Related papers: The Witten index for one-dimensional split-step qu…
We consider the Weyl quantum walk in 3+1 dimensions, that is a discrete-time walk describing a particle with two internal degrees of freedom moving on a Cayley graph of the group $\mathbb Z^3$, that in an appropriate regime evolves…
The role of classical noise in quantum walks (QW) on integers is investigated in the form of discrete dichotomic random variable affecting its reshuffling matrix parametrized as a SU2)/U(1) coset element. Analysis in terms of quantum…
An odd Fredholm module for a given invertible operator on a Hilbert space is specified by an unbounded so-called Dirac operator with compact resolvent and bounded commutator with the given invertible. Associated to this is an index pairing…
The development of quantum walks in the context of quantum computation, as generalisations of random walk techniques, led rapidly to several new quantum algorithms. These all follow unitary quantum evolution, apart from the final…
We give a topological classification of quantum walks on an infinite 1D lattice, which obey one of the discrete symmetry groups of the tenfold way, have a gap around some eigenvalues at symmetry protected points, and satisfy a mild locality…
The quantum walk (QW) is the term given to a family of algorithms governing the evolution of a discrete quantum system and as such has a founding role in the study of quantum computation. We contribute to the investigation of QW phenomena…
A coinless, discrete-time quantum walk possesses a Hilbert space whose dimension is smaller compared to the widely-studied coined walk. Coined walks require the direct product of the site basis with the coin space, coinless walks operate…
We perform the spectral analysis of the evolution operator U of quantum walks with an anisotropic coin, which include one-defect models, two-phase quantum walks, and topological phase quantum walks as special cases. In particular, we…
These notes are devoted to fluctuations of one-dimensional random walks. We discuss various approaches to first-passage times and to the corresponding conditional distributions. After discussion of some classical methods, such as reflection…
In this article we consider operators of the form $\partial_s\xi+A(s)\xi$ where $s$ lies in an interval $[-T,T]$ and $s\mapsto A(s)$ is continuous. Without boundary conditions these operators are not Fredholm. However, using interpolation…
We provide numerical evidence that the nonlinear searching algorithm introduced by Wong and Meyer \cite{meyer2013nonlinear}, rephrased in terms of quantum walks with effective nonlinear phase, can be extended to the finite 2-dimensional…
Quantum walks, both discrete and continuous, serve as fundamental tools in quantum information processing with diverse applications. This work introduces a hybrid quantum walk model that integrates the coin mechanism of discrete walks with…
Random walk algorithms are crucial for sampling and approximation problems in statistical physics and theoretical computer science. The mixing property is necessary for Markov chains to approach stationary distributions and is facilitated…
The time-evolution equation of a one-dimensional quantum walker is exactly mapped to the three-dimensional Weyl equation for a zero-mass particle with spin 1/2, in which each wave number k of walker's wave function is mapped to a point…
This paper describes an updated exponential Fourier based split-step method that can be applied to a greater class of partial differential equations than previous methods would allow. These equations arise in physics and engineering, a…
Universal quantum computation can be realised using both continuous-time and discrete-time quantum walks. We present a version based on single particle discrete-time quantum walk to realize multi-qubit computation tasks. The scalability of…
We construct concrete examples of time operators for both continuous and discrete-time homogeneous quantum walks, and we determine their deficiency indices and spectra. For a discrete-time quantum walk, the time operator can be self-adjoint…
There are presently two models for quantum walks on graphs. The "coined" walk uses discrete time steps, and contains, besides the particle making the walk, a second quantum system, the coin, that determines the direction in which the…
We examine the coherent propagation of the one-dimensional Frenkel exciton (correlated electron-hole pair system) based on a model of a quantum walker in multi-dimensional Hilbert space. The walk is governed by a non-Hermitian coin flip…
We study the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain. In the unitary (i.e., non-random) case, we allow any…