Quantum-assisted Recursive Algorithm Improves Exact Cover Problem Solutions at Shallow Circuit Depth

The exact cover problem, a computationally challenging task with widespread applications, frequently overwhelms even advanced algorithms. Xiao-Hui Ni, Jia-Cheng Fan from Beijing University of Posts and Telecommunications, Ling-Xiao Li, and colleagues demonstrate a significant advance with their Quantum-Assisted Recursive Algorithm (QARA). This new approach cleverly combines the strengths of both classical and quantum computing, alternating between classical simplification and quantum pruning to tackle complex instances. The team’s method extracts key information from quantum states to guide the reduction of the problem, and importantly, achieves a roughly 60% improvement in finding exact solutions compared to existing algorithms, offering a promising pathway towards solving previously intractable problems on current and near-term quantum hardware.

QARA employs a recursive strategy, breaking down large problems into smaller, more manageable subproblems, and then progressively refining the solution. At the heart of QARA lies the Quantum Approximate Optimisation Algorithm, or QAOA, which is used to solve these individual subproblems. The algorithm’s performance is further enhanced by an adaptive mixer allocation strategy, dynamically adjusting the quantum circuit parameters to optimise the solution process.

This hybrid quantum-classical approach leverages the strengths of both computational paradigms, with the quantum computer tackling the subproblem optimisation and the classical computer managing the overall recursive process and combining the results. QARA is designed to address NP-hard problems, those that become computationally intractable as their size increases. The algorithm has demonstrated potential in solving problems such as the Maximum Independent Set, the Maximum Coverage Problem, the Tail Assignment Problem, and optimising Wireless Sensor Network Coverage. The algorithm employs a hybrid approach, alternating between classical and quantum pruning techniques to simplify the problem and improve solution quality, even with limited quantum resources. The process begins with classical pruning, a repeatable step that reduces the problem’s complexity, before invoking quantum pruning when classical methods reach their limit. During quantum pruning, the team extracts information from the quantum output state to identify the subset exhibiting the strongest selection bias, then uses this information to further reduce the problem using tailored reduction rules.

To ensure the effectiveness of quantum simplification, the study incorporates a local verification and rollback mechanism, allowing the algorithm to assess and correct potentially detrimental pruning steps. Following quantum pruning, classical pruning is reapplied to the reduced problem, and this alternating process repeats until a complete solution is found. Recognising that deeper quantum circuits require more resources, scientists focused on achieving better results with shallow circuit depths. The core of QARA lies in a novel hybrid approach that alternates between classical and quantum pruning techniques to recursively simplify the problem. Classical pruning, a pre-processing step, reduces the problem size by leveraging completeness and uniqueness constraints, lessening the workload for the subsequent quantum stage.

When classical pruning reaches its limit, QARA invokes quantum pruning, a process that extracts information from the output state of the Quantum Approximate Optimisation Algorithm, or QAOA, to identify the subset exhibiting the strongest selection bias. This identified subset then guides the pruning process using problem-tailored reduction rules, designed to prevent over-coverage of elements and enable the simultaneous reduction of multiple variables, a significant improvement over existing methods. Experiments conducted on 140 instances, with subset sizes ranging from 8 to 20, demonstrate that QARA achieves a probability of finding an exact solution approximately 60% higher than both QAOA and Recursive QAOA. Furthermore, the team incorporated a local verification and rollback mechanism to assess the effectiveness of each quantum simplification step, ensuring reliability and preventing erroneous reductions. QARA combines classical and quantum techniques in a recursive process, alternating between classical pruning to simplify the problem and quantum pruning to further reduce its complexity. The algorithm strategically uses classical methods to handle a portion of the computational workload, thereby minimising the demands on limited quantum resources. The key innovation of QARA lies in its ability to extract information from the quantum state to identify the most promising subsets for pruning, guided by problem-specific reduction rules.

A local verification and rollback mechanism further enhances the algorithm’s robustness by correcting potential errors in the pruning process. Numerical simulations demonstrate that QARA achieves a significantly higher probability of finding exact solutions compared to existing methods, while also maintaining solution quality and efficiency. This work establishes QARA as a powerful and scalable hybrid algorithm suitable for tackling the exact cover problem on current noisy intermediate-scale quantum devices.