[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81653-en":3,"doc-seo-81653-105":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},81653,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Linear-Time T-Gate Optimization via Random Abstraction","Linear-time and practical methods for minimizing T-gate counts in fault-tolerant quantum circuits. The approach targets the bottleneck that T gates (π/4 phase rotations) require expensive magic-state distillation and often dominate resource budgets. A randomized phase-folding algorithm is introduced with a probabilistically sound static analysis using constant-width bitstrings rather than symbolic expressions. A corresponding tool, tzap, greatly accelerates optimization while matching reductions of state-of-the-art benchmarks.","arXiv :2605 . 13929v2 [ cs .PL] 10 Jul 2026  \nLinear-Time 􀀩 -Gate Optimization via Random Abstraction AWS ALBARGHOUTHI, University of Wisconsin-Madison, USA  \nQuantum computers promise exponential speedups for problems in cryptography, chemistry, and optimization. Realizing this promise requires fault tolerance: physical qubits are noisy, so logical qubits must be encoded redundantly across many physical ones using quantum error-correcting codes. In most practical fault-tolerance schemes, 􀀩 gates—phase rotations by 􀁣/4—cannot be implemented transversally and instead require costly magic-state distillation protocols involving a complex set of operations. As a result, 􀀩 -gate count can dominate the resource budget of large-scale quantum computations, making 􀀩 -count minimization a central bottleneck on the path to quantum advantage. Existing 􀀩-count optimization tools, however, do not scale to the circuits that quantum advantage demands.  \nWe present theoretical and practical results on 􀀩 -gate optimization. On the theoretical side, we give alinear-time randomized algorithm for phase folding, based on a novel randomized static analysis. Our static analysis approximates the transition relation of a quantum circuit and is sound except with an arbitrarily small probability of error. Our key insight is a static analysis that does not track symbolic expressions, but propagates constant-width bitstrings down the circuit. On the practical side, our implementation, tzap, is multiple orders of magnitude faster than state-of-the-art tools—such as PyZX, VOQC, and Feynman—closely matches their 􀀩-count reductions on standard benchmarks, and can optimize circuits with millions of gates within seconds on a laptop computer.  \n􀂇 [https://github.com/qqq-wisc/tzap](https://github.com/qqq-wisc/tzap)  \n1 Introduction  \nQuantum computers promise exponential speedups for a range of practically important problems—from simulating quantum chemistry [25] to breaking public-key cryptography via Shor’s algorithm [30]. Realizing this power requires fault tolerance: physical qubits are inherently noisy, so logical qubits must be encoded redundantly across many physical ones using quantum errorcorrecting codes, and gates must be applied in a fault-tolerant manner.  \nIn most practical fault-tolerance schemes, we use the universal Clifford+􀀩 gate set. While Clifford gates are efficiently simulable classically and are typically much cheaper to implement faulttolerantly, 􀀩 gates—phase rotations by 􀁣/4—cannot be implemented transversally in many practical fault-tolerance schemes, and instead require costly magic-state-distillation protocols [8, 9, 15] involving an elaborate and complex sequence of operations. Each logical 􀀩 gate can demand thousands of physical operations. As a result, 􀀩 -gate count can dominate the resource budget of large-scale quantum computations, and minimizing the number of 􀀩 gates in a circuit is a central bottleneck on the path to quantum advantage.  \nThe challenge of 􀀩-count reduction has attracted substantial attention, and a range of techniques have been developed. One broad family of approaches relies on phase folding: when two rotation gates act on the same basis states, their phases can be combined and redundant gates eliminated. This idea has appeared in different guises across the literature [2, 3, 26], and has been shown to be highly effective in practice. A second family of approaches uses rewriting: ZX-calculus represents a quantum circuit as a graphical tensor-network diagram and applies a suite of local rewrite rules to reduce gate count [10]. Tools based on these ideas, such as Feynman [2, 3] and PyZX [24], achieve strong 􀀩-count reductions on standard benchmarks.  \nAuthor’s Contact Information: Aws Albarghouthi, University of Wisconsin-Madison, Madison, WI, USA, [aws@cs.wisc.edu](aws@cs.wisc.edu).  \n1.1 Fast, Scalable 􀀩 -Gate Optimization  \nWe study 􀀩 -count minimization with two goals in mind. On the theoretical side, we see","cbCaiaYQ3psefMkF","https://ap.wps.com/l/cbCaiaYQ3psefMkF","pdf",922944,4,1,30,"English","en",105,"# Introduction\n## Fast, Scalable T-Gate Optimization\n## The phase-folding optimization\n## Fast phase folding with randomized abstraction","[{\"question\":\"Why is minimizing T-gate count a major bottleneck in fault-tolerant quantum computing?\",\"answer\":\"T gates (π/4 phase rotations) cannot be implemented transversally in many fault-tolerant schemes and instead rely on costly magic-state distillation. As a result, T-gate count can dominate the resource budget for large-scale computations.\"},{\"question\":\"What is the core idea behind phase folding for T-gate optimization?\",\"answer\":\"Phase folding merges and eliminates redundant rotation gates when they act on the same qubit state at different points in the circuit. The method relies on proving that the involved rotations target identical basis-state behavior for all possible inputs.\"},{\"question\":\"How does the proposed randomized algorithm achieve linear-time performance?\",\"answer\":\"It uses a lightweight randomized static analysis that approximates the circuit transition relation with a structure that propagates constant-width bitstrings. The analysis is sound except with an arbitrarily small probability of error.\"}]",1784175182,76,{"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},"linear-time-t-gate-optimization-via-random-abstraction","",{"@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/linear-time-t-gate-optimization-via-random-abstraction/81653/",{"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},"Why is minimizing T-gate count a major bottleneck in fault-tolerant quantum computing?","Question",{"text":75,"@type":76},"T gates (π/4 phase rotations) cannot be implemented transversally in many fault-tolerant schemes and instead rely on costly magic-state distillation. As a result, T-gate count can dominate the resource budget for large-scale computations.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the core idea behind phase folding for T-gate optimization?",{"text":80,"@type":76},"Phase folding merges and eliminates redundant rotation gates when they act on the same qubit state at different points in the circuit. The method relies on proving that the involved rotations target identical basis-state behavior for all possible inputs.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed randomized algorithm achieve linear-time performance?",{"text":84,"@type":76},"It uses a lightweight randomized static analysis that approximates the circuit transition relation with a structure that propagates constant-width bitstrings. 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