Enhanced Maximum Independent Set Preparation with Rydberg Atoms Guided by the Spectral Gap

Abstract

Adiabatic quantum computation with Rydberg atoms provides a natural route for solving combinatorial optimization problems such as the maximum independent set (MIS). However, its performance is fundamentally limited by the reduction of the spectral gap with increasing system size and connectivity, which induces population leakage from the ground state during finite-time evolution. Here we introduce the Adjusted Detuning for Ground-Energy Leakage Blockade (ADGLB), a spectral-gap-guided schedule engineering method that modifies the laser detuning profile to suppress leakage without introducing additional Hamiltonian terms or iterative optimization loops. We experimentally benchmark ADGLB on a quasi-one-dimensional chain of $N=10$ atoms, and the MIS preparation probability increases substantially compared with the standard adiabatic schedule. Furthermore, we show that the schedule optimized for smaller instances can be directly applied to larger two-dimensional triangular lattices with $N=25$ and $N=37$. With a small heuristic offset, the method also remains effective for instances with higher hardness parameters. These findings demonstrate that spectral-gap-guided schedule engineering offers a scalable and hardware-efficient strategy for enhancing adiabatic quantum optimization on neutral-atom platforms.

Figures (6)

• Figure 1: AQC for MIS preparation with Rydberg atoms. (a) An example of a MIS encoded in a $N=7$ Rydberg-atom array with nearest-neighbor spacing $a$, where atoms within the Rydberg blockade radius $R_b$ cannot be simultaneously excited. The atomic ground state $\ket{g}$ and the Rydberg state $\ket{r}$ are represented by gray and red circles, respectively. (b) The standard AQC schedule of the Rabi frequency $\Omega (t)$ and the detuning $\delta (t)$. (c) The ground-state energy $E_0$ (blue solid line), the first excited-state energy $E_1$ (blue dashed line), and the spectral gap $\Delta E_{01}\equiv E_1 - E_0$ (red solid line) as a function of the detuning $\delta (t)$ during the linear sweep $(t_r\leq t \leq T-t_r)$ of the standard schedule shown in (b). The region around the minimum spectral gap $g_{\rm min} \equiv \min_{t \in [0, T]} \Delta E_{01}(t)$ near $\delta_{\text{min}}$ is highlighted in light blue. (d) The time evolution of the ground-state population $P_{E_0}$ (blue solid line) and the population of all excited states $P_{E_{k>0}} \equiv 1-P_{E_0}$ (red dashed line). The population leakage from the ground state is denoted by $\varepsilon_{P_{E_0}}$.
• Figure 2: MIS-state overlaps $\left | \bra{\rm MIS}E_k\rangle \right |^2$ with the ground state (blue line) and the first excited state (red line) for the $N=7$ Rydberg atom array in Fig. \ref{['Figure1']}(a) along the standard AQC schedule.
• Figure 3: (a) The ADGLB schedule constructed using Eqs. \ref{['ADGLB_delta_design_eq_zeta_jth_order']} and \ref{['ADGLB_delta_design_eq']} with $j=1, 1.5, 1.8,$ and $2$, for a quasi-one-dimensional chain of $N=10$ Rydberg atoms, $Q_{1D,10}$ (${\rm HP}=6.5$). (b) Numerical results for the ground-state population dynamics $P_{E_0}(t)$ under the standard schedule (black dashed line) and the ADGLB schedules with $j=1$ (blue solid line), $1.5$ (red solid line), $1.8$ (yellow solid line), and $2$ (purple solid line).
• Figure 4: Experimental MIS preparation for the quasi-one-dimensional chain $Q_{1D,10}$. (a) Probability histogram of independent sets of size $|{\rm MIS}|$ (yellow bar) and $|{\rm MIS}|-1$ (blue bars) under the standard schedule (279 valid shots). (b) Probability histogram of independent sets of size $|{\rm MIS}|$ (yellow bar) and $|{\rm MIS}|-1$ (blue bars) under the ADGLB schedule with $j=1.8$ (277 valid shots).
• Figure 5: Experimental MIS preparation in two-dimensional triangular lattices. (a) A two-dimensional diamond-shaped lattice $T_{D,25}$. (b) Probability histograms of independent sets of size $|{\rm MIS}|$ and $|{\rm MIS}|-1$ for $T_{D,25}$ under the standard (237 valid shots) and ADGLB (252 valid shots) schedules. (c) A two-dimensional hexagonal-shaped lattice $T_{H,37}$. (d) Probability histograms of independent sets of size $|{\rm MIS}|$ and $|{\rm MIS}|-1$ for $T_{D,25}$ under the standard (707 valid shots) and ADGLB (678 valid shots) schedules. In (b) and (d), yellow and purple bars denote independent sets of size $|{\rm MIS}|$, while blue and red bars denote independent sets of size $|{\rm MIS}|-1$.