Hubbard physics with Rydberg atoms: Using a quantum spin simulator to simulate strong fermionic correlations

We propose a hybrid quantum-classical method to investigate the equilibrium physics and the dynamics of strongly correlated fermionic models with spin-based quantum processors. Our proposal avoids the usual pitfalls of fermion-to-spin mappings thanks to a slave-spin method which allows to approximate the original Hamiltonian into a sum of self-correlated free fermions and spin Hamiltonians. Taking as an example a Rydberg-based analog quantum processor to solve the interacting spin model, we avoid the challenges of variational algorithms or Trotterization methods. We explore the robustness of the method to experimental imperfections by applying it to the half-filled, single-orbital Hubbard model on the square lattice in and out of equilibrium. We show, through realistic numerical simulations of current Rydberg processors, that the method yields quantitatively viable results even in the presence of imperfections: it allows to gain insights into equilibrium Mott physics as well as the dynamics under interaction quenches. This method thus paves the way to the investigation of physical regimes, whether out of equilibrium, doped, or multiorbital, that are difficult to explore with classical processors.

Figure 1

Slave-spin mapping. The Hubbard Hamiltonian (top) is mapped onto two self-consistently determined simpler problems: an efficiently solvable free-fermionic Hamiltonian with a renormalized hopping [bottom left, $H_{\mathrm{f}}\left(Q\right)$ in the text], and a transverse-field Ising Hamiltonian [bottom right, $H_{\mathrm{s}}\left(J\right)$ in the text], which we solve using a Rydberg-based quantum processor.

Figure 2

Mott transition observed with the slave-spin method on a realistic numerical simulation of Rydberg atoms processor. The characteristics of the processor considered are ${\tau }_{\text{max}}=4\mu \mathrm{s}, \gamma =0.02\mathrm{MHz}, N_{\mathrm{s}}=150, \epsilon ={\epsilon }^{\prime }=3\%$, and $5\times 5$ loops allowed (see Supplemental Material, Sec. III C [45]). The error bar is the standard error stemming from $\epsilon$ and $N_{\mathrm{s}}$.

Figure 3

Dynamical response of the quasiparticle weight after an interaction quench. $N=12$ spin cluster. Time evolution of $Z$ for (a) $U_{\mathrm{f}}=2\mathrm{MHz}$ and (b) $U_{\mathrm{f}}=25\mathrm{MHz}$. The red line shows the noiseless annealing solution and the blue line a realistic numerical simulation on Rydberg atoms processor ($\gamma =0.02\mathrm{MHz}, \epsilon ={\epsilon }^{\prime }=3\%, N_{\mathrm{s}}=150$ shots, realistic Ising interactions and a global detuning are imposed). (c) Fourier transform amplitude $\left|Z\right(f,U_{\mathrm{f}}\left)\right|$. The vertical black line shows the equilibrium critical value $U_c$ as computed from Fig. 2. (d) Impact of the hopping terms $t$ on the damping of the response of $Z$ after a quench ($U_{\mathrm{f}}=13\approx U_c$). The blue, orange, and green lines represent the result for $t=0.125$, 0.25, and 0.5 MHz, respectively.