# Dynamics of the quantum search and quench-induced first-order phase transitions

## TL;DR

This paper analyzes the excitation dynamics during a first-order quantum phase transition in a quantum search algorithm, revealing exponential convergence of the critical point and a Kibble-Zurek mechanism for first-order transitions.

## Contribution

It derives exact dynamics for the quench-induced QPT in quantum search, showing exponential scaling of kink density and critical point convergence, and explores success probabilities under different strategies.

## Key findings

- Critical point exponentially converges to thermodynamic limit.
- Kink density follows exponential scaling with evolution speed.
- Quantum domains walls and Kibble-Zurek mechanism are observed.

## Abstract

We investigate the excitation dynamics at a first-order quantum phase transition (QPT). More specifically, we consider the quench-induced QPT in the quantum search algorithm, which aims at finding out a marked element in an unstructured list. We begin by deriving the exact dynamics of the model, which is shown to obey a Riccati differential equation. Then, we discuss the probabilities of success by adopting either global or local adiabaticity strategies. Moreover, we determine the disturbance of the quantum criticality as a function of the system size. In particular, we show that the critical point exponentially converges to its thermodynamic limit even in a fast evolution regime, which is characterized by both entanglement QPT estimators and the Schmidt gap. The excitation pattern is manifested in terms of quantum domains walls separated by kinks. The kink density is then shown to follow an exponential scaling as a function of the evolution speed, which can be interpreted as a Kibble-Zurek mechanism for first-order QPTs.