# Short time behavior of continuous time quantum walks on graphs

## TL;DR

This paper investigates the initial behavior of continuous time quantum walks on graphs, revealing how graph topology influences short time dynamics and how symmetry and coherence affect these asymptotics.

## Contribution

It provides a detailed analysis of short time asymptotics of CTQWs, including effects of symmetry, potentials, and non-coherent effects, with analytical formulas validated numerically.

## Key findings

- Short time asymptotics follow power laws determined by graph topology.
- Time-reversal symmetry doubles the power-law exponent in transition probabilities.
- Breaking symmetry and introducing non-coherent effects significantly alter short time behavior.

## Abstract

Dynamical evolution of systems with sparse Hamiltonians can always be recognized as continuous time quantum walks (CTQWs) on graphs. In this paper, we analyze the short time asymptotics of CTQWs. In recent studies, it was shown that for the classical diffusion process the short time asymptotics of the transition probabilities follows power laws whose exponents are given by the usual combinatorial distances of the nodes. Inspired by this result, we perform a similar analysis for CTQWs both in closed and open systems, including time-dependent couplings. For time-reversal symmetric coherent quantum evolutions, the short time asymptotics of the transition probabilities is completely determined by the topology of the underlying graph analogously to the classical case, but with a doubled power-law exponent. Moreover, this result is robust against the introduction of on-site potential terms. However, we show that time-reversal symmetry breaking terms and non-coherent effects can significantly alter the short time asymptotics. The analytical formulas are checked against numerics, and excellent agreement is found. Furthermore, we discuss in detail the relevance of our results for quantum evolutions on particular network topologies.