Krylov complexity and orthogonal polynomials

• URL: http://arxiv.org/abs/2205.12815v1
• Date: Wed, 25 May 2022 14:40:54 GMT
• Title: Krylov complexity and orthogonal polynomials
• Authors: Wolfgang M\"uck and Yi Yang
• Abstract summary: 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 method of recursion.
• Score: 30.445201832698192
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

Related papers

• The Complexity of Algebraic Algorithms for LWE [0.0]
  We revisit the Arora-Ge model to study complexity of Gr"obner basis computations on LWE systems. We generalize the Gr"obner basis algorithm of Semaev & Tenti to arbitrary systems with a finite degree of regularity.
  arXiv  Detail & Related papers  (2024-02-12T17:59:26Z)
• Spectral quantization of discrete random walks on half-line, and orthogonal polynomials on the unit circle [0.0]
  We represent unitary evolution operator of the quantum walk in terms of Verbs on the unit circle. We show that the both Markovs systems and their measures are connected by the classical SzegHo map.
  arXiv  Detail & Related papers  (2023-06-21T13:41:51Z)
• 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)
• An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree [79.43134049617873]
  In this paper, we demonstrate an exponential separation between exact degree and approximate quantum query for a partial function. For an alphabet size, we have a constant versus separation complexity.
  arXiv  Detail & Related papers  (2023-01-22T22:08:28Z)
• 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)
• Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators [0.618778092044887]
  We introduce the general algebras characterizing a class of higher order superintegrable systems that separate in coordinates. The construction relies on underlying Heisenberg algebras and their defining higher order ladder operators.
  arXiv  Detail & Related papers  (2022-02-27T03:33:26Z)
• Learning Algebraic Recombination for Compositional Generalization [71.78771157219428]
  We propose LeAR, an end-to-end neural model to learn algebraic recombination for compositional generalization. Key insight is to model the semantic parsing task as a homomorphism between a latent syntactic algebra and a semantic algebra. Experiments on two realistic and comprehensive compositional generalization demonstrate the effectiveness of our model.
  arXiv  Detail & Related papers  (2021-07-14T07:23:46Z)
• Hilbert Spaces of Entire Functions and Toeplitz Quantization of Euclidean Planes [0.0]
  We extend the theory of Toeplitz quantization to include diverse and interesting non-commutative realizations of the classical Euclidean plane. The Toeplitz operators are geometrically constructed as special elements from this algebra. Various illustrative examples are computed.
  arXiv  Detail & Related papers  (2021-05-18T09:52:48Z)
• 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)
• A refinement of Reznick's Positivstellensatz with applications to quantum information theory [72.8349503901712]
  In Hilbert's 17th problem Artin showed that any positive definite in several variables can be written as the quotient of two sums of squares. Reznick showed that the denominator in Artin's result can always be chosen as an $N$-th power of the squared norm of the variables.
  arXiv  Detail & Related papers  (2019-09-04T11:46:26Z)

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.