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.

• The paper introduces ADGLB, a spectral-gap-guided detuning schedule that slows evolution near the minimum gap without adding Hamiltonian terms, variational optimization, or hardware resources.
• Numerical simulations increased final ground-state population for a 10-atom chain from 0.739 to as high as 0.981, while Aquila experiments improved MIS probabilities by up to 87% on tested larger instances.
• The paper shows that schedules learned on small instances can transfer to larger lattices, but harder graphs require heuristic detuning offsets and raise open questions about scalability, generality, and robustness to noise.

Motivation and context

Adiabatic quantum computation (AQC) on neutral-atom Rydberg arrays encodes the maximum independent set (MIS) of a unit-disk graph as the ground state of a blockade-constrained Rydberg Hamiltonian. The central performance bottleneck is well known: the minimum spectral gap $g_{\rm min}$ decreases with system size—superexponentially according to adiabatic-theorem bounds—and depends on instance hardness via the hardness parameter $${\rm HP} = R_{|\text{MIS}|-1}/(|\text{MIS}| R_{|\text{MIS}|})$$. For fixed total evolution time $T$, this gap closure produces nonadiabatic population leakage from the ground state, degrading MIS preparation fidelity. Existing remedies either add counterdiabatic Hamiltonian terms or rely on variational/Bayesian schedule optimization requiring repeated hardware executions (2602.17991).

The paper introduces ADGLB (Adjusted Detuning for Ground-Energy Leakage Blockade), a schedule-engineering method that reshapes only the laser detuning profile $\delta(t)$ using spectral-gap information obtained from exact diagonalization of small instances. It requires no additional Hamiltonian terms, no iterative quantum-classical loops, and no extra hardware resources.

Method

The authors work within an effective two-level subspace $\{\ket{E_0}, \ket{E_1}\}$, deriving from the Jansen–Ruskai–Seiler adiabatic theorem the effective Hamiltonian

$$H_{\rm ad} = \frac{\bra{E_1}\frac{dH_{\rm Ry}}{dt}\ket{E_0}}{\Delta E_{01}} s_y - \frac{\Delta E_{01}}{2} s_z ,$$

which makes explicit that nonadiabatic leakage scales with the ratio of Hamiltonian slew rate to instantaneous gap. Since the gap $\Delta E_{01}(t)$ is proportional to the fixed Rabi frequency $\Omega_0$ during the linear sweep, ADGLB keeps $\Omega(t)$ unchanged and instead sets $d\delta/dt \propto (\Delta E_{01})^j$: the detuning sweep slows near $${\rm HP} = R_{|\text{MIS}|-1}/(|\text{MIS}| R_{|\text{MIS}|})$$0 where the gap is minimal and accelerates elsewhere. Concretely, the sweep interval is split at $${\rm HP} = R_{|\text{MIS}|-1}/(|\text{MIS}| R_{|\text{MIS}|})$$1 and interpolated through a normalized cumulative integral of $${\rm HP} = R_{|\text{MIS}|-1}/(|\text{MIS}| R_{|\text{MIS}|})$$2 with power exponent $${\rm HP} = R_{|\text{MIS}|-1}/(|\text{MIS}| R_{|\text{MIS}|})$$3, which controls how aggressively the schedule concentrates time near $${\rm HP} = R_{|\text{MIS}|-1}/(|\text{MIS}| R_{|\text{MIS}|})$$4.

For the quasi-one-dimensional $${\rm HP} = R_{|\text{MIS}|-1}/(|\text{MIS}| R_{|\text{MIS}|})$$5-PXP chain $${\rm HP} = R_{|\text{MIS}|-1}/(|\text{MIS}| R_{|\text{MIS}|})$$6 ($${\rm HP} = R_{|\text{MIS}|-1}/(|\text{MIS}| R_{|\text{MIS}|})$$7, $${\rm HP} = R_{|\text{MIS}|-1}/(|\text{MIS}| R_{|\text{MIS}|})$$8), numerical Schrödinger evolution gives final ground-state populations of 0.955, 0.981, 0.963, and 0.940 for $${\rm HP} = R_{|\text{MIS}|-1}/(|\text{MIS}| R_{|\text{MIS}|})$$9, versus 0.739 under the standard schedule. An intermediate $T$0 performs best numerically, balancing slowdown near $T$1 against distortion of the rest of the evolution.

Experimental results on Aquila

Experiments were performed on QuEra's Aquila processor ($T$2Rb, $T$3 Rydberg state, $T$4 MHz, $T$5s), using $T$6. Key results:

Instance	$T$7	HP	$T$8 (standard)	$T$9 (ADGLB)
$\delta(t)$0	10	6.5	28%	38%
$\delta(t)$1	25	6.16	17.7%	24.2%
$\delta(t)$2	37	5.38	3.0%	5.6%
$\delta(t)$3	23	34.2	1.45%	2.54% (with offset)

Notably, the schedule optimized on the $\delta(t)$4 chain was transferred without modification to two larger triangular lattices built by stacking the same building blocks, yielding relative improvements of roughly 37% and 87% respectively despite the fact that direct diagonalization of their spectra is computationally intractable. This transferability indicates the ADGLB schedule captures structural features of instance hardness rather than system size per se.

Extension to higher-hardness instances

The method's main limitation is its dependence on diagonalization: for harder instances such as $\delta(t)$5 (two $\delta(t)$6 blocks joined by a linear chain, $\delta(t)$7), neither $\delta(t)$8 nor $\delta(t)$9 can be computed directly. The authors exploit an empirical trend observed in solvable chains ($\{\ket{E_0}, \ket{E_1}\}$0): as size and HP grow, $\{\ket{E_0}, \ket{E_1}\}$1 shrinks while $\{\ket{E_0}, \ket{E_1}\}$2 shifts upward. They therefore apply a heuristic detuning offset $\{\ket{E_0}, \ket{E_1}\}$3 at $\{\ket{E_0}, \ket{E_1}\}$4 to the transferred ADGLB schedule. At $\{\ket{E_0}, \ket{E_1}\}$5 the transferred schedule actually underperforms the standard schedule ($\{\ket{E_0}, \ket{E_1}\}$6 vs. $\{\ket{E_0}, \ket{E_1}\}$7)—a candid negative result showing naive schedule transfer fails when the gap structure shifts—but a scan over offsets recovers and exceeds baseline performance, peaking at $\{\ket{E_0}, \ket{E_1}\}$8 for $\{\ket{E_0}, \ket{E_1}\}$9 MHz.

Limitations and open questions

Several caveats are explicit in the paper. First, the primary ADGLB construction requires full knowledge of $$H_{\rm ad} = \frac{\bra{E_1}\frac{dH_{\rm Ry}}{dt}\ket{E_0}}{\Delta E_{01}} s_y - \frac{\Delta E_{01}}{2} s_z ,$$0, restricting it to instances small enough for exact diagonalization; the offset heuristic for larger systems is empirical and unguided beyond a monotonicity assumption about $$H_{\rm ad} = \frac{\bra{E_1}\frac{dH_{\rm Ry}}{dt}\ket{E_0}}{\Delta E_{01}} s_y - \frac{\Delta E_{01}}{2} s_z ,$$1. Second, hardware imperfections—SPAM errors of 10% and 5%, a $$H_{\rm ad} = \frac{\bra{E_1}\frac{dH_{\rm Ry}}{dt}\ket{E_0}}{\Delta E_{01}} s_y - \frac{\Delta E_{01}}{2} s_z ,$$2 kHz detuning error, and an 8 $$H_{\rm ad} = \frac{\bra{E_1}\frac{dH_{\rm Ry}}{dt}\ket{E_0}}{\Delta E_{01}} s_y - \frac{\Delta E_{01}}{2} s_z ,$$3s coherence time limited by laser phase noise—compress experimental gains relative to numerics and prevent resolution of differences among $$H_{\rm ad} = \frac{\bra{E_1}\frac{dH_{\rm Ry}}{dt}\ket{E_0}}{\Delta E_{01}} s_y - \frac{\Delta E_{01}}{2} s_z ,$$4 values, so the optimal $$H_{\rm ad} = \frac{\bra{E_1}\frac{dH_{\rm Ry}}{dt}\ket{E_0}}{\Delta E_{01}} s_y - \frac{\Delta E_{01}}{2} s_z ,$$5 is determined only classically. Third, the demonstration is confined to quasi-1D chains and triangular lattices built from a single motif; generalization to arbitrary unit-disk graphs with heterogeneous local structure remains untested. Finally, whether the improvement persists at sizes where leakage competes with decoherence over the full schedule, and whether the offset heuristic can be replaced by a principled estimator of $$H_{\rm ad} = \frac{\bra{E_1}\frac{dH_{\rm Ry}}{dt}\ket{E_0}}{\Delta E_{01}} s_y - \frac{\Delta E_{01}}{2} s_z ,$$6 for large instances, are questions the paper leaves open.

Conclusion

This work demonstrates that spectral-gap-guided detuning engineering is a practical, hardware-efficient lever for improving Rydberg AQC on near-term neutral-atom processors: it roughly doubles MIS preparation probability in some cases without added Hamiltonian complexity, additional shots, or longer anneals, and it transfers across instance sizes at fixed hardness. Its dependence on classical spectral information for the base schedule, and the need for heuristic correction on hard instances, delineate precisely where future methodological work must focus.