Quantum walks with tuneable self-avoidance in one dimension

Abstract

Quantum walks exhibit many unique characteristics compared to classical random walks. In the classical setting, self-avoiding random walks have been studied as a variation on the usual classical random walk. Here the walker has memory of its previous locations and preferentially avoids stepping back to locations where it has previously resided. Classical self-avoiding random walks have found numerous algorithmic applications, most notably in the modelling of protein folding. We consider the analogous problem in the quantum setting – a quantum walk in one dimension with tunable levels of self-avoidance. We complement a quantum walk with a memory register that records where the walker has previously resided. The walker is then able to avoid returning back to previously visited sites or apply more general memory conditioned operations to control the walk. We characterise this walk by examining the variance of the walker's distribution against time, the standard metric for quantifying how quantum or classical a walk is. We parameterise the strength of the memory recording and the strength of the memory back-action on the walker, and investigate their effect on the dynamics of the walk. We find that by manipulating these parameters, which dictate the degree of self-avoidance, the walk can be made to reproduce ideal quantum or classical random walk statistics, or a plethora of more elaborate diffusive phenomena. In some parameter regimes we observe a close correspondence between classical self-avoiding random walks and the quantum self-avoiding walk.

Disorder-free localization in quantum walks

“…We have studied disorder-free localization of a quantum walker coupled to local spins on a one-dimensional lattice. Similar to models of quantum walks on graphs coupled to spins living on nodes 36 or links 38 , the local spins in our model live on lattice sites and do not act on the coin. The Hamiltonian and the initial spin state we have chosen are translationally invariant, yet localization occurs merely due to the interactions between the walker and the local spins.…”

“…In general, even after starting with a product state at t = 0, it is impossible to factorize any degree of freedom completely at later times. At this point, it is worthy to note that the model we consider here is actually simpler in terms of its construction compared to the other similar models since we consider a one-dimensional position space and the spins have no direct action on the walker [36][37][38] .…”

Self-avoiding pruning random walk on signed network

“…The SAW was first introduced by chemist P. J. Froly to study the behaviour of polymers on lattice graph [47]. SAW has also been applied to detect protein-protein interaction [48], to detect network structure which is more efficient than classic random walk by avoiding previously visited nodes in each step [49], and to detect unidentified network traffic [50].…”

Neighborhood-History Quantum Walk

“…It is noteworthy that physical realizations of QWs have emerged, and as they develop further [7,8], they are useful for demonstrating and manipulating quantum behavior. Altogether, strong interest continues to grow in understanding and designing their lattice traversal properties [9][10][11][12][13][14][15][16] to achieve faster and more accurate search algorithms.…”