Closed-form expressions for the probability distribution of quantum walk on a line

• URL: http://arxiv.org/abs/2308.05213v1
• Date: Wed, 9 Aug 2023 20:32:40 GMT
• Title: Closed-form expressions for the probability distribution of quantum walk on a line
• Authors: Mahesh N. Jayakody and Eliahu Cohen
• Abstract summary: We derive expressions for the probability distribution of quantum walks on a line. The most general two-state coin operator and the most general (pure) initial state are considered. We retrieve the simulated probability distribution of Hadamard walk on a line.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: Theoretical and applied studies of quantum walks are abundant in quantum science and technology thanks to their relative simplicity and versatility. Here we derive closed-form expressions for the probability distribution of quantum walks on a line. The most general two-state coin operator and the most general (pure) initial state are considered in the derivation. The general coin operator includes the common choices of Hadamard, Grover, and Fourier coins. The method of Fibonacci-Horner basis for the power decomposition of a matrix is employed in the analysis. Moreover, we also consider mixed initial states and derive closed-form expression for the probability distribution of the Quantum walk on a line. To prove the accuracy of our derivations, we retrieve the simulated probability distribution of Hadamard walk on a line using our closed-form expressions. With a broader perspective in mind, we argue that our approach has the potential to serve as a helpful mathematical tool in obtaining precise analytical expressions for the time evolution of qubit-based systems in a general context.

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)
• Efficient quantum loading of probability distributions through Feynman propagators [2.56711111236449]
  We present quantum algorithms for the loading of probability distributions using Hamiltonian simulation for one dimensional Hamiltonians of the form $hat H= Delta + V(x) mathbbI$. We consider the potentials $V(x)$ for which the Feynman propagator is known to have an analytically closed form and utilize these Hamiltonians to load probability distributions into quantum states.
  arXiv  Detail & Related papers  (2023-11-22T21:41:58Z)
• Quantum tomography of helicity states for general scattering processes [55.2480439325792]
  Quantum tomography has become an indispensable tool in order to compute the density matrix $rho$ of quantum systems in Physics. We present the theoretical framework for reconstructing the helicity quantum initial state of a general scattering process.
  arXiv  Detail & Related papers  (2023-10-16T21:23:42Z)
• Third quantization of open quantum systems: new dissipative symmetries and connections to phase-space and Keldysh field theory formulations [77.34726150561087]
  We reformulate the technique of third quantization in a way that explicitly connects all three methods. We first show that our formulation reveals a fundamental dissipative symmetry present in all quadratic bosonic or fermionic Lindbladians. For bosons, we then show that the Wigner function and the characteristic function can be thought of as ''wavefunctions'' of the density matrix.
  arXiv  Detail & Related papers  (2023-02-27T18:56:40Z)
• Quantum Bayesian Inference in Quasiprobability Representations [0.0]
  Bayes' rule plays a crucial piece of logical inference in information and physical sciences alike. quantum versions of Bayes' rule have been expressed in the language of Hilbert spaces.
  arXiv  Detail & Related papers  (2023-01-05T08:16:50Z)
• The uniform measure for quantum walk on hypercube: a quantum Bernoulli noises approach [6.855885246744849]
  We present a quantum Bernoulli noises approach to quantum walks on hypercubes. We introduce a discrete-time quantum walk model on a general hypercube. We establish two limit theorems showing that the averaged probability distribution of the walk even converges to the uniform probability distribution.
  arXiv  Detail & Related papers  (2022-11-15T07:24:02Z)
• Theory of Quantum Generative Learning Models with Maximum Mean Discrepancy [67.02951777522547]
  We study learnability of quantum circuit Born machines (QCBMs) and quantum generative adversarial networks (QGANs) We first analyze the generalization ability of QCBMs and identify their superiorities when the quantum devices can directly access the target distribution. Next, we prove how the generalization error bound of QGANs depends on the employed Ansatz, the number of qudits, and input states.
  arXiv  Detail & Related papers  (2022-05-10T08:05:59Z)
• Why we should interpret density matrices as moment matrices: the case of (in)distinguishable particles and the emergence of classical reality [69.62715388742298]
  We introduce a formulation of quantum theory (QT) as a general probabilistic theory but expressed via quasi-expectation operators (QEOs) We will show that QT for both distinguishable and indistinguishable particles can be formulated in this way. We will show that finitely exchangeable probabilities for a classical dice are as weird as QT.
  arXiv  Detail & Related papers  (2022-03-08T14:47:39Z)
• Bernstein-Greene-Kruskal approach for the quantum Vlasov equation [91.3755431537592]
  The one-dimensional stationary quantum Vlasov equation is analyzed using the energy as one of the dynamical variables. In the semiclassical case where quantum tunneling effects are small, an infinite series solution is developed.
  arXiv  Detail & Related papers  (2021-02-18T20:55:04Z)
• Theory of Ergodic Quantum Processes [0.0]
  We consider general ergodic sequences of quantum channels with arbitrary correlations and non-negligible decoherence. We compute the entanglement spectrum across any cut, by which the bipartite entanglement entropy can be computed exactly. Other physical implications of our results are that most Floquet phases of matter are metastable and that noisy random circuits in the large depth limit will be trivial as far as their quantum entanglement is concerned.
  arXiv  Detail & Related papers  (2020-04-29T18:00:03Z)
• Hilbert space average of transition probabilities [0.0]
  We show that the transition probability of two random uniformly distributed states is connected to the spectral statistics of the considered operator. We will demonstrate our quite general result numerically for a kicked spin chain in the integrable resp. chaotic regime.
  arXiv  Detail & Related papers  (2020-02-21T16:25:33Z)

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.