Photons Walking the Line: A quantum walk with adjustable coin operations

可调深度量子电路用于离散时间量子行走中位置相关硬币算符

Adjustable-depth quantum circuit for position-dependent coin operators of discrete-time quantum walks

本文提出一种可调深度量子电路，用于精确实现离散时间量子行走中的位置相关硬币算符，通过并行和顺序组合硬币算符包，在深度和辅助比特之间提供灵活权衡。

Quantum random walks without walking

Quantum random walks have received much interest due to their non-intuitive dynamics, which may hold the key to a new generation of quantum algorithms. What remains a major challenge is a physical realization that is experimentally viable and not limited to special connectivity criteria. We present a scheme for walking on arbitrarily complex graphs, which can be realized using a variety of quantum systems such as a BEC trapped inside an optical lattice. This scheme is particularly elegant since the walker is not required to physically step between the nodes; only flipping coins is sufficient.

Quantum walking in curved spacetime: discrete metric

A discrete-time quantum walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs have familiar physics PDEs as their continuum limit. Some slight generalization of them (allowing for prior encoding and larger neighbourhoods) even have the curved spacetime Dirac equation, as their continuum limit. In the $(1+1)-$dimensional massless case, this equation decouples as scalar transport equations with tunable speeds. We characterise and construct all those QWs that lead to scalar transport with tunable speeds. The local coin operator dictates th…

Asymptotic reduced density matrix of discrete-time quantum walks

In this article we show that for any quantum walker with \textit{m}-dimensional coin subspace, we have $m^2\times m^2$ specific constant matrix $\mathcal{C}$ where it completely determines the asymptotic reduced density matrix of the walker. We show that for any initial state with $P_0$ projector, reduced density matrix, can be obtained by $Tr_1\left(P_0\otimes I\;\mathcal{C}\right)$ or equivalently $Tr_2\left(I\otimes P_0\;\mathcal{C}\right)$. It is worth to mention that characteristic matrix $\mathcal{C}$ is independent of the initial state and just depends on coin operator, so by finding th…

Correcting for Potential Barriers in Quantum Walk Search

A randomly walking quantum particle searches in Grover's $Θ(\sqrt{N})$ iterations for a marked vertex on the complete graph of $N$ vertices by repeatedly querying an oracle that flips the amplitude at the marked vertex, scattering by a "coin" flip, and hopping. Physically, however, potential energy barriers can hinder the hop and cause the search to fail, even when the amplitude of not hopping decreases with $N$. We correct for these errors by interpreting the quantum walk search as an amplitude amplification algorithm and modifying the phases applied by the coin flip and oracle such that the …

Adjustable self-loop on discrete-time quantum walk and its application in spatial search

How self-loops on vertices affect quantum walks is an interesting issue, and self-loops play important roles in quantum walk based algorithms. However, the original model that adjusting the effect of self-loops by changing their number has limitations. For example, the effect of self-loops cannot be adjusted continuously, for their number must be an integer. In this paper, we proposed a model of adjustable self-loop on discrete-time quantum walk, whose weight is controlled by a real parameter in the coin operator. The proposed method not only generalises the situations where the number of self…