Efficient Hamiltonian Engineering for Adiabatic MIS Algorithms

Abstract: We present a hybrid adiabatic algorithm for maximum independent set (MIS) using Rydberg atom arrays. We engineer local controls that preferentially excite atoms with few neighbors, which represent graph nodes with small degrees. Numerical simulations show that the designed pulses accelerate convergence to the MIS state and suppress population in trap states. We obtain higher success probabilities than traditional global controls and a $25\%$ reduction in fidelity decay rate as problem hardness increases.

• The paper presents a degree-dependent local detuning method that accelerates convergence and suppresses trap-state populations in adiabatic MIS algorithms.
• It employs efficient O(N²) classical preprocessing to sort vertex degrees and compute a critical detuning parameter, ensuring the MIS remains the ground state.
• Numerical results demonstrate a 25% reduction in fidelity decay rate and sub-linear scaling improvements, highlighting the method's practical scalability on Rydberg atom arrays.

Efficient Hamiltonian Engineering for Adiabatic MIS Algorithms: Technical Summary

Context and Motivation

The Maximum Independent Set (MIS) problem—a canonical NP-hard graph optimization task—is central to both theoretical computer science and near-term quantum computational proposals. Recent advances in programmable Rydberg atom arrays have enabled direct mapping of MIS to many-body quantum states, facilitating experimental and algorithmic exploration. However, adiabatic quantum computation (AQC) for MIS suffers from rapidly shrinking spectral gaps as system size increases, limiting scalability and algorithmic efficiency. The paper "Efficient Hamiltonian Engineering for Adiabatic MIS Algorithms" (2605.16944) presents a methodical augmentation to standard AQC approaches, leveraging vertex-level graph information to engineer degree-dependent local controls, thereby accelerating convergence and suppressing trap-state populations.

Algorithmic Innovations

Local-Degree Adiabatic Quantum Computation (LD-AQC)

The central innovation is the transition from global homogeneous detuning to degree-dependent local detuning profiles for atom array control. The LD-AQC protocol identifies vertices with lower degree as more likely to be part of the MIS, substantiated by analytic and empirical statistics over graph ensembles. The algorithm preprocesses the input graph to produce a sorted degree table and computes a critical detuning parameter $a^*$ using efficient root-finding, ensuring energetic ordering such that the MIS remains the ground state while energetically penalizing non-connected independent sets (trap states).

Hamiltonian Structure

The AQC Hamiltonian:

$$H = \sum_i \Omega(t) (\sigma^x_i) - \sum_i \Delta_i(t) n_i + \sum_{(i,j) \in E} V_{ij} n_i n_j$$

is modified by letting $\Delta_i(t)$ scale with vertex degree, favoring excitation of less-connected atoms. This demands single-qubit addressability, achievable with current atom array hardware.

Energetic Gap Engineering

LD-AQC lifts degeneracies in the $\ket{\text{IS}}$ band, especially between connected and disconnected states in the $|\text{MIS}|-1$ excitation manifold. The central algorithm (Alg. 0) finds $a^*$ such that the gap between manifolds with $k$ and $k-1$ excitations is strictly positive, ensuring monotonic energy ordering and preventing manifolds from overlapping.

Complexity and Classical Preprocessing

The classical runtime of the preprocessing and detuning profile computation is bounded by $O(N^2)$, dominated by adjacency list construction and degree sorting. This polynomial overhead is minimal relative to quantum hardware runtime.

Numerically Verified Performance

Scaling and Success Probability

Benchmarking on ensembles of random graphs (11-12 vertices), LD-AQC consistently achieves higher MIS ground-state fidelity than traditional AQC. Numerical results demonstrate:

• 25% reduction in fidelity decay rate across increasing problem hardness.
• Sub-linear speedup: The success probability metric (SPM, defined as $-\log(1-P_{MIS})$) for LD-AQC scales as $$H = \sum_i \Omega(t) (\sigma^x_i) - \sum_i \Delta_i(t) n_i + \sum_{(i,j) \in E} V_{ij} n_i n_j$$0, yielding an explicit scaling relation:

$$H = \sum_i \Omega(t) (\sigma^x_i) - \sum_i \Delta_i(t) n_i + \sum_{(i,j) \in E} V_{ij} n_i n_j$$1

• The average error rate is reduced by nearly a factor of three compared to global AQC.
• The generalized hardness parameter $$H = \sum_i \Omega(t) (\sigma^x_i) - \sum_i \Delta_i(t) n_i + \sum_{(i,j) \in E} V_{ij} n_i n_j$$2 correlates strongly (mutual information: 1.23) with minimal spectral gap, outstripping alternative hardness metrics.

Robustness Across Control Profiles

Comparative analysis of linear, exponential, and power-law detuning profiles finds the LD-AQC framework robust to the specific form, with power-law profiles providing optimal average performance.

Mechanistic Insights

The improvement is attributed to energetic penalization of disconnected IS states, i.e., trap configurations. By raising the energy of these states, LD-AQC suppresses their occupation during adiabatic evolution. Detailed statistical analyses verify energetic ordering, fidelity distributions, and reduction in effective hardness, substantiating the mechanistic claims.

Implications and Extensions

Practical Implications

For quantum hardware utilizing Rydberg atom arrays or similarly addressable qubit platforms, LD-AQC offers a scalable, hardware-compatible approach to MIS and other combinatorial optimization tasks. Efficient classical preprocessing is feasible for moderate system sizes, and the approach leverages intrinsic graph structure for algorithmic acceleration.

Theoretical Implications

The method provides a template for leveraging vertex-level heuristics in quantum optimization, generalizing to weighted MIS, MAX-cut, and potentially broader classes of NP-hard problems. The rigorous definition and validation of the generalized hardness parameter inform algorithmic complexity assessment and shed light on spectral gap engineering—a key theoretical bottleneck in quantum optimization.

Future Directions

• Quantum optimal control can further refine detuning profiles, exploiting more complex heuristics and schedules.
• Extension to larger graphs could utilize tensor network simulation techniques or next-generation quantum hardware.
• Application to other platforms and problems—such as electric charging optimization or integer factorization—is straightforward, given the method’s generality.

Conclusion

Efficient Hamiltonian engineering through degree-dependent local detuning profiles markedly improves adiabatic quantum computation for MIS, both in fidelity and scaling. Theoretical and empirical analyses establish the approach’s robustness, efficiency, and mechanistic validity. The results lay a foundation for more sophisticated quantum optimization protocols, underscoring the utility of integrating classical graph analytics into quantum algorithm design.