Quantum-classical Algorithms Advance Combinatorial Optimization with Fair Ground-State Sampling

Combinatorial optimisation problems, which underpin many areas of science and engineering, often possess numerous equally valid solutions, creating a challenge for algorithms seeking to find them all without bias. Yuichiro Nakano and Keisuke Fujii, from The University of Osaka and RIKEN, investigate how hybrid quantum-classical Markov chain Monte Carlo (MCMC) algorithms can overcome this challenge, achieving unbiased sampling even when dealing with complex systems. Their work demonstrates that by combining the power of quantum dynamics with classical acceptance steps, these algorithms correct for inherent biases in quantum approaches and restore near-uniform sampling across all valid solutions. Applying this method to both simple Ising models and more complex random problems, the researchers show it achieves fairness comparable to, and in some cases exceeding, established classical techniques, while also offering efficient solution counting, a significant step towards developing truly versatile optimisation tools.

This presents a significant hurdle for algorithms seeking to explore all options without introducing bias. Their work demonstrates that by combining the power of quantum dynamics with classical acceptance steps, these algorithms correct for inherent biases in quantum approaches and restore near-uniform sampling across all valid solutions. Applying this method to both simple Ising models and more complex random problems, the researchers show it achieves fairness comparable to, and in some cases exceeding, established classical techniques, while also offering efficient solution counting, a significant step towards developing truly versatile optimisation tools.,.

Hybrid Quantum-Classical Markov Chain Monte Carlo Sampling

Scientists have developed a novel hybrid quantum-classical Markov chain Monte Carlo (MCMC) methodology to address biased sampling in quantum optimisation algorithms, particularly for combinatorial problems with multiple optimal solutions. This work pioneers a technique where quantum dynamics functions solely as a proposal step within a classical MCMC framework, enforcing detailed balance through classical acceptance criteria. This approach aims to correct the inherent sampling bias present in quantum heuristics like quantum annealing and the quantum approximate optimisation algorithm (QAOA). Researchers validated the method using small Ising models, demonstrating its ability to restore near-uniform sampling across degenerate ground states, effectively mitigating the bias stemming from quantum dynamics.

The team then applied this hybrid MCMC to random k-SAT problems near the satisfiability threshold, assessing its performance on more complex instances. For random 2-SAT problems, the methodology combines QAOA-assisted neural proposals with single spin-flip updates, achieving fairness comparable to that of the PT-ICM algorithm, a well-established classical method. Significantly, for random 3-SAT problems, where classical techniques often falter, the hybrid MCMC still attains approximately uniform sampling, demonstrating its robustness in challenging scenarios. To evaluate the efficiency of solution counting, scientists measured the number of transitions required, finding it comparable to that of the WalkSAT algorithm, a leading classical approach for SAT solving. This innovative methodology provides a viable framework for both fair sampling and efficient solution enumeration, offering a significant advancement in the field of quantum-enhanced optimisation.,.

MCMC Corrects Bias in Quantum Sampling

Scientists have demonstrated a novel hybrid quantum-classical Markov chain Monte Carlo (MCMC) method for fair sampling in combinatorial optimisation problems, particularly those with multiple optimal solutions. This work addresses a key challenge in quantum optimisation, where standard quantum heuristics often induce biased sampling, favouring certain solutions over others. The team investigated whether incorporating classical MCMC steps could mitigate this bias and restore near-uniform sampling, even with a simple transverse-field driver. Experiments with small-scale Ising models confirmed that MCMC post-processing effectively corrects the sampling bias inherent in the initial quantum dynamics, achieving near-uniform sampling across all ground states.

Extending this approach to more complex problems, researchers applied the hybrid MCMC method to random 2-SAT instances, finding that combining quantum-assisted neural proposals with single spin-flip updates achieves fairness comparable to that of the established PT-ICM algorithm. Significantly, the method maintains approximately uniform sampling even for random 3-SAT problems, where classical algorithms like PT-ICM are no longer effective. Data shows the hybrid MCMC method requires a number of transitions comparable to WalkSAT for solution counting, demonstrating its efficiency in enumerating all optimal solutions. These results demonstrate the hybrid quantum-classical MCMC framework provides a viable pathway to fair sampling and efficient solution enumeration, offering a practical algorithm for near-term quantum devices without requiring complex hardware or circuit designs.,.

Hybrid MCMC Corrects Biases, Enables Fair Sampling

This research demonstrates that hybrid quantum-classical Markov chain Monte Carlo (MCMC) algorithms can achieve fair sampling even when applied to complex combinatorial optimisation problems with multiple ground states. The team successfully showed that incorporating quantum dynamics as a proposal step, combined with classical acceptance criteria, corrects for biases inherent in purely quantum approaches. This correction restores near-uniform sampling across all ground states, a crucial feature for accurate solutions and reliable enumeration of possibilities. The study applied this hybrid MCMC method to both small Ising models and larger, random 3-SAT problems, demonstrating its effectiveness in scenarios where traditional classical methods struggle.

Results indicate that the hybrid approach achieves fairness comparable to established classical algorithms for certain problem types, and still maintains approximate uniformity even when those classical methods become impractical. Furthermore, the number of computational steps required for solution counting is comparable to that of efficient classical algorithms like WalkSAT, suggesting a viable path towards scalable solution enumeration. The authors acknowledge that further research is needed to assess the method’s performance across a wider range of optimisation challenges and explore its scalability. Despite these limitations, this research represents a significant step forward in developing fair and efficient sampling methods for combinatorial optimisation, bridging the gap between quantum and classical computation.