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

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 domain 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.

Figure 1

Probability of success, $P_0\left(s\right)$, as a function of the normalized time $s$ for $n=10$ qubits for several dimensionless running rates ${\tau }_{\text{GA}}$, under a global adiabaticity strategy.

Figure 2

Probability of success, $P_0\left(s\right)$, as a function of the normalized time $s$ for $n=10$ qubits for several dimensionless running rates ${\tau }_{\text{LA}}$, under a local adiabaticity strategy.

Figure 3

QPT estimator ${\mathrm{\Delta }}_E^{\left(n\right)}\left(s\right)$ for $n=8$ under local adiabatic evolution for several dimensionless running rates ${\tau }_{\text{LA}}$.

Figure 4

Estimator for $n\in \left[8,64\right]$ and dimensionless time ${\tau }_{\text{LA}}=0.1$ in the local adiabatic regime as a function of the normalized time $s$. The final plateau height at $s=1$ can be fit as $ln\left[{\mathrm{\Delta }}_E^{\left(n\right)}(s=1)\right]=-0.18n-3.5-6.7/n$. The precursor $s_m$ of the critical point $s_c$ exponentially converges as $ln[s_m-s_c]=-0.35n-0.95$.

Figure 5

Schmidt gap ${\mathrm{\Delta }}_G^{\left(n\right)}\left(s\right)$ for $n\in \left[8,64\right]$ and dimensionless time ${\tau }_{\text{LA}}=0.1$ in the local adiabatic regime as a function of the normalized time $s$. The precursor $s_m$ of the critical point $s_c$ exponentially converges as $ln[s_m-s_c]=-0.34n-1.96$.

Figure 6

Kink density $d_k\left(s\right)$ as a function of the normalized time $s$ for $n=8$ and $n=64$ qubits, where fast and slow speed regimes in terms of the dimensionless time ${\tau }_{\text{LA}}$ are considered.

Figure 7

Kink density $d_k$ for $s=1$ as a function of the dimensionless speed $1/{\tau }_{\text{LA}}$ for $n=64$ qubits. The plot can be fit by the curve $d_k=1/2{(1-exp[-a/{\tau }_{\text{LA}}])}^{b{\tau }_{\text{LA}}}$, with $a=0.73$ and $b=0.37$. In the inset, we see a log plot and its best linear fit, which shows that a power law cannot describe the kink density behavior.