[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85752-en":3,"doc-seo-85752-105":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},85752,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","End-to-End Hybrid Quantum-Classical Sampling Workflow for Discrete Markov Random Fields","Sampling from discrete Markov random fields (MRFs) is a challenging inference task, and the study analyzes amplitude-encoded i.i.d. sampling for small discrete MRFs. In the regime where the 2^n probabilities are precomputable classically, circuit execution yields independent samples (τ≈1), enabling direct comparison against classical MCMC. Across 60 instances and multiple graph families, modern classical samplers largely close the ESS gap. After amortizing O(2^n) preprocessing, exact inverse-CDF sampling attains far higher ESS/s, showing no wall-clock advantage. Additional experiments evaluate MPS scaling and VQC fidelity versus MPS.","An End-to-End Hybrid Quantum–Classical Sampling Workflow for Discrete Markov Random Fields: A  \nReproducible Case Study  \nArul Rhik Mazumder  \nSchool of Computer Science  \nCarnegie Mellon University  \nPittsburgh, PA, USA  \n[arulm@cs.cmu.edu](arulm@cs.cmu.edu)  \narXiv :2607 .09893v1 [ quant-ph] 10 Jul 2026  \nAbstract—Sampling from discrete Markov random fields (MRFs) is a well-known hard problem in probabilistic inference. We study amplitude-encoded i.i.d. sampling for small discrete MRFs, scoped to the regime where the 2n target probabilities can be precomputed classically—so no quantum exponential speedup is possible, but the structural property that each circuit execution returns an independent sample (τ ≈ 1) can be cleanly compared against classical MCMC alternatives. Across 60 instances spanning five graph families (barbell, barbell-path, chain, Erds–Rnyi, two-clique) with 1,000-step burn-in and 3,000 retained samples, Quantum/SingleSite-Gibbs, Quantum/Block-Gibbs, Quantum/Tuned-Block-Gibbs, and Quantum/Parallel-Tempering ESS ratios have means of 16.35, 7.29, 1.82, and 1.79 respectively, showing that modern classical samplers substantially close—and may eliminate—the ESS gap relative to amplitude-encoded sampling. When the O(2n ) classical preprocessing required by the amplitude encoder is amortized into wall-clock time, exact inverse-CDF sampling reaches a mean of 17 ,683 ,598 ESS/s versus 487 ,706 ESS/s for the quantum sampler (36× on mean rates, 153 × mean perinstance), confirming no wall-clock advantage in this regime. The contribution is therefore not a speedup claim but (i) a clean characterization of MCMC autocorrelation costs under a fixed protocol and (ii) a reproducible benchmark of amplitudeencoded state preparation for discrete MRFs at n = 8 , 10 , 12. We further report a multi-trial matrix product state (MPS) scaling study (three seeds per point, n up to 40) showing χ = 32 achieves F = 0 .721 ± 0.059 at n = 40, and a matched-budget variational quantum circuit (VQC) vs. MPS comparison at n = 8 , 10 , 12 where VQC fidelities fall below MPS at every point ((FVQC , FMPS ) = (0 .306 , 0.990) ,(0 .210 , 0.958) ,(0 .165 , 0.878) at compressions 10.7×, 34.1 ×, 113.8×)—a negative result for shallow hardware-efficient anstze. Code and data: [https://github](https://github). com/arulrhikm/QuantumDGM.  \nIndex Terms—amplitude encoding, graphical models, hardwareefficient ansatz, Markov random fields, Monte Carlo methods, quantum machine learning, quantum sampling, variational quantum circuits  \nI. INTRODUCTION  \nA. Problem Selection and Motivation  \nMarkov random fields (MRFs) encode conditional independence structures through undirected graphs and are fundamental to computer vision [1], computational biology [2], statistical  \nphysics, and generative modeling [3] . The central computational challenge is drawing samples from the joint distribution  \nPθ (x) = Z~~ ~~1(θ) exp  yC θC,y ϕC,y(x) , (1) where C denotes the set of maximal cliques of the graph, ϕC,yare indicator sufficient statistics, and the partition function Z (θ)  \nrequires summing over 2n configurations for binary variables. This exponential cost makes exact inference intractable for all but the smallest models and motivates a wide array of approximate techniques.  \nWe selected MRF sampling because classical MCMC [4] suffers slow mixing and correlated samples on loopy graphs, while variational inference [5] and loopy belief propagation [6] have their own accuracy or convergence limits. The amplitudeencoding path studied here does not circumvent O(2n ) classical preprocessing; it is an i.i.d. sample generator after Pθ has been enumerated. We therefore ask (i) how much autocorrelation cost classical MCMC pays under a fixed protocol (§IV-A) and (ii) whether any quantum structural property beyond i.i.d. output yields practical advantage (§V-A) .  \nB. Data Structure and Architecture  \nOur datasets consist of synthetic MRFs with controlled graph structures used in th","cbCaiaFHqi3pI3UP","https://ap.wps.com/l/cbCaiaFHqi3pI3UP","pdf",923946,1,9,"English","en",105,"# Abstract\n# I. Introduction\n## A. Problem Selection and Motivation\n## B. Data Structure and Architecture\n## C. Evaluation Methodology","[{\"question\":\"What sampling setting does the workflow target for discrete MRFs?\",\"answer\":\"The work focuses on amplitude-encoded i.i.d. sampling for small discrete Markov random fields, in a regime where all 2^n target probabilities can be precomputed classically.\"},{\"question\":\"How does the paper compare quantum sampling against classical methods?\",\"answer\":\"It compares independent-sample performance using MCMC autocorrelation/ESS-based metrics across multiple MRF families, including Quantum variants such as SingleSite/Block-Gibbs and classical alternatives with tuned ESS ratios.\"},{\"question\":\"What conclusion is drawn about speedup in the studied regime?\",\"answer\":\"No wall-clock advantage is claimed: when the classical preprocessing cost required by amplitude encoding is amortized, exact inverse-CDF sampling achieves much higher ESS/s than the quantum sampler.\"}]",1784206026,23,{"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},"end-to-end-hybrid-quantum-classical-sampling-workflow-for-discrete-markov-random-fields","",{"@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/end-to-end-hybrid-quantum-classical-sampling-workflow-for-discrete-markov-random-fields/85752/",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 sampling setting does the workflow target for discrete MRFs?","Question",{"text":75,"@type":76},"The work focuses on amplitude-encoded i.i.d. sampling for small discrete Markov random fields, in a regime where all 2^n target probabilities can be precomputed classically.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper compare quantum sampling against classical methods?",{"text":80,"@type":76},"It compares independent-sample performance using MCMC autocorrelation/ESS-based metrics across multiple MRF families, including Quantum variants such as SingleSite/Block-Gibbs and classical alternatives with tuned ESS ratios.",{"name":82,"@type":73,"acceptedAnswer":83},"What conclusion is drawn about speedup in the studied regime?",{"text":84,"@type":76},"No wall-clock advantage is claimed: when the classical preprocessing cost required by amplitude encoding is amortized, exact inverse-CDF sampling achieves much higher ESS/s than the quantum 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