Abstract
We theoretically analyze the case of noisy Quantum walks (QWs) by introducing four qubit decoherence models into the coin degree of freedom of linear and cyclic QWs. These models include flipping channels (bit flip, phase flip and bit-phase flip), depolarizing channel, phase damping channel and generalized amplitude damping channel. Explicit expressions for the probability distribution of QWs on a line and on a cyclic path are derived under localized and delocalized initial states. We show that QWs which begin from a delocalized state generate mixture probability distributions, which could give rise to useful algorithmic applications related to data encoding schemes. Specifically, we show how the combination of delocalzed initial states and decoherence can be used for computing the binomial transform of a given set of numbers. However, the sensitivity of QWs to noisy environments may negatively affect various other applications based on QWs.
Highlights
Quantum walks (QWs), which differ much from their classical counterparts, have gained substantial attention of the scientific community by becoming an effective ideation and testing ground in various areas of science
We are interested in decoherence models defined on the coin degree of freedom that trigger quantum-to-classical transition in quantum walks on a line and a cycle
We have studied the dynamics of linear and cyclic quantum walks under the influence of a noisy environment
Summary
Quantum walks (QWs), which differ much from their classical counterparts, have gained substantial attention of the scientific community by becoming an effective ideation and testing ground in various areas of science. Space and time depended coin operators are used in [34,37] to achieve classical-like behavior in QWs. A phenomenological decoherence model is introduced in [21,23] for QWs on a line by defining a completely positive map on the coin degree of freedom. A phenomenological decoherence model is introduced in [21,23] for QWs on a line by defining a completely positive map on the coin degree of freedom This model contains decoherence schemes like pure dephasing and weak measurements on coin space and characterizes a QW with a coin subjected to decoherence. The influence of these decoherence models on linear and cyclic QWs is studied separately, but general conclusions are eventually drawn
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