Abstract
Continuous time quantum walks provide an important framework for designing new algorithms and modelling quantum transport and state transfer problems. Often, the graph representing the structure of a problem contains certain symmetries that confine the dynamics to a smaller subspace of the full Hilbert space. In this work, we use invariant subspace methods, that can be computed systematically using the Lanczos algorithm, to obtain the reduced set of states that encompass the dynamics of the problem at hand without the specific knowledge of underlying symmetries. First, we apply this method to obtain new instances of graphs where the spatial quantum search algorithm is optimal: complete graphs with broken links and complete bipartite graphs, in particular, the star graph. These examples show that regularity and high-connectivity are not needed to achieve optimal spatial search. We also show that this method considerably simplifies the calculation of quantum transport efficiencies. Furthermore, we observe improved efficiencies by removing a few links from highly symmetric graphs. Finally, we show that this reduction method also allows us to obtain an upper bound for the fidelity of a single qubit transfer on an XY spin network.
Highlights
Quantum walks[1,2,3,4,5,6,7] are an important framework to model quantum dynamics, with applications ranging from quantum computation to quantum transport
We show that the algorithm runs optimally on the complete graph with imperfections in the form of broken links, and for complete bipartite graphs (CBG)
In the context of quantum transport, we show that breaking a link from the complete graph can affect severely the dynamics if one starts with a localized initial state
Summary
Quantum walks[1,2,3,4,5,6,7] are an important framework to model quantum dynamics, with applications ranging from quantum computation to quantum transport. We use invariant subspace methods, that can be computed systematically using the Lanczos algorithm[24], to obtain a reduced model that fully describes the evolution of the probability amplitude at the node we are interested in. Any problem in quantum mechanics wherein the dynamics is described by a time independent Hamiltonian can be mapped to a CTQW on a weighted line, where the nodes are the elements of the Lanczos basis In this way, we explore the notion of invariant subspaces to systematically reduce the dimension of the Hamiltonian that completely describes the dynamics relevant to our problem. We use this method to obtain new results on several CTQW problems, as well as re-derive some other known results in a simpler manner
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