In this work, we have systematically investigated the Krylov complexity of curvature perturbation for the modified dispersion relation in inflation, using the algorithm in closed system and open system. Our analysis could be applied to the most inflationary models. Following the Lanczos algorithm, we find the very early universe is an infinite, many-body, and maximal chaotic system. Our numerics shows that the Lanczos coefficient and Lyapunov index of the standard dispersion relation are mainly determined by the scale factor. As for the modified case, it is nearly determined by the momentum. In a method of the closed system, we discover that the Krylov complexity will show irregular oscillation before the horizon exits. The modified case will present faster growth after the horizon exists. Since the whole universe is an open system, the approach of an open system is more realistic and reliable. Then, we construct the exact wave function which is very robust only requiring the Lanczos coefficient proportional to $n$ (main quantum number). Based on it, we find the Krylov complexity and Krylov entropy could nicely recover in the case of a closed system under the weak dissipative approximation, in which our analysis shows that the evolution of Krylov complexity will not be the same with the original situation. We also find the inflationary period is a strong dissipative system. Meanwhile, our numerics clearly shows the Krylov complexity will grow during the whole inflationary period. But for the small scales, there will be a peak after the horizon exits. Our analysis reveals that the dramatic change in background (inflation) will significantly impact the evolution of Krylov complexity. Since the curvature perturbation will transit from the quantum level to the classical level.
Geometric-phase control of Krylov complexity in adiabatic dynamics
Beihang University（北京航空航天大学）；China West Normal University（西华师范大学）；Jishou University（吉首大学）；Peng Huanwu Collaborative Center for Research and Education, Beihang University（北京航空航天大学彭桓武研究院）
Krylov Complexity for Open Quantum System: Dissipation and Decoherence
Indian Institute of Technology, Gandhinagar（印度理工学院甘德纳格尔分校）；University of Cape Town（开普敦大学）；The National Institute for Theoretical and Computational Sciences（理论与计算科学国家研究所）
Explicit Connections Between Krylov and Nielsen Complexity
The International Solvay Institutes（索尔维国际研究所）；School of Physics, Peking University（北京大学物理学院）；Center of High Energy Physics, Peking University（北京大学高能物理中心）
Krylov Complexity and Mixed-State Phase Transition
The Institute for Solid State Physics, The University of Tokyo（东京大学固体物理研究所）；Graduate School of Informatics, Nagoya University（名古屋大学情报学研究科）；Department of Physics, College of Humanities and Sciences, Nihon University（日本大学人文学部理科系物理学教室）
Institute for Theoretical and Mathematical Physics, Lomonosov Moscow State University（莫斯科国立大学理论与数学物理研究所）；Department of Physics, Indian Institute of Technology Roorkee（鲁尔基印度理工学院物理系）
Krylov complexity and oscillatory quantum dynamics in two-field inflation
Department of Physics, Guizhou Minzu University（贵州民族大学物理系）；Department of Physics, College of Physics, Mechanical and Electrical Engineering, Jishou University（吉首大学物理科学与机械电子工程学院物理系）
Toward a Refinement of Krylov Complexity: Scrambling, Classical Operator Growth and Replicas
National Taiwan University（国立台湾大学）；National Center for Theoretical Sciences（国家理论科学中心）；National Sun Yat-sen University（国立中山大学）；Gwangju Institute of Science and Technology（光州科学技术院）
Emergence of Krylov complexity through quantum walks: An exploration of the quantum origins of complexity
Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences（中国科学院大学卡尔费理论物理研究所）；Nordita, KTH Royal Institute of Technology and Stockholm University（诺尔迪塔，皇家理工学院与斯德哥尔摩大学）
Krylov Complexity Under Hamiltonian Deformations and Toda Flows
University of Luxembourg（卢森堡大学）；Mie University（三重大学）；Kanazawa University（金泽大学）；Vrije Universiteit Brussel (VUB)（布鲁塞尔自由大学）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；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）；Donostia International Physics Center（圣塞巴斯蒂安国际物理中心）
