Generalized Krylov Complexity

• URL: http://arxiv.org/abs/2507.23739v1
• Date: Thu, 31 Jul 2025 17:23:03 GMT
• Title: Generalized Krylov Complexity
• Authors: Amin Faraji Astaneh, Niloofar Vardian,
• Abstract summary: We extend the concept of Krylov complexity to include general unitary evolutions involving multiple generators.<n>This generalization serves as a measure of the complexity of states associated with continuous symmetries within a model.
• Score: 0.0
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: We extend the concept of Krylov complexity to include general unitary evolutions involving multiple generators. This generalization enables us to formulate a framework for generalized Krylov complexity, which serves as a measure of the complexity of states associated with continuous symmetries within a model. Furthermore, we investigate scenarios where different directions of transformation lead to varying degrees of complexity, which can be compared to geometric approaches to understanding complexity, such as Nielsen complexity. In this context, we introduce a generalized orthogonalization algorithm and delineate its computational framework, which is structured as a network of orthogonal blocks rather than a simple linear chain. Additionally, we provide explicit evaluations of specific illustrative examples to demonstrate the practical application of this framework.

Related papers

• Growth of block diagonal operators and symmetry-resolved Krylov complexity [0.0]
  This work addresses how the growth of invariant operators is influenced by their underlying symmetry structure.<n>We introduce the symmetry-resolved Krylov complexity, which captures the time evolution of each block into which an operator, invariant under a given symmetry, can be decomposed.
  arXiv  Detail & Related papers  (2025-07-02T18:00:00Z)
• A Quantum Computational Perspective on Spread Complexity [0.0]
  We establish a direct connection between spread complexity and quantum circuit complexity by demonstrating that spread complexity emerges as a limiting case of a circuit complexity framework built from two fundamental operations: time-evolution and superposition.<n>Our approach leverages a computational setup where unitary gates and beam-splitting operations generate target states, with the minimal cost of synthesis yielding a complexity measure that converges to spread complexity in the infinitesimal time-evolution limit.
  arXiv  Detail & Related papers  (2025-06-08T19:04:42Z)
• Inducing Systematicity in Transformers by Attending to Structurally Quantized Embeddings [60.698130703909804]
  Transformers generalize to novel compositions of structures and entities after being trained on a complex dataset. We propose SQ-Transformer that explicitly encourages systematicity in the embeddings and attention layers. We show that SQ-Transformer achieves stronger compositional generalization than the vanilla Transformer on multiple low-complexity semantic parsing and machine translation datasets.
  arXiv  Detail & Related papers  (2024-02-09T15:53:15Z)
• Discovering modular solutions that generalize compositionally [55.46688816816882]
  We show that identification up to linear transformation purely from demonstrations is possible without having to learn an exponential number of module combinations. We further demonstrate empirically that meta-learning from finite data can discover modular policies that generalize compositionally in a number of complex environments.
  arXiv  Detail & Related papers  (2023-12-22T16:33:50Z)
• Krylov complexity in a natural basis for the Schrödinger algebra [0.0]
  We investigate operator growth in quantum systems with two-dimensional Schr"odinger group symmetry. Cases such as the Schr"odinger algebra which is characterized by a semi-direct sum structure are complicated. We compute Krylov complexity for this algebra in a natural orthonormal basis.
  arXiv  Detail & Related papers  (2023-06-05T18:00:03Z)
• Building Krylov complexity from circuit complexity [4.060731229044571]
  We show that Krylov complexity can be rigorously established from circuit complexity when dynamical symmetries exist. Multiple Krylov complexity may be exploited jointly to fully describe operator dynamics.
  arXiv  Detail & Related papers  (2023-03-13T17:59:43Z)
• DIFFormer: Scalable (Graph) Transformers Induced by Energy Constrained Diffusion [66.21290235237808]
  We introduce an energy constrained diffusion model which encodes a batch of instances from a dataset into evolutionary states. We provide rigorous theory that implies closed-form optimal estimates for the pairwise diffusion strength among arbitrary instance pairs. Experiments highlight the wide applicability of our model as a general-purpose encoder backbone with superior performance in various tasks.
  arXiv  Detail & Related papers  (2023-01-23T15:18:54Z)
• A universal approach to Krylov State and Operator complexities [0.0]
  In our formalism, the Krylov complexity is defined in terms of the density matrix of the associated state. This unified definition of complexity enables us to extend the notion of Krylov complexity to subregion or mixed state complexities.
  arXiv  Detail & Related papers  (2022-12-20T19:00:12Z)
• The Dynamics of Riemannian Robbins-Monro Algorithms [101.29301565229265]
  We propose a family of Riemannian algorithms generalizing and extending the seminal approximation framework of Robbins and Monro. Compared to their Euclidean counterparts, Riemannian algorithms are much less understood due to lack of a global linear structure on the manifold. We provide a general template of almost sure convergence results that mirrors and extends the existing theory for Euclidean Robbins-Monro schemes.
  arXiv  Detail & Related papers  (2022-06-14T12:30:11Z)
• Detailed Account of Complexity for Implementation of Some Gate-Based Quantum Algorithms [55.41644538483948]
  In particular, some steps of the implementation, as state preparation and readout processes, can surpass the complexity aspects of the algorithm itself. We present the complexity involved in the full implementation of quantum algorithms for solving linear systems of equations and linear system of differential equations.
  arXiv  Detail & Related papers  (2021-06-23T16:33:33Z)
• Bilinear Classes: A Structural Framework for Provable Generalization in RL [119.42509700822484]
  Bilinear Classes is a new structural framework which permits generalization in reinforcement learning. The framework incorporates nearly all existing models in which a sample complexity is achievable. Our main result provides an RL algorithm which has sample complexity for Bilinear Classes.
  arXiv  Detail & Related papers  (2021-03-19T16:34:20Z)

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.