Symmetries are crucial for tailoring parametrized quantum circuits to applications, due to their capability to capture the essence of physical systems. In this work, we shift the focus away from incorporating symmetries in the circuit design and towards symmetry-aware training of variational quantum algorithms. For this, we introduce the concept of projected derivatives of parametrized quantum circuits, in particular the equivariant and covariant derivatives. We show that the covariant derivative gives rise to the quantum Fisher information and quantum natural gradient. This provides an operational meaning for the covariant derivative, and allows us to extend the quantum natural gradient to all continuous symmetry groups. Connecting to traditional particle physics, we confirm that our covariant derivative is the same as the one introduced in physical gauge theory. This work provides tools for tailoring variational quantum algorithms to symmetries by incorporating them locally in derivatives, rather than into the design of the circuit.
Parametrized-circuit-free quantum regression with variance regularization
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术学院）；Donostia International Physics Center（圣塞巴斯蒂安国际物理中心）；Faculty of Physics, M.V. Lomonosov Moscow State University（莫斯科国立大学物理系）
Estimates of loss function concentration in noisy parametrized quantum circuits
University of Trieste（的里雅斯特大学）；Istituto Nazionale di Fisica Nucleare, Trieste Section（意大利国家核物理研究所，的里雅斯特分部）；European Organization for Nuclear Research (CERN)（欧洲核子研究组织（CERN））
Phase transitions in parametrized quantum circuits
RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所跨学科理论与数学科学中心）；RIKEN Center for Computational Science (R-CCS)（理化学研究所计算科学中心）；RIKEN Center for Quantum Computing (RQC)（理化学研究所量子计算中心）；RIKEN Pioneering Research Institute (PRI)（理化学研究所先锋研究所）；RIKEN Cluster for Pioneering Research (CPR)（理化学研究所先锋研究集群）；RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所新兴物质科学中心）
Global Optimization for Parametrized Quantum Circuits
Singapore University of Technology and Design（新加坡科技设计大学）；Centre for Quantum Technologies（量子技术中心）；Archimedes Research Unit on AI, Marousi, Greece（希腊马里乌西阿尔基米德人工智能研究部）；Quantum Technologies Group, Innovation & Ventures, HSBC（汇丰银行创新与风险投资部量子技术组）；Czech Technical University in Prague（布拉格捷克理工大学）
Probabilistic Design of Parametrized Quantum Circuits through Local Gate Modifications
The Edward S. Rogers Sr. Department of Electrical \& Computer Engineering, University of Toronto 10 King's College Rd Toronto ON M5S 3G8 Canada；Department of Chemical；Physical Sciences, University of Toronto Mississauga 3359 Mississauga Road Mississauga ON L5L 1C6 Canada；Department of Computer Science, University of Toronto Toronto ON M5S 2E4 Canada；Vector Institute for Artificial Intelligence 108 College St W1140 Toronto ON M5G 0C6 Canada；The Edward S. Rogers Sr. Department of Electrical \& Computer Engineering, University of Toronto 10 King's College Rd Toronto ON M5G 0C6 Canada；Department of Chemistry, University of Calgary 2500 University Drive NW Calgary AB T2N 1N4 Canada；Department of Computer Science, University of Toronto 40 St. George Street Toronto ON M5S 2E4 Canada；The Edward S. Rogers Sr. Department of Electrical \& Computer Engineering, University of Toronto；Physical Sciences, University of Toronto Mississauga；Department of Computer Science, University of Toronto；Vector Institute for Artificial Intelligence；Department of Chemistry, University of Calgary
Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Theoretical Division, Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室理论部）；Center for Nonlinear Studies, Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室非线性研究中心）
Parametrized Quantum Circuit Learning for Quantum Chemical Applications
The Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, 10 King's College Road, Toronto, Ontario, Canada M5S 3G4（电气与计算机工程系，多伦多大学）；Department of Chemical and Physical Sciences, University of Toronto Mississauga, 3359 Mississauga Road, Mississauga, Ontario, Canada L5L 1C6（化学与物理科学系，多伦多大学滑铁卢分校）
Volkswagen Data:Lab（大众数据实验室）；Ludwig Maximilian University（路德维希·马克西米利安大学）；Leiden University（莱顿大学）；Google Research（谷歌研究院）；Volkswagen Group Machine Learning Research Lab（大众集团机器学习研究实验室）；Eötvös Loránd University（厄特沃什·罗兰大学）
Extended parameter shift rules with minimal derivative variance for parameterized quantum circuits
Beijing International Center for Mathematical Research, Peking University（北京大学数学中心）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
Tangent Space Excitation Ansatz for Quantum Circuits
Sun Yat-sen University（中山大学）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；University of Vienna（维也纳大学）；The University of Hong Kong（香港大学）；Freie Universität Berlin（柏林自由大学）
