an initial state with nonnegligible ground-state overlap. In one dimension, a gap further implies an area-law for the ground-state entanglement [19 , 20] and a polynomial-time classical algorithm for computing the ground state [21] . In general, however, deciding whether a family of Hamiltonians is gapped is undecidable, even for translation-invariant nearest-neighbor interactions in two dimensions [22] .  \n∗ [chaithanyarss@unm.edu](chaithanyarss@unm.edu)  \n†[juntakahashi@issp.u-tokyo.ac.jp](juntakahashi@issp.u-tokyo.ac.jp)  \nIn this paper, we prove a uniform lower bound on the field-induced spectral gap for a broad class of Hamiltonians, which we refer to as Lee-Yang Hamiltonians (Definition 1), thus giving an efficient adiabatic quantum algorithm for the corresponding ground state energy problem.  \nTheorem (main result, Theorem 11) . Let H be a Lee-Yang Hamiltonian on n qubits and let h > 0. Then the ground state of Hh = H − hP Zi is nondegenerate, and the spectral gap above it is at least h/4 .  \nThe Hamiltonian family we consider acts on n qubits with 2-local terms:  \nDefinition (Lee-Yang Hamiltonians, Definition 1) .  \nH = XHij where Hij = −wzijZiZj + wxijXiXj + wyijYiYj + wxijyXiYj + wyijxYiXj and ij  \n∀ij, wzij ≥ 12 h 􀀀wxij − wyij􀀁 2 +􀀀wxijy + wyijx 􀀁 2i ~~1~~2 + 12 h 􀀀wxij + wyij􀀁 2 +􀀀wxijy − wyijx 􀀁 2i ~~1~~2 .  \n1.1 Algorithmic implications  \nOur result answers an open question that has attracted considerable attention in recent years about the complexity of the bipartite quantum Max-Cut problem [23 , 24 , 25 , 26 , 27 , 28 , 29] . We make progress on this question by placing th","cbCaibQy1lhVe4vR","https://ap.wps.com/l/cbCaibQy1lhVe4vR","pdf",535609,4,1,23,"English","en",105,"# Abstract\n# Introduction\n# Main Result\n## Algorithmic implications\n## Related works","[{\"question\":\"What do the Lee-Yang theorem and its quantum extensions state about partition function zeros?\",\"answer\":\"For a broad class of graph Hamiltonians, partition function zeros in the complex magnetic-field plane lie only on the imaginary axis.\"},{\"question\":\"What spectral gap bound is proven for Lee-Yang Hamiltonians under a uniform Z-field?\",\"answer\":\"For Hh = H − hP Zi with h \\u003e 0, the ground state is nondegenerate and the spectral gap above it is at least h/4.\"},{\"question\":\"How does the proof establish the gap?\",\"answer\":\"It uses zero-freeness of the partition function (Asano and Suzuki-Fisher) to show exponential decay of imaginary-time correlations for any product of Z-operators.\"}]",1784202026,58,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"spectral-gap-of-lee-yang-hamiltonians","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/spectral-gap-of-lee-yang-hamiltonians/85244/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What do the Lee-Yang theorem and its quantum extensions state about partition function zeros?","Question",{"text":75,"@type":76},"For a broad class of graph Hamiltonians, partition function zeros in the complex magnetic-field plane lie only on the imaginary axis.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What spectral gap bound is proven for Lee-Yang Hamiltonians under a uniform Z-field?",{"text":80,"@type":76},"For Hh = H − hP Zi with h > 0, the ground state is nondegenerate and the spectral gap above it is at least h/4.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proof establish the gap?",{"text":84,"@type":76},"It uses zero-freeness of the partition function (Asano and Suzuki-Fisher) to show exponential decay of imaginary-time correlations for any product of Z-operators.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]