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arXiv:1505.01435 · Pramana

An effective Hamiltonian approach to quantum random walk

Debajyoti Sarkar, Niladri Paul, Kaushik Bhattacharya, Tarun Kanti Ghosh

Published Feb 8, 2017• 15 citations
arXivHTMLPDF
DOI
10.1007/s12043-016-1340-5

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•May 1, 2022•0 citations
Abstract We study the effect of random scattering in quantum walks on a finite graph and compare it with the effect of repeated measurements. To this end, a constructive approach is employed by introducing a localized and a delocalized basis for the underlying Hilbert space. This enables us to design Hamiltonians whose eigenvectors are either localized or delocalized. By presenting some specific examples we demonstrate that the localization of eigenvectors restricts the transition probabilities on the graph and leads to a removal of energy states from the quantum walk in the monitored evolution. We conclude that repeated measurements as well as random scattering provide efficient tools for controlling quantum walks.
We study the effects of quantum noise in hybrid quantum-classical solver for sparse systems of linear equations using quantum random walks, applied to stoquastic Hamiltonian matrices. In an ideal noiseless quantum computer, sparse matrices achieve solution vectors with lower relative error than dense matrices. However, we find quantum noise reverses this effect, with overall error increasing as sparsity increases. We identify invalid quantum random walks as the cause of this increased error and propose a revised linear solver algorithm which improves accuracy by mitigating these invalid walks.
We study continuous time quantum walk on a random comb graph with infinite teeth. Due to localization effects along the spine, the walk cannot go to infinity in the spine direction, while it can escape to infinity along the teeth of the comb. Starting from an initial vertex, the walk has a nonzero probability to stay trapped in a finite region. These results are obtained by studying the spectrum and eigenstates of the random Hamiltonian for the graphand analysing its properties. We use both analytic and numerical methods, many of which come from the theory of Anderson localization in one dimension.
Recent findings of Markov violations challenge Markov random walk processes for decision making. On the other hand, quantum walk processes explain these Markov violations in a natural way, but they have only been applied to binary alternative decision making. In this work, we propose a general framework for extending quantum walk processes to multi-alternative decision making. The multi-alternative quantum walk model operates in a direct sum space of the alternatives, with Hamiltonian built for each pair of alternatives to model context effects. Order effects come naturally from the matrix non-commutativity of the Hamiltonians. Future works built on this framework can connect parameters of the models with the expected utilities.