Can Constrained Quantum Annealing Be Effective in Noisy Quantum Annealers?

We investigate the performance of penalty-based quantum annealing (PQA) and constrained quantum annealing (CQA) in solving the graph partitioning problem under various noise models, including depolarizing, bit-flip, and phase-flip noise. We found that even in the absence of noise, the relative superiority of PQA or CQA is highly problem-dependent. PQA generally demonstrates relatively stable success probabilities, while CQA's performance varies significantly across problem instances. Notably, CQA experiences leakage from the constraint-satisfying subspace under most noise models, with the exception of phase-flip noise, where no leakage occurs exactly. The dependence of success probability on noise strength reveals that PQA consistently outperforms CQA in the weak noise regime, whereas CQA achieves higher success probabilities under strong phase-flip noise. However, even in this case, the overall success probability remains low, suggesting that CQA is not yet practical for noisy quantum annealers without the development of error-corrected quantum annealing and autonomous error correction techniques to address leakage from the constraint-satisfying subspace. To confine the system within the constraint subspace in CQA, error correction methods and cat code encoding will likely be required, both of which can suppress bit-flip errors.