[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82440-en":3,"doc-seo-82440-105":29,"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":13,"seo_description":14,"update_tm":27,"read_time":28},82440,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","A Quantum Path to Partial Differential Equations","A Quantum Path to Partial Differential Equations develops a quantum-computing framework for solving and analyzing PDEs through circuit-level building blocks. It begins with quantum state modeling, measurement, and efficient state preparation, then builds core primitives such as quantum Fourier transform, phase estimation, inner-product estimation (swap/Hadamard tests), postselection, amplitude amplification, and block encodings. The document connects finite-difference operators to quantum block encodings, derives PDE algorithm workflows, and illustrates worked examples like the Poisson equation and Schrödinger dynamics on discrete Laplacians.","· · ·  \narXiv :2607 .09639v1 [ quant-ph] 10 Jul 2026  \nA Quantum Path to Partial Differential Equations  \n  ⋄    \nXiantao Li  \nJuly 9, 2026  \nContents  \nPreface v  \n1 Basic Quantum Elements for PDE Algorithms 1  \n1. 1 An overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1  \n1.2 Quantum states as normalized vectors . . . . . . . . . . . . . . . . . . . . . . . . 2  \n1.2.1 Gates as unitary matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 2  \n1.2.2 Pauli matrices and Pauli strings . . . . . . . . . . . . . . . . . . . . . . . 3  \n1.2.3 Tensor products and register notation . . . . . . . . . . . . . . . . . . . . 4  \n1.3 A short norm dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4  \n1.4 Measurement and observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5  \n1.5 Grid functions, amplitude encoding, and the L2–ℓ2 scaling . . . . . . . . . . . . . 7  \n1.5. 1 The norm conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7  \n1.5.2 Inner products of functions . . . . . . . . . . . . . . . . . . . . . . . . . . 8  \n1.6 State preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8  \n1.6. 1 Grover–Rudolph preparation for smooth densities . . . . . . . . . . . . . . 8  \n1.7 The quantum Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10  \n1.8 Quantum phase estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11  \n1.9 Estimating inner products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12  \n1.9.1 The swap test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12  \n1.9.2 The Hadamard test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13  \n1. 10 Postselection and amplitude amplification . . . . . . . . . . . . . . . . . . . . . . 14  \n1.10. 1 Postselection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15  \n1.10.2 Amplitude amplification ............................ 16  \n1.10.3 Quantities of interest and recovering the missing norm ........... 16  \n1.10.4 Amplitude estimation ............................. 17  \n1.11 Block encodings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18  \nCONTENTS ii  \n1.11.1 A small calculus of block encodings ...................... 18  \n1.11.2 Linear combination of unitaries . . . . . . . . . . . . . . . . . . . . . . . . 19  \n1.11.3 Sparse-matrix block encodings ........................ 20  \n1.12 Quantum singular value transformation ....................... 22  \n1.12. 1 Example: inverse filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23  \n1.12.2 Diagonal matrices and the cost of inversion ................. 24  \n1.12.3 Example: heat semigroups and exponential filters .............. 24  \n1.12.4 Example: oscillatory propagators . . . . . . . . . . . . . . . . . . . . . . . 25  \n1.13 Finite-difference operators as block encodings . . . . . . . . . . . . . . . . . . . . 26  \n1.13. 1 Forward difference operator . . . . . . . . . . . . . . . . . . . . . . . . . . 26  \n1.13.2 Three-point Laplacian in one dimension . . . . . . . . . . . . . . . . . . . 27  \n1.13.3 Two-dimensional five-point Laplacian . . . . . . . . . . . . . . . . . . . . . 28  \n1.13.4 General d-dimensional periodic Laplacian .................. 29  \n1.14 From finite differences to PDE algorithms ...................... 29  \n1.14.1 Initial and forcing data ............................ 29  \n1.14.2 Operator functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  \n1.14.3 Normalization and success probabilities . . . . . . . . . . . . . . . . . . . 30  \n1.14.4 Reading out useful information ........................ 30  \n1.15 A worked example: Poisson equation on a periodic grid .............. 30  \n1.16 A worked example: Schrödinger dynamics with the three-point Laplacian .... 31  \n1.17 How to read query and gate counts . . . . . . . . . . . . .","cbCainsWUhzNWjAL","https://ap.wps.com/l/cbCainsWUhzNWjAL","pdf",1226775,1,140,"English","en",105,"# Preface\n# Basic Quantum Elements for PDE Algorithms\n## An overview\n## Quantum states as normalized vectors\n## Gates as unitary matrices\n## Pauli matrices and Pauli strings\n## Tensor products and register notation\n## Measurement and observables\n## Grid functions, amplitude encoding, and the L2–ℓ2 scaling\n## State preparation\n## Quantum Fourier transform\n## Quantum phase estimation\n## Estimating inner products\n## Postselection and amplitude amplification\n## Block encodings\n## Quantum singular value transformation\n## Finite-difference operators as block encodings\n## From finite differences to PDE algorithms\n## Worked examples: Poisson and Schrödinger dynamics\n## Summary\n# Quantum Algorithms for Elliptic PDEs\n## Classical discretization and the direct QLSA route\n## First-order factorization and Hermitian dilation\n## QFT-based spectral filtering for special elliptic problems\n## Elliptic eigenvalue problems, mass lumping, and phase estimation\n## What to remember","[{\"question\":\"What quantum primitives are introduced to support PDE algorithms?\",\"answer\":\"The document covers quantum states and observables, quantum Fourier transform and phase estimation, inner-product estimation via swap and Hadamard tests, postselection and amplitude amplification, amplitude estimation, and block encodings.\"},{\"question\":\"How are finite-difference operators turned into quantum resources?\",\"answer\":\"Finite-difference operators are expressed as block encodings, including examples like forward differences and various-dimensional Laplacians, then mapped into broader PDE algorithm workflows through normalization, success probabilities, and readout of useful information.\"},{\"question\":\"Which specific PDE problem examples are discussed?\",\"answer\":\"Worked examples include the Poisson equation on a periodic grid and Schrödinger dynamics using the three-point Laplacian.\"}]",1784180385,353,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"a-quantum-path-to-partial-differential-equations","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/a-quantum-path-to-partial-differential-equations/82440/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 quantum primitives are introduced to support PDE algorithms?","Question",{"text":75,"@type":76},"The document covers quantum states and observables, quantum Fourier transform and phase estimation, inner-product estimation via swap and Hadamard tests, postselection and amplitude amplification, amplitude estimation, and block encodings.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are finite-difference operators turned into quantum resources?",{"text":80,"@type":76},"Finite-difference operators are expressed as block encodings, including examples like forward differences and various-dimensional Laplacians, then mapped into broader PDE algorithm workflows through normalization, success probabilities, and readout of useful information.",{"name":82,"@type":73,"acceptedAnswer":83},"Which specific PDE problem examples are discussed?",{"text":84,"@type":76},"Worked examples include the Poisson equation on a periodic grid and Schrödinger dynamics using the three-point Laplacian.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]