Interacting quantum walk on a graph

Abstract

We introduce an elementary quantum system consisting of a set of spins on a graph and a particle hopping between its nodes. The quantum state is build sequentially, applying a unitary transformation that couples neighboring spins and, at each node, couples the local spin with the particle. We observe the relaxation of the system towards a stationary paramagnetic or ferromagnetic state, and demonstrate that it is related to eigenvectors thermalization and random matrix statistics. The relation between these macroscopic properties and interaction generated entanglement is discussed.

Thermal state entanglement entropy on a quantum graph

“…The operator U associated with a graph, in particular random graphs, has a set of chaotic eigenvectors and eigenvalues, well described by the unitary Gaussian ensemble (Ref. [32]), which in addition to support the eigenstate thermal hypothesis of quantum isolated systems [14,15], demonstrates the ability of the quantum walk dynamics to create highly entangled states. In the next section we investigate the structure of the thermal state relating it with the topology of the graph through the entanglement entropy.…”

“…Using a model of an interacting quantum system introduced in [32], we investigated the entanglement structure of the thermal state. The model do not contain dimensional parameters: it is essentially defined by the graph.…”

“…We consider quantum systems [32] defined on graphs G = (V, E), consisting in a set x ∈ V of |V | vertices (| · | denotes the cardinal of a set), and a set e ∈ E of |E| edges; the set of neighbors of the node x is,…”

“…In a previous work [32] we investigated the entanglement properties of graph quantum states, obtained by the interaction of a particle walking between neighboring nodes and the network of spins. The model can be viewed as a unitary construction of a quantum state associated to a graph by the interaction of a walker with the spins (which physically define the graph connectivity).…”

Disorder-free localization in quantum walks

“…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] .…”