arXiv 1105.07052011-05-03quant-phgr-qchep-th

Two-Site Quantum Random Walk

Stan Gudder, Rafael D. Sorkin

备注：: 28 pages
发表信息：: General Relativity and Gravitation 2011; 43(12): 3451-3475
DOI：: 10.1007/s10714-011-1245-z

英文摘要：

We study the measure theory of a two-site quantum random walk. The truncated decoherence functional defines a quantum measure $μ_n$ on the space of $n$-paths, and the $μ_n$ in turn induce a quantum measure $μ$ on the cylinder sets within the space $Ω$ of untruncated paths. Although $μ$ cannot be extended to a continuous quantum measure on the full $σ$-algebra generated by the cylinder sets, an important question is whether it can be extended to sufficiently many physically relevant subsets of $Ω$ in a systematic way. We begin an investigation of this problem by showing that $μ$ can be extended to a quantum measure on a "quadratic algebra" of subsets of $Ω$ that properly contains the cylinder sets. We also present a new characterization of the quantum integral on the $n$-path space.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Homogeneous open quantum walks on the line: criteria for site recurrence and absorption

In this work, we study open quantum random walks, as described by S. Attal et al. These objects are given in terms of completely positive maps acting on trace-class operators, leading to one of the simplest open quantum versions of the recurrence problem for classical, discrete-time random walks. This work focuses on obtaining criteria for site recurrence of nearest-neighbor, homogeneous walks on the integer line, with the description presented here making use of recent results of the theory of open walks, most particularly regarding reducibility properties of the operators involved. This allo…

2021-01-11math-phmath.MPmath.PR

Persistence of unvisited sites in quantum walks on a line

We analyze the asymptotic scaling of persistence of unvisited sites for quantum walks on a line. In contrast to the classical random walk there is no connection between the behaviour of persistence and the scaling of variance. In particular, we find that for a two-state quantum walks persistence follows an inverse power-law where the exponent is determined solely by the coin parameter. Moreover, for a one-parameter family of three-state quantum walks containing the Grover walk the scaling of persistence is given by two contributions. The first is the inverse power-law. The second contribution …

2016-03-16quant-ph

Probing the sign of on-site Hubbard interaction by two-particle quantum walks

We consider two identical bosons propagating on a one-dimensional lattice and address the prob- lem of discriminating whether their mutual on-site interaction is attractive or repulsive. We suggest a probing scheme based on the properties of the corresponding two-particle quantum walks, and show that the sign of the interaction introduces specific and detectable features in the dynamics of quantum correlations, thus permitting to discriminate between the two cases. We also discuss how these features are connected to the band-structure of the Hubbard Hamiltonian, and prove that discrimination m…

2017-10-27quant-ph

Site recurrence of open and unitary quantum walks on the line

We study the problem of site recurrence of discrete time nearest neighbor open quantum random walks (OQWs) on the integer line, proving basic properties and some of its relations with the corresponding problem for unitary (coined) quantum walks (UQWs). For both kinds of walks our discussion concerns two notions of recurrence, one given by a monitoring procedure, another in terms of Pólya numbers, and we study their similarities and differences. In particular, by considering UQWs and OQWs induced by the same pair of matrices, we discuss the fact that recurrence of these walks are related by an …

2016-09-29math-phmath.MPmath.PR

Fluctuations of Quantum Random Walks on Circles

Temporal fluctuations in the Hadamard walk on circles are studied. A temporal standard deviation of probability that a quantum random walker is positive at a given site is introduced to manifest striking differences between quantum and classical random walks. An analytical expression of the temporal standard deviation on a circle with odd sites is shown and its asymptotic behavior is considered for large system size. In contrast with classical random walks, the temporal fluctuation of quantum random walks depends on the position and initial conditions, since temporal standard deviation of the …

2003-09-29quant-ph

Noise-enhanced quantum transport on a closed loop using quantum walks

We study the effect of noise on the transport of a quantum state from a closed loop of $n-$sites with one of the sites as a sink. Using a discrete-time quantum walk dynamics, we demonstrate that the transport efficiency can be enhanced with noise when the number of sites in the loop is small and reduced when the number of sites in the loop grows. By using the concept of measurement induced disturbance we identify the regimes in which genuine quantum effects are responsible for the enhanced transport.

2014-01-08quant-ph

更多相关论文收起更多论文

IBM量子计算机上的测量诱导量子行走

Measurement induced quantum walks on an IBM Quantum Computer

本研究在IBM量子计算机上实现并验证了双位点测量诱导量子行走，精确观测到首次探测返回时间的量子化及简并点附近跃迁时间的增强。

2022-10-18quant-ph

Quantum walks and quantum search on graphene lattices

Quantum walks have been very successful in the development of search algorithms in quantum information, in particular in the development of spatial search algorithms. However, the construction of continuous-time quantum search algorithms in two-dimensional lattices has proved difficult, requiring additional degrees of freedom. Here, we demonstrate that continuous-time quantum walk search is possible in two-dimensions by changing the search topology to a graphene lattice, utilising the Dirac point in the energy spectrum. This is made possible by making a change to standard methods of marking a …

2015-07-01quant-ph

线上连续时间开放量子行走的位点复现性

Site Recurrence for continuous-time open quantum walks on the line

本文研究线上连续时间开放量子行走的位点复现性，引入 SJK-复现性并证明其与传统复现性等价，进而给出有限维硬币且辅助 Lindblad 算子具有单一稳态时的完整判据，并补全内部自由度为 2 维的情形。

2023-05-31quant-ph

Universal Behavior of Quantum Walks with Long-Range Steps

Quantum walks with long-range steps $R^{-γ}$ ($R$ being the distance between sites) on a discrete line behave in similar ways for all $γ\geq2$. This is in contrast to classical random walks, which for $γ>3$ belong to a different universality class than for $γ\leq 3$. We show that the average probabilities to be at the initial site after time $t$ as well as the mean square displacements are of the same functional form for quantum walks with $γ=2$, 4, and with nearest neighbor steps. We interpolate this result to arbitrary $γ\geq2$.

2007-11-23cond-mat.stat-mechquant-ph

一维格上持久随机游走与量子游走之间插值游走具有反正弦律的正态方差混合

Normal variance mixture with arcsine law of an interpolating walk between persistent random walk and quantum walk

提出一维格上单参数插值量子游走与持久随机游走的模型，证明其极限分布服从反正弦律的正态方差混合。

2025-03-04quant-ph

Dynamical Localization for d-Dimensional Random Quantum Walks

We consider a d-dimensional random quantum walk with site-dependent random coin operators. The corresponding transition coefficients are characterized by deterministic amplitudes times independent identically distributed site-dependent random phases. When the deterministic transition amplitudes are close enough to those of a quantum walk which forbids propagation, we prove that dynamical localization holds for almost all random phases. This instance of Anderson localization implies that all quantum mechanical moments of the position operator are uniformly bounded in time and that spectral loca…

2012-04-05math-phmath.MPquant-ph

Mapping a finite and an infinite Hadamard quantum walk onto a unique case of a random walk process

A new model that maps a quantum random walk described by a Hadamard operator to a particular case of a random walk is presented. The model is represented by a Markov chain with a stochastic matrix, i.e., all the transition rates are positive, although the Hadamard operator contains negative entries. Using a proper transformation that is applied to the random walk distribution after n steps, the probability distributions in space of the two quantum states |1>, |0> are revealed. These show that a quantum walk can be entirely mapped to a particular case of a higher dimension of a random walk mode…

2020-11-17quant-ph

Renormalization and Scaling in Quantum Walks

We show how to extract the scaling behavior of quantum walks using the renormalization group (RG). We introduce the method by efficiently reproducing well-known results on the one-dimensional lattice. As a nontrivial model, we apply this method to the dual Sierpinski gasket and obtain its exact, closed system of RG-recursions. Numerical iteration suggests that under rescaling the system length, $L^{\prime}=2L$, characteristic times rescale as $t^{\prime}=2^{d_{w}}t$ with the exact walk exponent $d_{w}=\log_{2}\sqrt{5}=1.1609\ldots$. Despite the lack of translational invariance, this is very cl…

2014-09-28cond-mat.stat-mechquant-ph

Temporal Fluctuations of Continuous-Time Quantum Random Walks on Circles

This work deals with both instantaneous uniform mixing property and temporal standard deviation for continuous-time quantum random walks on circles in order to study their fluctuations comparing with discrete-time quantum random walks, and continuous- and discrete-time classical random walks.

2004-07-31quant-ph

Slow transport by continuous time quantum walks

Continuous time quantum walks (CTQW) do not necessarily perform better than their classical counterparts, the continuous time random walks (CTRW). For one special graph, where a recent analysis showed that in a particular direction of propagation the penetration of the graph is faster by CTQWs than by CTRWs, we demonstrate that in another direction of propagation the opposite is true; In this case a CTQW initially localized at one site displays a slow transport. We furthermore show that when the CTQW's initial condition is a totally symmetric superposition of states of equivalent sites, the tr…

2004-10-29cond-mat.stat-mechquant-ph

What is random about a quantum random walk?

We use simple deterministic dynamical systems as coins in studying quantum walks. These dynamical systems can be chosen to display, in the classical limit, a range of behaviors from the integrable to chaotic, or deterministically random. As an example of an integrable coin we study the Fourier walk that generalizes the Hadamard walk and show that the walker slows down with coin dimensionality, which controls the effective Planck constant. Introducing multi-Harper maps as deterministic models of random walks we study the effect of coin chaos on the quantum walk. We also demonstrate that breakin…

2003-05-06cond-matnlin.CDquant-ph

Localization and Recurrence of Quantum Walk in Periodic Potential on a Line

We present numerical study of a model of quantum walk in periodic potential on the line. We take the simple view that different potentials affect differently the way the coin state of the walker is changed. For simplicity and definiteness, we assume the walker's coin state is unaffected at sites without potential, and is rotated in an unbiased way according to Hadamard matrix at sites with potential. This is the simplest and most natural model of a quantum walk in a periodic potential with two coins. Six generic cases of such quantum walks were studied numerically. It is found that of the six …

2014-11-28cond-mat.othermath-phmath.MP