Möbius quantum walk

Abstract

By adding an extra Hilbert space to Hadamard Quantum Walk on Cycles (QWC), we presented a new type of QWCs called Möbius Quantum Walk (MQW). The new space configuration enables the particle to rotate around the axis of movement. We defined factor α as the Möbius factor which is number of rotations per cycle. So by α = 0 we have normal QWC, while α = 0 defines new type of QWC (namely Möbius Quantum Walk). Specially α = 1 2 defines a structure similar to Möbius strip. We analytically investigated this new type of QW and found that by tuning the parameter α we can reach uniform distribution for any number of nodes, while it is impossible for QWC. The effects of α on limiting distribution have been investigated and an explicit formula for non-uniform cases has been derived as well.

Discrete Quantum Walks on Graphs and Digraphs

“…The aim of this section is to lift a shunt-decomposition walk on X to a walk on its cover X φ . Our construction generalizes the shunt-decomposition model due to Aharonov et al [2], as well as the Möbius walk defined by Moradi and Annabestani [53].…”

“…In this chapter, we introduce quantum walks on unitary covers of digraphs; they can be seen as quantum walks on the base digraphs with enlarged state spaces. Our model generalizes the shunt-decomposition walks [2], as well as the Möbius quantum walks [53]. In the Möbius walk, the walker can rotate around the axis of movement while walking on the cycle, and the extra rotation space allows uniform average vertex mixing to occur with optimized mixing time [53].…”

“…Our model generalizes the shunt-decomposition walks [2], as well as the Möbius quantum walks [53]. In the Möbius walk, the walker can rotate around the axis of movement while walking on the cycle, and the extra rotation space allows uniform average vertex mixing to occur with optimized mixing time [53].…”

“…We explain the connection between this walk and the ones in [2] and [53]. If r = 1 and φ is the identity map, then X φ = X and the shift matrix is simply Thus x ⊗ y ⊗ z is an eigenvector of U for the eigenvalue α if and only if x is an eigenvector of DC for the eigenvalue α.…”

Asymptotic reduced density matrix of discrete-time quantum walks

“…The power and applications of QW motivated researchers to introduce different models of QWs. Many coins QW [7], QW on higher dimension [8], QW on a circle [9], coin less QW [10], Mobius QW [11] and staggered QW [12] are just a few examples of these models.…”