Krylov complexity, a quantum complexity measure which uniquely characterizes the spread of a quantum state or an operator, has recently been studied in the context of quantum chaos. However, the definitiveness of this measure as a chaos quantifier is in question in light of its strong dependence on the initial condition. This article clarifies the connection between the Krylov complexity dynamics and the initial operator or state. We find that the Krylov complexity depends monotonically on the inverse participation ratio (IPR) of the initial condition in the eigenbasis of the Hamiltonian. We explain the reversal of the complexity saturation levels observed in \href{https://doi.org/10.1103/PhysRevE.107.024217}{ Phys.Rev.E.107,024217, 2023} using the initial spread of the operator in the Hamiltonian eigenbasis. IPR dependence is present even in the fully chaotic regime, where popular quantifiers of chaos, such as out-of-time-ordered correlators and entanglement generation, show similar behavior regardless of the initial condition. Krylov complexity averaged over many initial conditions still does not characterize chaos.
Quantum-to-classical correspondence in Krylov complexity
Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales（布宜诺斯艾利斯大学，理学院）；CONICET - Universidad de Buenos Aires, Instituto de Física de Buenos Aires (IFIBA)（阿根廷国家科学研究委员会-布宜诺斯艾利斯大学，布宜诺斯艾利斯物理研究所）
Krylov's State Complexity and Information Geometry in Qubit Dynamics
University at Albany-SUNY（奥尔巴尼州立大学纽约州立大学）；Department of Nanoscale Science and Engineering, University at Albany-SUNY（奥尔巴尼州立大学纽约州立大学纳米科学与工程系）；Department of Physics, University at Albany-SUNY（奥尔巴尼州立大学纽约州立大学物理系）
Krylov complexity is a novel approach to study how an operator spreads over a specific basis. Recently, it has been stated that this quantity has a long-time saturation that depends on the amount of chaos in the system. Since this quantity not only depends on the Hamiltonian but also on the chosen operator, in this work we study the level of generality of this hypothesis by studying how the saturation value varies in the integrability to chaos transition when different operators are expanded. To do this, we work with an Ising chain with a transverse-longitudinal magnetic field and compare the …
A phase space approach to the wavefunction spreading and operator growth in the Krylov basis
Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；Department of Physics and Photon Science, Gwangju Institute of Science and Technology（光州科学技术研究院物理与光子科学系）；Research Center for Photon Science Technology, Gwangju Institute of Science and Technology（光州科学技术研究院光子科学技术研究中心）
Krylov complexity has emerged as a new probe of operator growth in a wide range of non-equilibrium quantum dynamics. However, a fundamental issue remains in such studies: the definition of the distance between basis states in Krylov space is ambiguous. Here, we show that Krylov complexity can be rigorously established from circuit complexity when dynamical symmetries exist. Whereas circuit complexity characterizes the geodesic distance in a multi-dimensional operator space, Krylov complexity measures the height of the final operator in a particular direction. The geometric representation of ci…
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）
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）
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）
We study the operator growth problem and its complexity in the many-body localization (MBL) system from the Lanczos algorithm perspective. Using the Krylov basis, the operator growth problem can be viewed as a single-particle hopping problem on a semi-infinite chain with the hopping amplitudes given by the Lanczos coefficients. We find that, in the MBL systems, the Lanczos coefficients scale as $\sim n/\ln(n)$ asymptotically, same as in the ergodic systems, but with an additional even-odd alteration and an effective randomness. We use a simple linear extrapolation scheme as an attempt to extra…
Krylov complexity measures operator growth with respect to a basis, which is adapted to the Heisenberg time evolution. The construction of that basis relies on the Lanczos algorithm, also known as the recursion method. The mathematics of Krylov complexity can be described in terms of orthogonal polynomials. We provide a pedagogical introduction to the subject and work out analytically a number of examples involving the classical orthogonal polynomials, polynomials of the Hahn class, and the Tricomi-Carlitz polynomials.
Quantum Cosmology in Krylov Space: Complexity and Entropy
Department of Physics and Astronomy, Louisiana State University（路易斯安那州立大学物理与天文系）；Center for Computation and Technology, Louisiana State University（路易斯安那州立大学计算与技术中心）
Bounds on Entanglement Dynamics from Krylov-Space Spreading
International Institute of Information Technology Hyderabad（海德拉巴国际信息技术学院）；Harish-Chandra Research Institute（哈里什-钱德拉研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）
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）
We study the spectral properties of two classes of random matrix models: non-Gaussian RMT with quartic and sextic potentials, and RMT with Gaussian noise. We compute and analyze the quantum Krylov complexity and the spectral form factor for both of these models. We find that both models show suppression of the spectral form factor at short times due to decoherence effects, but they differ in their long-time behavior. In particular, we show that the Krylov complexity for the non-Gaussian RMT and RMT with noise deviates from that of a Gaussian RMT. We discuss the implications and limitations of …
We develop computational tools necessary to extend the application of Krylov complexity beyond the simple Hamiltonian systems considered thus far in the literature. As a first step toward this broader goal, we show how the Lanczos algorithm that iteratively generates the Krylov basis can be augmented to treat coherent states associated with the Jacobi group, the semi-direct product of the 3-dimensional real Heisenberg-Weyl group $H_{1}$, and the symplectic group, $Sp(2,\mathbb{R})\simeq SU(1,1)$. Such coherent states are physically realized as squeezed states in, for example, quantum optics. W…
