The challenge of efficiently solving complex optimisation problems drives ongoing research into quantum algorithms, and a recent study by Kristina Bell, Adam Lowe, and Emily Coles, all from QinetiQ, alongside colleagues, directly compares different approaches within the Quantum Approximate Optimisation Algorithm (QAOA). Most QAOA investigations rely on simplified, quadratic representations of problems, but these can demand a large number of qubits as complexity increases. This research addresses a critical gap by directly comparing quadratic and higher-order formulations applied to the same routing problem, revealing that the higher-order approach delivers improved solution quality and better scalability. Importantly, the team also developed a factoring method that significantly reduces the number of quantum gates needed to implement the higher-order version, making it more practical for deployment on current and near-term quantum computers.
The study systematically evaluates the performance of each algorithm, considering solution quality and computational complexity. Data was gathered by running each algorithm multiple times on each test problem, allowing for a robust assessment of consistency and reliability. The analysis focuses on key metrics including circuit depth, the number of two-qubit gates required, and the value of the objective function achieved.
The team meticulously compared the solutions found by each algorithm to the known optimal solutions for each test problem, providing a clear measure of their effectiveness. Algorithms that consistently approach the optimal solution with minimal computational resources are considered superior. The study also examines the consistency of each algorithm by analysing the variation in results across multiple runs, revealing how sensitive they are to initial conditions or random noise. By comparing the performance of QUBO, HUBO, and factored HUBO, researchers identify strengths and weaknesses of each approach, and determine which algorithms are best suited for specific types of optimisation problems. Further analysis, including calculating average performance and standard deviation, provides a more detailed understanding of algorithm behaviour. The ARP serves as a valuable benchmark for evaluating optimisation algorithms. Researchers directly compared a standard Quadratic Unconstrained Binary Optimisation (QUBO) formulation with a Higher-Order Unconstrained Binary Optimisation (HUBO) approach, using the same underlying ARP to ensure a fair comparison. The team engineered a specific problem instance to assess the scaling of both formulations in terms of qubit count and computational complexity.
Scientists implemented both QUBO and HUBO formulations and ran them on small example problems, meticulously measuring solution quality and analysing the trade-offs between qubit scaling, circuit depth, and the number of two-qubit gates. Experiments were conducted on IBM quantum hardware to evaluate the practical implications of each approach, providing insights into the feasibility of implementing these algorithms on near-term devices. The study pioneered a factoring method designed to reduce the circuit depth of the HUBO version, mitigating the increased gate count often associated with higher-order representations. This factoring method achieved a significant reduction in the number of two-qubit gates, particularly when the HUBO formulation was run on real IBM hardware. The team then compared the performance of the factored HUBO version with the standard QUBO implementation, evaluating which approach offered a more appropriate balance between solution quality and computational resources under current hardware constraints. Experiments demonstrate that the higher-order formulation yields improved solution quality and better scaling in terms of qubit requirements, although it initially demands more two-qubit gates. To mitigate the increased gate count, the team implemented a factoring method to reduce circuit complexity, achieving a substantial reduction in the number of two-qubit gates when the algorithm was run on actual quantum hardware. This is particularly important because two-qubit gates are a major source of noise in quantum systems, directly impacting the reliability of calculations.
The study carefully balanced the trade-off between qubit scaling and two-qubit gate scaling, demonstrating that the higher-order formulation, despite its initial complexity, can deliver a higher success probability and improved solution quality on IBM quantum hardware. Specifically, the team found that while the higher-order formulation requires more gates than the standard quadratic approach, the improvement in qubit scaling is sufficient to overcome this disadvantage. The factoring method proved highly effective in reducing the two-qubit gate count, further enhancing the practicality of the higher-order formulation for implementation on current gate-based quantum computers. This work highlights the potential of higher-order formulations to provide a more efficient approach to solving certain problems in the near term. The team investigated both quadratic and higher-order representations, addressing a noted gap in previous work which largely focused on quadratic forms. Results demonstrate that utilising a higher-order representation yields improved solution quality and better scalability in terms of the number of qubits required, despite necessitating a greater number of two-qubit gates in the quantum circuit. To mitigate the increased gate count, the researchers developed a factoring method specifically tailored to the higher-order formulation.
This method achieves a significant reduction in the number of two-qubit gates when the algorithm is implemented on actual quantum hardware, representing a practical step towards more efficient quantum computation. The findings suggest that, while more complex, higher-order representations offer a promising pathway for tackling optimisation problems with improved performance on near-term quantum devices. Future work will focus on further refining the factoring method and exploring its application to a wider range of problem instances. This continued development aims to fully realise the potential benefits of higher-order representations within the QAOA framework and advance the field of quantum optimisation.