[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-118549-en":3,"doc-seo-118549-105":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":4,"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":27,"seo_description":14,"update_tm":28,"read_time":29},118549,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","Expressivity and Complexity Analysis of Variational Quantum Circuits for Quantum Machine Learning","Variational quantum circuits (VQCs) offer promising computational speedups for quantum machine learning, while potentially reducing space complexity. A central challenge is selecting circuit architectures with sufficient expressivity to represent the target structure of complex patterns. Expressivity characterizes how effectively a VQC maps input state vectors to distinguishable output states within Hilbert space. When the desired patterns fall outside the circuit’s representational regime, model performance degrades, motivating systematic expressivity and complexity evaluation.","UC Irvine  \nUC Irvine Electronic Theses and Dissertations  \nTitle  \nExpressivity and Complexity Analysis of Variational Quantum Circuits for Quantum Machine Learning  \nPermalink  \n[https://escholarship.org/uc/item/2nc5x4dg](https://escholarship.org/uc/item/2nc5x4dg)  \nAuthor  \nHu, Zhihan  \nPublication Date  \n2025  \nPeer reviewed|Thesis/dissertation  \n[eScholarship.org](eScholarship.org) Powered by the California Digital Library  \nUniversity of California  \nUNIVERSITY OF CALIFORNIA,  \nIRVINE  \nExpressivity and Complexity Analysis of Variational Quantum Circuits for Quantum  \nMachine Learning  \nTHESIS  \nsubmitted in partial satisfaction of the requirements for the degree of  \nMASTER OF SCIENCE  \nin Computer Engineering  \nby  \nZhihan Hu  \nThesis Committee:  \nAssistant Professor Maxim Shcherbakov, Chair Chancellor’s Professor Syed Jafar  \nProfessor Nader Bagherzadeh  \n© 2025 Zhihan Hu  \nTABLE OF CONTENTS  \nPage  \nLIST OF FIGURES iii  \nLIST OF TABLES iv  \nACKNOWLEDGMENTS v  \nABSTRACT OF THE THESIS vi  \n1 Introduction 1  \n2 Experimental Setup 4  \n2.1 VQC Designs ................................... 4  \n2.2 Random State Vector Generalization ...................... 6  \n2.3 Expressivity Estimation ............................. 7  \n2.4 Degree of Entanglement of a State Vector ................... 8  \n2.5 Participation Ratio of a State Vector ...................... 8  \n3 Numerical Experiments on Expressivity 9  \n3.1 Number of Qubits ................................. 9  \n3.2 Expressivity vs. Circuit Architect ........................ 12  \n3.3 Degree of Entanglement ............................. 15  \n3.4 Participation Ratio ................................ 18  \n4 Complexity Analysis on the Circuit 20  \n5 Result and Discussion 25  \nBibliography 27  \nLIST OF FIGURES  \nPage  \n2.1 (a)-(t): 20 different VQC designs based on an 3-qubit system. L1-L20 is used as the labels for each circuit, ordered by the number of learnable parameters 5  \n3.1 (a): Expressivity vs Number of Qubits of circuits L1-L5: The expressivity  \ndecreases as num of qubit increases to 4, but then improves as Number of  \nQubits continues to increase. (b): Expressivity vs Number of Qubits of circuits  \nL6-L10: The expressivity decreases as Number of Qubits increases to 5, but  \nthen improves as Number of Qubits continues to increase. (c): Expressivity vs  \nNumber of Qubits of circuits L11-L15: The expressivity decreases as Number  \nof Qubits increases to 5, but then improves as Number of Qubits continues  \nto increase. (d): Expressivity vs Number of Qubits of circuits L16-L20: The  \nexpressivity decreases as Number of Qubits increases to 5, but then improves as Number of Qubits continues to increase.................... 10  \n3.2 Expressivity of different VQC designs given same set of input state vectors  \nwith number of qubit equal to 6 . The graph shows a general trend in the  \nimprovement of Expressivity as the number of variational parameters of a VQC grows .................................... 13  \n3.3 (a)-(t) shows the Expressivity [vs. degree](vs. degree) of entanglement for each circuit  \nL1-L20 respectively based on a 6-qubit system. A general trend of expres  \nsivity improvement can be observed as the degree of entanglement increases. Additionally, the variance of the expressivity is larger when the degree of entanglement is small compared to when the degree of entanglement is big.. 17  \n3.4 (a)-(t) Expressivity vs. participation ratio for each of the circuit L1-L20 . A  \nmore complex (non-linear) but generally improving trend of expressivity as  \nparticipation ratio increases. Compared to the degree of entanglement, the variance is comparatively large, meaning a larger uncertainty......... 19  \n4.1 L1-L15 the decomposition of each circuit. Where all the circuits are decomposed into arbitrary single qubit U and CX gates ............... 23  \n4.2 L1-L15 the decomposition of each circuit. Where all the circuits are decomposed into arbitrary single qubit U and CX gates ......","cbCaikmF9lSOeB49","https://ap.wps.com/l/cbCaikmF9lSOeB49","pdf",1228279,1,37,"English","en",105,"# List of Figures\n# List of Tables\n# Acknowledgments\n# Abstract of the Thesis\n# Introduction\n# Experimental Setup\n## VQC Designs\n## Random State Vector Generalization\n## Expressivity Estimation\n## Degree of Entanglement of a State Vector\n## Participation Ratio of a State Vector\n# Numerical Experiments on Expressivity\n## Number of Qubits\n## Expressivity vs. Circuit Architect\n## Degree of Entanglement\n## Participation Ratio\n# Complexity Analysis on the Circuit\n# Result and Discussion\n# Bibliography","[{\"question\":\"What does expressivity mean in variational quantum circuits for quantum machine learning?\",\"answer\":\"Expressivity is the VQC capability to map input state vectors to different state vectors in Hilbert space, enabling the model to represent complex patterns.\"},{\"question\":\"Why does insufficient expressivity reduce quantum machine learning performance?\",\"answer\":\"If the accurate patterns lie in a regime the VQC cannot represent, the implemented model cannot map the inputs to the needed states, resulting in low performance.\"},{\"question\":\"What kinds of analyses are performed on variational quantum circuits in this thesis?\",\"answer\":\"The thesis conducts numerical experiments to estimate expressivity and evaluates circuit complexity, including measures related to entanglement and participation ratio.\"}]","Expressivity and Complexity Analysis of Variational Quantum Circuits for Quantum Machine Learning | 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does expressivity mean in variational quantum circuits for quantum machine learning?","Question",{"text":75,"@type":76},"Expressivity is the VQC capability to map input state vectors to different state vectors in Hilbert space, enabling the model to represent complex patterns.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why does insufficient expressivity reduce quantum machine learning performance?",{"text":80,"@type":76},"If the accurate patterns lie in a regime the VQC cannot represent, the implemented model cannot map the inputs to the needed states, resulting in low performance.",{"name":82,"@type":73,"acceptedAnswer":83},"What kinds of analyses are performed on variational quantum circuits in this thesis?",{"text":84,"@type":76},"The thesis conducts numerical experiments to estimate expressivity and evaluates circuit complexity, including measures related to entanglement and participation 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