[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82493-en":3,"doc-seo-82493-105":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},82493,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253",8,"Research & Report","Quantum Simulation of Stokes Flow via Schrödingerisation and Artificial Compressibility","Simulating incompressible Stokes flow is central to microfluidics and low-Reynolds-number hydrodynamics, yet classical solvers become prohibitively expensive because the saddle-point structure of the velocity–pressure formulation grows costly with dimension. The work introduces a quantum algorithm using Schrödingerisation for the Stokes equations, augmented by artificial compressibility regularization. An explicit quantum circuit encodes the resulting regularized system, and a rigorous complexity analysis shows an exponential speedup with dimensionality. Numerical simulations on Qiskit support validity and scalability.","arXiv :2607 .00281v1 [math .NA] 1 Jul 2026  \nQuantum Simulation of Stokes Flow via Schrödingerisation and  \nArtificial Compressibility  \nShi Jin, Jiaqi Tang, Qilong Zhai, Lei Zhang  \nAbstract  \nSimulating incompressible Stokes flow is essential for studies in microfluidics and low-Reynoldsnumber hydrodynamics. However, the computational cost of resolving the associated saddle-point problem grows prohibitively with the dimensionality of the problem. In this work, we present a quantum algorithm based on the Schrödingerisation technique for the Stokes equations, incorporating an artificial compressibility regularization. The core of our approach is the design of an explicit quantum circuit that encodes the resulting regularized system. The artificial compressibility formulation provides a unified framework for the system, which is then efficiently mapped to a quantum circuit via the Schrödingerisation procedure. A rigorous complexity analysis demonstrates the quantum computational advantage of our algorithms in high-dimensional settings, notably an exponential speedup in problem dimensionality. The validity and scalability of the proposed method are corroborated by numerical simulations performed on Qiskit.  \nKeywords: Stokes flow, Schrödingerisation, Artificial compressibility, Quantum circuit, Complexity analysis.  \n1. Introduction.  \nThe time-dependent Stokes problem governs creeping incompressible flow in complex geometries such as porous media [7, 9, 10] . With broad applications spanning petroleum engineering, biomedical transport, heat conduction, and microfluidic systems, this model provides a foundational framework for low-Reynolds-number hydrodynamics. A fundamental form is to find the velocity field u (t, x) and pressure field p(t, x) satisfying the time-dependent Stokes system subject to periodic boundary conditions:  \nut − a∆u − ∇p = f ,∇ · u = 0 , u ( · , 0) = u0 ,  \nin Ω × (0, T],  \nin Ω × (0, T],  \nin Ω ,  \n(1.1)  \n(1.2)  \n(1.3)  \nwhere Ω is a polygonal or polyhedral domain in Rd , f denotes a momentum source term, a > 0 is the kinematic viscosity. In the subsequent analysis, we assume that f and u0 are given and sufficiently smooth.  \nThe numerical solution of the Stokes equations has been extensively studied using classical methods such as finite element methods [28, 30, 32] and finite volume methods [3, 31] . The fundamental challenge persists across these classical schemes: the saddle-point nature of the Stokes system, which mandates satisfying the incompressibility constraint exactly at the discrete level. This coupling not only complicates the design of stable discretizations but also leads to large, ill-conditioned linear systems that are expensive to solve, especially in high dimensions. One influential strategy is the  \nartificial compressibility method [6] . The core idea is to relax the strict incompressibility condition by introducing a pseudo pressure, effectively replacing continuity equation with an artificial compressibility equation. This transformation has led to the development of robust numerical schemes that significantly improve the computational efficiency and accuracy [2, 23] . While traditional approaches are well-established for low-dimensional problems, the extension to high-dimensional settings remains computationally prohibitive, as the required resources scale exponentially with the number of dimensions.  \nTo overcome this fundamental dimensionality bottleneck, alternative computational paradigms are urgently needed. Quantum computing represents a paradigm shift for computational mathematics [8], offering potential exponential speedups for problems in linear algebra and differential equations [24, 27] . While Hamiltonian simulation techniques have been highly successful for unitary Schrödinger-type dynamics [1, 4, 11, 19, 20, 21, 33], a fundamental gap remains in handling physically critical systems with non-unitary, dissipative, or non-Hermitian characteristics, as exemplified by the","cbCaiq953ra1svnF","https://ap.wps.com/l/cbCaiq953ra1svnF","pdf",3752655,1,32,"English","en",105,"# Introduction\n## Stokes problem and saddle-point challenge\n## Classical artificial compressibility approach\n## Quantum computing and Schrödingerisation unitarization\n## Proposed quantum algorithm and organization","[{\"question\":\"What computational bottleneck motivates the proposed method for Stokes flow?\",\"answer\":\"Classical discretizations must handle the saddle-point coupling between velocity and pressure while enforcing incompressibility at the discrete level, leading to large ill-conditioned linear systems whose cost grows prohibitively in high dimensions.\"},{\"question\":\"How does the method reformulate the incompressibility constraint?\",\"answer\":\"It incorporates artificial compressibility regularization, relaxing the strict incompressibility condition by replacing the continuity equation with an artificial compressibility equation coupled to a pseudo pressure.\"},{\"question\":\"What role does Schrödingerisation play in the quantum algorithm?\",\"answer\":\"Schrödingerisation converts the resulting non-unitary Stokes dynamics into a Schrödinger-type system in an augmented Hilbert space, enabling an efficient mapping to an explicit quantum circuit representation.\"}]",1784180904,81,{"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},"quantum-simulation-of-stokes-flow-via-schrodingerisation-and-artificial-compressibility","",{"@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/quantum-simulation-of-stokes-flow-via-schrodingerisation-and-artificial-compressibility/82493/",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 computational bottleneck motivates the proposed method for Stokes flow?","Question",{"text":75,"@type":76},"Classical discretizations must handle the saddle-point coupling between velocity and pressure while enforcing incompressibility at the discrete level, leading to large ill-conditioned linear systems whose cost grows prohibitively in high dimensions.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the method reformulate the incompressibility constraint?",{"text":80,"@type":76},"It incorporates artificial compressibility regularization, relaxing the strict incompressibility condition by replacing the continuity equation with an artificial compressibility equation coupled to a pseudo pressure.",{"name":82,"@type":73,"acceptedAnswer":83},"What role does Schrödingerisation play in the quantum algorithm?",{"text":84,"@type":76},"Schrödingerisation converts the resulting non-unitary Stokes dynamics into a Schrödinger-type system in an augmented Hilbert space, enabling an efficient mapping to an explicit quantum circuit representation.","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 & 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