Digital Quantum Simulation of the Holstein Model in Trapped Ions

We propose the implementation of the Holstein model by means of digital methods in a linear chain of trapped ions. We show how the simulation fidelity scales with the generation of phononic excitations. We propose a decomposition and a stepwise trapped-ion implementation of the Holstein Hamiltonian. Via numerical simulations, we study how the protocol is affected by realistic gates. Finally, we show how measurements of the size of the simulated polaron can be performed.

Figure 1

(a) Behavior of the fidelity loss $1-F(t)=1-|⟨{\Psi }_E(t)|{\Psi }_S(t)⟩{|}^2$, for a two site configuration, as a function of the electron-phonon coupling strength $g$, for ${\omega }_0=h/4$. As the coupling $g$ increases, more phonons are created, the Hilbert space describing the dynamics enlarges, and the fidelity decreases for a fixed number of approximant gates ($r=10$ here). (b) Dependence of the fidelity loss in the number of sites. Here $g=0.3h$, ${\omega }_0=0.5h$, and ten symmetric steps are considered ($r=10$). The initial state of both plots corresponds to a configuration in which an electron is injected in the site $N/2$ ($N$ even) or $(N+1)/2$ ($N$ odd), and there are no phonons.

Figure 2

Dynamics for the $3+1$ ions configuration of the NN $XX$ Hamiltonian. Dotted curves stand for $⟨{\sigma }_z^i{⟩}_E$ for the exact dynamics, and solid curves stand for $⟨{\sigma }_z^i{⟩}_I$ for realistic ion interactions ($i=1$, 2, 3 for the first, second, and third ion). The parameters are chosen in order to have maxima in the fidelity $F(t)=|⟨{\Psi }_E(t)|{\Psi }_I(t)⟩{|}^2$ of $\sim 0.995$ (top black curve) at time steps of $\sim 333$ ${\nu }_{1}t$. These time steps can be chosen as Trotter steps.

Figure 3

Fidelity loss for $3+1$ ion configuration, involving Trotter simulation with perfect gates and realistic ion interactions, for two and three symmetric Trotter steps.