A quantum walk with a delocalized initial state: contribution from a coin-flip operator

A unit evolution step of discrete-time quantum walks is determined by both a coin-flip operator and a position-shift operator. The behavior of quantum walkers after many steps delicately depends on the coin-flip operator and an initial condition of the walk. To get the behavior, a lot of long-time limit distributions for the quantum walks starting with a localized initial state have been derived. In the present paper, we compute limit distributions of a 2-state quantum walk with a delocalized initial state, not a localized initial state, and discuss how the walker depends on the coin-flip operator. The initial state induced from the Fourier series expansion, which is called the $(\alpha,\beta)$ delocalized initial state in this paper, provides different limit density functions from the ones of the quantum walk with a localized initial state.Comment: International Journal of Quantum Information, Vol.11, No.5, 1350053 (2013

Limit theorems for a localization model of 2-state quantum walks

We consider 2-state quantum walks (QWs) on the line, which are defined by two matrices. One of the matrices operates the walk at only half-time. In the usual QWs, localization does not occur at all. However, our walk can be localized around the origin. In this paper, we present two limit theorems, that is, one is a stationary distribution and the other is a convergence theorem in distribution.Comment: International Journal of Quantum Information, Vol.9, No.3, pp.863-874 (2011

Localization of the Grover walks on spidernets and free Meixner laws

A spidernet is a graph obtained by adding large cycles to an almost regular tree and considered as an example having intermediate properties of lattices and trees in the study of discrete-time quantum walks on graphs. We introduce the Grover walk on a spidernet and its one-dimensional reduction. We derive an integral representation of the $n$-step transition amplitude in terms of the free Meixner law which appears as the spectral distribution. As an application we determine the class of spidernets which exhibit localization. Our method is based on quantum probabilistic spectral analysis of graphs.Comment: 32 page

Quantum walks: a comprehensive review

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing Journa

Polymorphisms in NFkB, PXR, LXR and risk of colorectal cancer in a prospective study of Danes

<p>Abstract</p> <p>Background</p> <p>Transcription factors and nuclear receptors constitute a link between exposure to heterocyclic amines and polycyclic aromatic hydrocarbons from meat and tobacco smoke and colorectal cancer (CRC) risk. The aim of this study was to investigate if polymorphisms in nuclear factor kappa-B, pregnane X receptor, and liver X receptor were associated with risk of CRC, and to investigate possible interactions with lifestyle factors such as smoking, meat consumption, and NSAID use.</p> <p>Methods</p> <p>The polymorphisms nuclear factor kappa-B (<it>NFkB, NFKB1) </it>-94 insertion/deletion ATTG (rs28362491), pregnane X receptor (<it>PXR, NR1I2) </it>A-24381C (rs1523127), C8055T (rs2276707), A7635G (rs6785049), liver X receptor (<it>LXR-β, NR1H3) </it>C-rs1405655T, T-rs2695121C were assessed together with lifestyle factors in a nested case-cohort study of 378 CRC cases and 756 random participants from the Danish prospective Diet, Cancer and Health study of 57,053 persons.</p> <p>Results</p> <p>Carriers of <it>NFkB </it>-94deletion were at 1.45-fold higher risk of CRC than homozygous carriers of the insertion allele (incidence rate ratio (IRR) = 1.45, 95% confidence interval (95% CI): 1.10-1.92). There was interaction between this polymorphism and intake of red and processed meat in relation to CRC risk. Carriers of <it>NFkB </it>-94deletion were at 3% increased risk pr 25 gram meat per day (95% CI: 0.98-1.09) whereas homozygous carriers of the insertion were not at increased risk (p for interaction = 0.03). <it>PXR </it>and <it>LXR </it>polymorphisms were not associated with CRC risk. There was no interaction between use of nonsteroid antiinflammatory drugs (NSAID) or smoking status and <it>NFkB</it>, <it>PXR </it>or <it>LXR </it>polymorphisms.</p> <p>Conclusions</p> <p>A polymorphism in <it>NFkB </it>was associated with CRC risk and there was interaction between this polymorphism and meat intake in relation to CRC risk. This study suggests a role for NFkB in CRC aetiology.</p