Noisy quantum walk
A quantum walk (QW) is the quantum analogue of the classical random walk. In a classical walk, the random walker moves to the adjacent sites (either left or right) at each step with a prescribed probability. An unbiased walker will cover positions following a Gaussian distribution. Feynman (1986) realized the limitation of classical computing in simulating quantum systems, hinting towards the possibility of using quantum walk to resolve this problem. A quantum walker allows superposition of possible paths which leads to interference, resulting in a different probability distribution compared to the classical walk. The variance of quantum walk spreads faster (linearly with time/steps) in contrast to the square root of time-step of the classical walk. This property of QW has found numerous applications for superior quantum algorithms for quantum computing, quantum search engine and quantum communications. Fundamental properties of different formulations of QWs are introduced and generalized to accommodate noisy conditions. Given the broad class of dynamics, it is important to characterize the degree of quantumness or non-classicality of QWs in the presence of noises or disorders. Several useful measures including the exponent of variance scaling behavior related to the ballistic transport, generalized information entropies and their relations to information backflow and non-Markovianity for describing space-coin states entanglement will be described. We demonstrate the validity of these measures for the discrete-time QW in a variety of noisy quantum channels. Noisy Intermediate-Scale Quantum (NISQ) strategy is a middle path between the isolated quantum and fault tolerant setups to realize various applications that may be subjected to noises, imperfections and coupling with the fluctuating environment, resulting in decoherence. The prospect for making use of a noisy quantum walk for quantum-like transport is also explored.