We analyze the properties of Krylov state complexity in qubit dynamics, considering a single qubit and a qubit pair. A geometrical picture of the Krylov complexity is discussed for the single-qubit case, whereas it becomes non-trivial for the two-qubit case. Considering the particular case of interacting Rydberg two-level atoms, we show that the Krylov basis obtained using an effective Hamiltonian minimizes the time-averaged spread complexity compared to that which is obtained from the original Hamiltonian. We further generalize the latter property to an arbitrary Hamiltonian in which the entire Hilbert space comprises of two subspaces provided a weak coupling between them.
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'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）
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）
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 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 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.
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 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）
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）
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）
Towards 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）
Quantum complexity, suitably defined, has been suggested as an important probe of late-time dynamics of black holes, particularly in the context of AdS/CFT. A notion of quantum complexity can be effectively captured by quantifying the spread of an operator in Krylov space as a consequence of time evolution. Complexity is expected to behave differently in chaotic many-body systems, as compared to integrable ones. In this paper we investigate Krylov complexity for the case of interacting integrable models at finite size and find that complexity saturation is suppressed as compared to chaotic sys…
