Complexity Growth in Integrable and Chaotic Models

• URL: http://arxiv.org/abs/2101.02209v2
• Date: Tue, 27 Apr 2021 02:30:47 GMT
• Title: Complexity Growth in Integrable and Chaotic Models
• Authors: Vijay Balasubramanian, Matthew DeCross, Arjun Kar, Cathy Li, Onkar Parrikar
• Abstract summary: We use the SYK family of models with $N$ Majorana fermions to study the complexity of time evolution. We study how this linear growth is eventually truncated by the appearance and accumulation of conjugate points.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We use the SYK family of models with $N$ Majorana fermions to study the complexity of time evolution, formulated as the shortest geodesic length on the unitary group manifold between the identity and the time evolution operator, in free, integrable, and chaotic systems. Initially, the shortest geodesic follows the time evolution trajectory, and hence complexity grows linearly in time. We study how this linear growth is eventually truncated by the appearance and accumulation of conjugate points, which signal the presence of shorter geodesics intersecting the time evolution trajectory. By explicitly locating such "shortcuts" through analytical and numerical methods, we demonstrate that: (a) in the free theory, time evolution encounters conjugate points at a polynomial time; consequently complexity growth truncates at $O(\sqrt{N})$, and we find an explicit operator which "fast-forwards" the free $N$-fermion time evolution with this complexity, (b) in a class of interacting integrable theories, the complexity is upper bounded by $O({\rm poly}(N))$, and (c) in chaotic theories, we argue that conjugate points do not occur until exponential times $O(e^N)$, after which it becomes possible to find infinitesimally nearby geodesics which approximate the time evolution operator. Finally, we explore the notion of eigenstate complexity in free, integrable, and chaotic models.

Related papers

• Krylov Complexity as a Probe for Chaos [0.7373617024876725]
  We show that the dynamics towards saturation precisely distinguish between chaotic and integrable systems. For chaotic models, the saturation value of complexity reaches its infinite time average at a finite saturation time. In integrable models, complexity approaches the infinite time average value from below at a much longer timescale.
  arXiv  Detail & Related papers  (2024-08-19T17:52:42Z)
• Self-healing of Trotter error in digital adiabatic state preparation [52.77024349608834]
  We prove that the first-order Trotterization of a complete adiabatic evolution has a cumulative infidelity that scales as $mathcal O(T-2 delta t2)$ instead of $mathcal O(T2delta t2)$ expected from general Trotter error bounds. This result suggests a self-healing mechanism and explains why, despite increasing $T$, infidelities for fixed-$delta t$ digitized evolutions still decrease for a wide variety of Hamiltonians.
  arXiv  Detail & Related papers  (2022-09-13T18:05:07Z)
• Complexity in the Lipkin-Meshkov-Glick Model [0.0]
  We study complexity in a spin system with infinite range interaction. Exact expressions for the Nielsen complexity (NC) and the Fubini-Study complexity (FSC) are derived.
  arXiv  Detail & Related papers  (2022-04-13T13:11:58Z)
• Complex Evolutional Pattern Learning for Temporal Knowledge Graph Reasoning [60.94357727688448]
  TKG reasoning aims to predict potential facts in the future given the historical KG sequences. The evolutional patterns are complex in two aspects, length-diversity and time-variability. We propose a new model, called Complex Evolutional Network (CEN), which uses a length-aware Convolutional Neural Network (CNN) to handle evolutional patterns of different lengths.
  arXiv  Detail & Related papers  (2022-03-15T11:02:55Z)
• Entanglement and correlations in fast collective neutrino flavor oscillations [68.8204255655161]
  Collective neutrino oscillations play a crucial role in transporting lepton flavor in astrophysical settings. We study the full out-of-equilibrium flavor dynamics in simple multi-angle geometries displaying fast oscillations. We present evidence that these fast collective modes are generated by the same dynamical phase transition.
  arXiv  Detail & Related papers  (2022-03-05T17:00:06Z)
• Bounds on quantum evolution complexity via lattice cryptography [0.0]
  We address the difference between integrable and chaotic motion in quantum theory as manifested by the complexity of the corresponding evolution operators. Complexity is understood here as the shortest geodesic distance between the time-dependent evolution operator and the origin within the group of unitaries.
  arXiv  Detail & Related papers  (2022-02-28T16:20:10Z)
• Operator Complexity for Quantum Scalar Fields and Cosmological Perturbations [0.0]
  We study the complexity of the unitary evolution of quantum cosmological perturbations in de Sitter space. The complexity of cosmological perturbations scales as the square root of the (exponentially) growing volume of de Sitter space.
  arXiv  Detail & Related papers  (2021-10-15T20:37:36Z)
• On Function Approximation in Reinforcement Learning: Optimism in the Face of Large State Spaces [208.67848059021915]
  We study the exploration-exploitation tradeoff at the core of reinforcement learning. In particular, we prove that the complexity of the function class $mathcalF$ characterizes the complexity of the function. Our regret bounds are independent of the number of episodes.
  arXiv  Detail & Related papers  (2020-11-09T18:32:22Z)
• Operator complexity: a journey to the edge of Krylov space [0.0]
  Krylov complexity, or K-complexity', quantifies this growth with respect to a special basis. We study the evolution of K-complexity in finite-entropy systems for time scales greater than the scrambling time.
  arXiv  Detail & Related papers  (2020-09-03T18:10:20Z)
• Relevant OTOC operators: footprints of the classical dynamics [68.8204255655161]
  The OTOC-RE theorem relates the OTOCs summed over a complete base of operators to the second Renyi entropy. We show that the sum over a small set of relevant operators, is enough in order to obtain a very good approximation for the entropy. In turn, this provides with an alternative natural indicator of complexity, i.e. the scaling of the number of relevant operators with time.
  arXiv  Detail & Related papers  (2020-07-31T19:23:26Z)
• Measuring Model Complexity of Neural Networks with Curve Activation Functions [100.98319505253797]
  We propose the linear approximation neural network (LANN) to approximate a given deep model with curve activation function. We experimentally explore the training process of neural networks and detect overfitting. We find that the $L1$ and $L2$ regularizations suppress the increase of model complexity.
  arXiv  Detail & Related papers  (2020-06-16T07:38:06Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.