Quantum-classical dynamical distance and quantumness of quantum walks

Abstract

We introduce a fidelity-based measure, D QC (t), to quantify the differences in the dynamics of classical versus quantum walks over a graph. We provide universal, graph-independent, analytic expressions of this quantumclassical dynamical distance, showing that at short times D QC (t) is proportional to the coherence of the walker, i.e., a genuine quantum feature, whereas at long times it depends only on the size of the graph. At intermediate times, D QC (t) does depend on the graph topology through its algebraic connectivity. Our results show that the difference in the dynamical behavior of classical and quantum walks is entirely due to the emergence of quantum features at short times. In the long-time limit, quantumness and the different nature of the generators of the dynamics, e.g., the open-system nature of classical walks and the unitary nature of quantum walks, are instead contributing equally.

Modeling Quantum Dot Systems as Random Geometric Graphs with Probability Amplitude-Based Weighted Links

“…The Laplacian acts as a node to node transition matrix. The Hamiltonian of the CTQW can be written as [ 68 , 75 , 77 , 82 , 84 , 85 , 96 , 97 , 98 , 99 , 100 , 101 ].…”

“…We characterize the network’s transport efficiency by using the average return probability , defined as [ 85 ] where the operator , stated in Equation ( 12 ), is the unitary time evolution operator governing the evolution of the probability amplitudes. Please note that, as shown in a number of papers [ 68 , 75 , 77 , 82 , 84 , 85 , 96 , 97 , 98 , 99 , 100 , 101 ], the Hamiltonian of the network is the Laplacian matrix (also called Connectivity matrix in some contexts).…”

Approaching Disordered Quantum Dot Systems by Complex Networks with Spatial and Physical-Based Constraints

“…In our approach, the matrix elements take different values as they depend on the involved overlap integrals and Boltzmann factors ( ) and, as shown throughout the paper, they play a natural role in the probability for an electron to hop from one node to another. The Laplacian acts as a node-to-node transition matrix so that the Hamiltonian of the CTQW can be written as [ 50 , 58 , 60 , 65 , 67 , 68 , 93 , 94 , 95 , 96 , 97 , 98 ].…”

“…Regarding this, we can characterize the network’s transport efficiency by using the average return probability (ARP), , defined as [ 68 ] where the operator , presented in Equation ( 50 ), is the unitary time evolution operator governing the evolution of the probability amplitudes. Please note that, as shown in a number of papers [ 50 , 58 , 60 , 65 , 67 , 68 , 93 , 94 , 95 , 96 , 97 , 98 ], the Hamiltonian of the network is the Laplacian matrix (also called the connectivity matrix in some contexts). We have also shown that , the matrix form of the TB Hamiltonian in the second quantization—Equation ( 44 )—with , meaning that the unitary time evolution operator in quantum mechanics is equivalent to …”

Quantum spatial search in two-dimensional waveguide arrays

“…Quantum random walks on graphs exhibit, in general, a different behaviour than their classical counterparts [2][3][4][5]. A continuous-time quantum walk (CTQW), for example, spreads on the line quadratically faster than its classical analogue, whereas there are particular graphs, such as the glued-tree graph [6] where such "speed-up" is even exponential.…”