Spectral quantization of discrete random walks on half-line, and orthogonal polynomials on the unit circle

• URL: http://arxiv.org/abs/2306.12265v2
• Date: Sun, 1 Sep 2024 20:57:08 GMT
• Title: Spectral quantization of discrete random walks on half-line, and orthogonal polynomials on the unit circle
• Authors: Adam Doliwa, Artur Siemaszko,
• Abstract summary: 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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We define quantization scheme for discrete-time random walks on the half-line consistent with Szegedy's quantization of finite Markov chains. Motivated by the Karlin and McGregor description of discrete-time random walks in terms of polynomials orthogonal with respect to a measure with support in the segment $[-1,1]$, we represent the unitary evolution operator of the quantum walk in terms of orthogonal polynomials on the unit circle. We find the relation between transition probabilities of the random walk with the Verblunsky coefficients of the corresponding polynomials of the quantum walk. We show that the both polynomials systems and their measures are connected by the classical Szeg\H{o} map. Our scheme can be applied to arbitrary Karlin and McGregor random walks and generalizes the so called Cantero-Gr\"{u}nbaum-Moral-Vel\'{a}zquez method. We illustrate our approach on example of random walks related to the Jacobi polynomials. Then we study quantization of random walks with constant transition probabilities where the corresponding polynomials on the unit circle have two-periodic real Verblunsky coefficients. We present geometric construction of the spectrum of such polynomials (in the general complex case) which generalizes the known construction for the Geronimus polynomials. In the Appendix we present the explicit form, in terms of Chebyshev polynomials of the second kind, of polynomials orthogonal on the unit circle and polynomials orthogonal on the real line with coefficients of arbitrary period.

Related papers

• Quantum channels, complex Stiefel manifolds, and optimization [45.9982965995401]
  We establish a continuity relation between the topological space of quantum channels and the quotient of the complex Stiefel manifold. The established relation can be applied to various quantum optimization problems.
  arXiv  Detail & Related papers  (2024-08-19T09:15:54Z)
• Quantum Random Walks and Quantum Oscillator in an Infinite-Dimensional Phase Space [45.9982965995401]
  We consider quantum random walks in an infinite-dimensional phase space constructed using Weyl representation of the coordinate and momentum operators. We find conditions for their strong continuity and establish properties of their generators.
  arXiv  Detail & Related papers  (2024-06-15T17:39:32Z)
• Continuous-time open quantum walks in one dimension: matrix-valued orthogonal polynomials and Lindblad generators [0.0]
  We study continuous-time open quantum walks in one dimension through a matrix focusing on nearest-neighbor transitions. Recent results for quantum walks are adapted in order to apply the folding trick to continuous-time birth-death chains on the integers.
  arXiv  Detail & Related papers  (2023-11-27T23:12:51Z)
• Polynomial-time Solver of Tridiagonal QUBO and QUDO problems with Tensor Networks [41.94295877935867]
  We present an algorithm for solving tridiagonal Quadratic Unconstrained Binary Optimization (QUBO) problems and Quadratic Unconstrained Discrete Optimization (QUDO) problems with one-neighbor interactions. Our method is based on the simulation of a quantum state to which we will apply an imaginary time evolution and perform a series of partial traces to obtain the state of maximum amplitude.
  arXiv  Detail & Related papers  (2023-09-19T10:45:15Z)
• 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)
• Grothendieck inequalities characterize converses to the polynomial method [1.024113475677323]
  A surprising 'converse to the method' of Aaronson et al. (CCC16) shows that any bounded quadratic can be computed exactly by a 1-query up to a universal multiplicative factor related to the Grothendieck constant.
  arXiv  Detail & Related papers  (2022-12-16T16:26:04Z)
• Krylov complexity and orthogonal polynomials [30.445201832698192]
  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.
  arXiv  Detail & Related papers  (2022-05-25T14:40:54Z)
• On the general family of third-order shape-invariant Hamiltonians related to generalized Hermite polynomials [0.0]
  This work reports and classifies the most general construction of rational quantum potentials in terms of the generalized Hermites. It is achieved by exploiting the intrinsic relation between third-order shape-invariant Hamiltonians and the fourth Painlev'e equation.
  arXiv  Detail & Related papers  (2022-03-10T20:45:37Z)
• Hilbert-space geometry of random-matrix eigenstates [55.41644538483948]
  We discuss the Hilbert-space geometry of eigenstates of parameter-dependent random-matrix ensembles. Our results give the exact joint distribution function of the Fubini-Study metric and the Berry curvature. We compare our results to numerical simulations of random-matrix ensembles as well as electrons in a random magnetic field.
  arXiv  Detail & Related papers  (2020-11-06T19:00:07Z)
• Positive maps and trace polynomials from the symmetric group [0.0]
  We develop a method to obtain operator inequalities and identities in several variables. We give connections to concepts in quantum information theory and invariant theory.
  arXiv  Detail & Related papers  (2020-02-28T17:43:37Z)
• 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.