Exploring critical states of the quantum Rabi model via Hamiltonian variational ansätze

Abstract

Characterizing quantum critical states towards the thermodynamic limit is essential for understanding phases of matter. The power of quantum simulators for preparing the critical states relies crucially on the structure of quantum circuits and in return provides new insight into the critical states. Here, we explore the critical states of the quantum Rabi model~(QRM) by preparing them variationally with Hamiltonian variational ansätze~(HVA), in which the intricated interplay among different quantum fluctuations can be parameterized at different levels. We find that the required circuit depth scales linearly with the effective system size, suggesting that HVA can efficiently capture the behavior of critical states of QRM towards the thermodynamic limit. Moreover, we reveal that HVA gradually squeeze the initial state to the target critical state, with a number of blocks increasing only linearly with the effective system size. Our work suggests variational quantum algorithm as a new probe for the complicated critical states.

Figures (7)

• Figure 1: Illustration of VQE with hybrid-variables. The black and red lines represent the qubit and the continuous-variable, respectively. The HVA consists of blocks of parameterized evolutions of $H_1$, $H_2$,$H_3$, a decomposition of the Hamiltonian of the QRM.
• Figure 2: Relationship between the logarithmic infidelity $\log(1-F)$ and the number of circuit layers $p$, under various $\Omega$. The parameters are set at the critical point $g=1$, with $\omega_0=0.1$, $\Omega=2^{i}$ ($i=2,3,4\cdots$) , and $\lambda=\frac{g\sqrt{\omega_0\Omega}}{2}$.
• Figure 3: The linear scaling relation between the required circuit depth $p^*$ for accurately preparing the critical state and the effective system size $\log_2(\Omega)$.
• Figure 4: Wigner probability distribution. The process of state generation with successive blocks ($j=1,2,5,7,10$) along the quantum circuit. The horizontal and vertical axes represent the position $Q$ and momentum $P$, respectively. The initially symmetric, disk-like Wigner distribution is progressively squeezed and rotated toward the target state as the circuit depth increases. Here, the parameters are $\Omega=64$, $\omega_0=0.1$ and $\lambda=1.26$. The total number of blocks is $10$.
• Figure 5: Fock number distribution for quantum states generated with successive blocks in the quantum circuit. The parameters are $\Omega=64$, $\omega_0=0.1$ and $\lambda=1.26$. The total number of blocks is $10$.