Analysis of Decoherence in Linear and Cyclic Quantum Walks

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.

Topological Characterisation of Multi-Buffer Simulation

Multi-buffer simulation is an extension of simulation preorder that can be used to approximate inclusion of languages recognised by Büchi automata up to their trace closures. DUPLICATOR can use some bounded or unbounded buffers to simulate SPOILER’s move. It has been shown that multi-buffer simulation can be characterised with the existence of a continuous function. In this paper, we show that such a characterisation can be refined to a more restricted case, that is, to the one where DUPLICATOR only uses bounded buffers, by requiring the function to be Lipschitz continuous instead of only continuous. This characterisation however only holds for some restricted classes of automata. One of the automata should only produce words where each letter cannot commute unboundedly. We show that this property can be syntactically characterised with cyclic-path-connectedness, a refinement of syntactic condition on automata that have regular trace closure. We further show that checking cyclic-path-connectedness is indeed co-NP-complete.