A method of approximation of discrete Schr\"odinger equation with the normalized Laplacian by discrete-time quantum walk on graphs

• URL: http://arxiv.org/abs/2308.13741v1
• Date: Sat, 26 Aug 2023 02:57:47 GMT
• Title: A method of approximation of discrete Schr\"odinger equation with the normalized Laplacian by discrete-time quantum walk on graphs
• Authors: Kei Saito, Etsuo Segawa
• Abstract summary: We propose a class of continuous-time quantum walk models on graphs induced by a certain class of discrete-time quantum walk models. The induced continuous-time quantum walk is an extended version of the (free) discrete-Schr"odinger equation driven by the normalized Laplacian. We show that each discrete-time quantum walk with an appropriate setting of the parameter $epsilon$ in the long time limit identifies with its induced continuous-time quantum walk.
• Score: 0.10878040851637999
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose a class of continuous-time quantum walk models on graphs induced by a certain class of discrete-time quantum walk models with the parameter $\epsilon\in [0,1]$. Here the graph treated in this paper can be applied both finite and infinite cases. The induced continuous-time quantum walk is an extended version of the (free) discrete-Schr\"odinger equation driven by the normalized Laplacian: the element of the weighted Hermitian takes not only a scalar value but also a matrix value depending on the underlying discrete-time quantum walk. We show that each discrete-time quantum walk with an appropriate setting of the parameter $\epsilon$ in the long time limit identifies with its induced continuous-time quantum walk and give the running time for the discrete-time to approximate the induced continuous-time quantum walk with a small error $\delta$. We also investigate the detailed spectral information on the induced continuous-time quantum walk.

Related papers

• Constant-Time Quantum Search with a Many-Body Quantum System [39.58317527488534]
  We consider a many-body quantum system that naturally effects parallel queries. We show that its parameters can be tuned to search a database in constant time.
  arXiv  Detail & Related papers  (2024-08-09T22:57:59Z)
• Quantum walks, the discrete wave equation and Chebyshev polynomials [1.0878040851638]
  A quantum walk is the quantum analogue of a random walk. We show that quantum walks can speed up the spreading or mixing rate of random walks on graphs.
  arXiv  Detail & Related papers  (2024-02-12T17:15:19Z)
• Dilation theorem via Schr\"odingerisation, with applications to the quantum simulation of differential equations [29.171574903651283]
  Nagy's unitary dilation theorem in operator theory asserts the possibility of dilating a contraction into a unitary operator. In this study, we demonstrate the viability of the recently devised Schr"odingerisation approach.
  arXiv  Detail & Related papers  (2023-09-28T08:55:43Z)
• Exponentially decaying velocity bounds of quantum walks in periodic fields [0.0]
  We consider a class of discrete-time one-dimensional quantum walks, associated with a CMV unitary matrix, in the presence of a local field. We show that for a certain range for $t$, the corresponding velocity can be made arbitrarily small by introducing a periodic local field with a sufficiently large period.
  arXiv  Detail & Related papers  (2023-02-03T17:15:15Z)
• Measurement phase transitions in the no-click limit as quantum phase transitions of a non-hermitean vacuum [77.34726150561087]
  We study phase transitions occurring in the stationary state of the dynamics of integrable many-body non-Hermitian Hamiltonians. We observe that the entanglement phase transitions occurring in the stationary state have the same nature as that occurring in the vacuum of the non-hermitian Hamiltonian.
  arXiv  Detail & Related papers  (2023-01-18T09:26:02Z)
• Algebraic Compression of Quantum Circuits for Hamiltonian Evolution [52.77024349608834]
  Unitary evolution under a time dependent Hamiltonian is a key component of simulation on quantum hardware. We present an algorithm that compresses the Trotter steps into a single block of quantum gates. This results in a fixed depth time evolution for certain classes of Hamiltonians.
  arXiv  Detail & Related papers  (2021-08-06T19:38:01Z)
• A crossover between open quantum random walks to quantum walks [0.0]
  The walk connects an open quantum random walk and a quantum walk with parameters $Min mathbbN$ controlling a decoherence effect. We analytically show that a typical behavior of quantum walks appears even in a small gap of the parameter from the open quantum random walk.
  arXiv  Detail & Related papers  (2020-07-02T07:42:24Z)
• Second-order topological insulator in a coinless discrete-time quantum walk [3.7528520149256006]
  We construct a two-dimensional coinless quantum walk to simulate second-order topological insulator with zero-dimensional corner states. We show that both of the corner and edge states can be observed through the probability distribution of the walker. We propose a possible experimental implementation to realize this discrete-time quantum walk in a three-dimensional integrated photonic circuits.
  arXiv  Detail & Related papers  (2020-03-19T09:07:34Z)
• Quantum Statistical Complexity Measure as a Signalling of Correlation Transitions [55.41644538483948]
  We introduce a quantum version for the statistical complexity measure, in the context of quantum information theory, and use it as a signalling function of quantum order-disorder transitions. We apply our measure to two exactly solvable Hamiltonian models, namely: the $1D$-Quantum Ising Model and the Heisenberg XXZ spin-$1/2$ chain. We also compute this measure for one-qubit and two-qubit reduced states for the considered models, and analyse its behaviour across its quantum phase transitions for finite system sizes as well as in the thermodynamic limit by using Bethe ansatz.
  arXiv  Detail & Related papers  (2020-02-05T00:45:21Z)
• Jumptime unraveling of Markovian open quantum systems [68.8204255655161]
  We introduce jumptime unraveling as a distinct description of open quantum systems. quantum jump trajectories emerge, physically, from continuous quantum measurements. We demonstrate that quantum trajectories can also be ensemble-averaged at specific jump counts.
  arXiv  Detail & Related papers  (2020-01-24T09:35:32Z)
• Discrete-Time Quantum Walks on Oriented Graphs [0.0]
  We define discrete-time quantum walks on arbitrary oriented graphs. We introduce a parameter, called alpha, that quantifies the amount of orientation.
  arXiv  Detail & Related papers  (2020-01-13T01:42:42Z)

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.