[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85569-en":3,"doc-seo-85569-105":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},85569,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Polylogarithmic-Weight Dicke States in QAC0 and Arbitrary Symmetric States in QAC0f","An n-qubit Dicke state |Dnk⟩ is the uniform superposition of all bit strings with Hamming weight k, and it is central to quantum speedups and symmetric-state synthesis. The work studies QAC0, a minimal constant-depth quantum analogue of AC0, defined via arbitrary-width Toffoli gates instead of global fanout. It presents a QAC0 construction for Dicke states of polylog(n) weight, yielding the first super-constant-weight preparation without FANOUTn, and extends to exact synthesis in QAC0f using limited-fanout tools and QRAM-indexing.","arXiv :2604 . 15298v2 [ quant-ph] 13 Jul 2026  \nPolylogarithmic-Weight Dicke States in QAC 0 and Arbitrary  \nSymmetric States in QAC0f  \nLucas Gretta ∗ Meghal Gupta† Malvika Raj Joshi‡  \nAbstract  \nAn n-qubit Dicke state of weight k , |Dnk⟩, is the uniform superposition over all n-bit strings of Hamming weight k. Dicke states are central to quantum algorithms exhibiting speedups, such as Decoded Quantum Interferometry (Jordan et al. , Nature, 2025) . In the NISQ era, quantum hardware is constrained by both depth and locality, motivating the question of which global operations suffice to prepare such states. QAC0 , the quantum analogue of AC0 , minimally extends local O(1)-depth quantum circuits by allowing arbitrary-width Toffoli (reversible AND) gates. We show that Dicke states of polylog(n) weight can be prepared in QAC0 . This gives the first QAC0 construction of any super-constant-weight n-qubit Dicke state, since previous constructions relied on the much more powerful FANOUTn gate. In general, we show that any weight-k Dicke state can be constructed using FANOUTmin(k,n−k) gates. Combined with recent hardness results, this yields a tight characterization: for k ≤ n/2, |Dnk⟩ can be prepared in QAC0 if and only if FANOUTk ∈ QAC0 .  \nWe develop a limited-fanout state-synthesis toolkit for QAC0 that yields further constantdepth, poly (n)-ancilla constructions. Every n-qubit symmetric state supported on Hamming weight ≤ k can be prepared using FANOUTk gates. Moreover, every O(log n)-qubit state can be prepared using quantum random-access memory (QRAMn ), which refers to a coherent indexing gate |x⟩ |i⟩ |0⟩ 7→ |x⟩ |i⟩ |xi ⟩, a potentially weaker resource than FANOUTn (QRAMn ∈ QAC0f) . Thus, every symmetric state can be synthesized exactly in QAC0f .  \nContents  \n1 Introduction 2  \n1.1 Our results ......................................... 4  \n1.2 Prior work .......................................... 5  \n2 Overview 6  \n2.1 Our construction ...................................... 6  \n2.1.1 Constructing a state with constant fidelity with |Dnk⟩ .............. 7  \n∗ University of California at Berkeley. Email: [lucas_gretta@berkeley.edu](lucas_gretta@berkeley.edu. Supported)[. Supported](lucas_gretta@berkeley.edu. Supported) by NSF Award CCF- 2231095  \n†University of California at Berkeley. Email: [meghal@berkeley.edu](meghal@berkeley.edu. Supported)[. Supported](meghal@berkeley.edu. Supported) by NSF GRFP.  \n‡University of California at Berkeley. Email: [malvika@berkeley.edu](malvika@berkeley.edu. Supported)[. Supported](malvika@berkeley.edu. Supported) by UC Berkeley EECS Fellowship.  \n2.1.2 Adding back the missing strings.......................... 8  \n2.2 Parameter choices and putting it together......................... 8  \n2.3 Extending to arbitrary symmetric states ......................... 10  \n3 Preliminaries 11  \n3.1 Additional Notation .................................... 12  \n3.2 Known procedures in QAC0 ................................ 13  \n3.2.1 Applications of FANOUT ............................... 14  \n3.2.2 Amplitude amplification .............................. 14  \n3.3 Useful distributions and associated quantum states ................... 15  \n4 Limited-fanout state-synthesis toolkit 16  \n4.1 Arbitrary quantum states in constant depth ....................... 16  \n4.2 Amplitude manipulations ................................. 19  \n4.3 Controlled-unitaries .................................... 22  \n5 Constant-depth symmetric states 24  \n5.1 Intermediate distributional states ............................. 25  \n5.1.1 Dicke occupancy states .............................. 27  \n5.2 Dicke States ......................................... 31  \n5.3 General symmetric states ................................. 32  \n6 Acknowledgments 32  \nA Deferred Proofs 35  \nA.1 Limited-fanout symmetric states synthesis ........................ 36  \n1 Introduction  \nQAC0 circuits are a fundamental model of shallow quantum computation consisting of c","cbCaiuc7sb7DScNw","https://ap.wps.com/l/cbCaiuc7sb7DScNw","pdf",528944,4,1,39,"English","en",105,"# Introduction\n## Our results\n## Prior work\n# Overview\n## Our construction\n### Constructing a state with constant fidelity with |Dnk⟩\n### Adding back the missing strings\n## Parameter choices and putting it together\n## Extending to arbitrary symmetric states\n# Preliminaries\n## Additional Notation\n## Known procedures in QAC0\n# Limited-fanout state-synthesis toolkit\n## Arbitrary quantum states in constant depth\n## Amplitude manipulations\n## Controlled-unitaries\n# Constant-depth symmetric states\n## Dicke States\n## General symmetric states\n# Acknowledgments\n## Deferred Proofs","[{\"question\":\"What is the main problem addressed about QAC0 circuits?\",\"answer\":\"The paper asks which global operations suffice to prepare Dicke and other symmetric states when quantum hardware is limited by constant depth and locality in the QAC0 model.\"},{\"question\":\"How are polylogarithmic-weight Dicke states prepared according to the paper?\",\"answer\":\"It provides a QAC0 construction that prepares Dicke states of polylog(n) weight, avoiding reliance on the much stronger FANOUTn gate used in earlier approaches.\"},{\"question\":\"What condition determines whether |Dnk⟩ can be prepared in QAC0?\",\"answer\":\"For k ≤ n/2, the work gives a tight characterization: |Dnk⟩ can be prepared in QAC0 if and only if FANOUTk is in QAC0.\"}]",1784204666,98,{"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},"polylogarithmic-weight-dicke-states-in-qac0-and-arbitrary-symmetric-states-in-qac0f","",{"@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/polylogarithmic-weight-dicke-states-in-qac0-and-arbitrary-symmetric-states-in-qac0f/85569/",{"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 is the main problem addressed about QAC0 circuits?","Question",{"text":75,"@type":76},"The paper asks which global operations suffice to prepare Dicke and other symmetric states when quantum hardware is limited by constant depth and locality in the QAC0 model.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are polylogarithmic-weight Dicke states prepared according to the paper?",{"text":80,"@type":76},"It provides a QAC0 construction that prepares Dicke states of polylog(n) weight, avoiding reliance on the much stronger FANOUTn gate used in earlier approaches.",{"name":82,"@type":73,"acceptedAnswer":83},"What condition determines whether |Dnk⟩ can be prepared in QAC0?",{"text":84,"@type":76},"For k ≤ n/2, the work gives a tight characterization: |Dnk⟩ can be prepared in QAC0 if and only if FANOUTk is in 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