Overparameterization Statistically Necessary for Quantum Approximate Optimization Algorithm Performance

Quantum approximate optimization algorithms represent a promising avenue for tackling complex computational challenges, yet understanding the resources required for effective problem-solving remains a significant hurdle. Daniil Rabinovich, Andrey Kardashin, and Soumik Adhikary, all from the Skolkovo Institute of Science and Technology, investigate how the size of the quantum circuit, specifically, the degree of overparametrization, affects performance. Their work, inspired by techniques in classical machine learning, focuses on the Quantum Approximate Optimization Algorithm (QAOA) and its ability to solve combinatorial optimization problems like MAX-CUT and MAX-2-SAT. The team demonstrates that overparametrization is crucial for achieving exact solutions with MAX-CUT, while surprisingly, underparametrized circuits often perform adequately for MAX-2-SAT, suggesting QAOA holds potential even with the limitations of current quantum hardware.

The minimal quantum resources necessary to obtain a solution to complex problems remains an open question. This work investigates the impact of overparameterization on the performance of variational algorithms, drawing inspiration from concepts within classical machine learning. The study focuses specifically on the quantum approximate optimization algorithm (QAOA), a prominent variational quantum algorithm designed to solve combinatorial optimization problems. The core of the study centers on understanding how well QAOA performs on random graph instances of the MAX-CUT problem, and identifying factors that influence its success. The team employs a layer-by-layer approach to building the QAOA circuit, progressively adding layers and initializing new layer parameters based on previous ones. This approach can sometimes lead to the optimization getting stuck in local minima, where adding a layer increases the energy error instead of decreasing it.

To address this, the researchers implemented two post-processing techniques. First, they correct the energy error by ensuring it never increases as layers are added. Second, they enforce convergence by considering a run converged if it reaches the target accuracy at a certain depth, applying that result to all deeper layers. Experiments used random graphs with varying numbers of vertices and edge probabilities, with multiple optimization runs performed for each instance to account for the algorithm’s inherent randomness. The primary metrics used to evaluate performance were the energy error, representing the difference between the QAOA result and the optimal solution, and the success probability, measuring the fraction of runs that converge to the target accuracy.

Graphical rules were developed for tracking Pauli backpropagation, offering a tool for further analysis, and the team observed specific patterns in the optimal parameters for the QAOA circuit. These findings highlight the importance of careful experimental design, data analysis, and post-processing to obtain meaningful results. The research demonstrates that the layer-by-layer optimization strategy can be problematic, and that data post-processing techniques are essential for obtaining reliable results and ensuring accurate performance measurements. The success of QAOA depends on the specific graph instance, the number of layers used, and the optimization strategy employed, suggesting that further research is needed to understand the optimal parameters and strategies for QAOA.

Overparametrization Enables Exact MAX-CUT Solutions

Researchers have investigated the impact of circuit complexity on the performance of quantum algorithms, specifically QAOA. This work explores whether increasing the number of parameters in a quantum circuit, known as overparametrization, is necessary or sufficient to find effective solutions to challenging optimization problems. The study focuses on two representative problems, MAX-CUT and MAX-2-SAT, to understand how overparametrization affects QAOA’s ability to deliver accurate results. The findings reveal a striking difference between the two problems. For MAX-CUT, overparametrization is demonstrably both necessary and sufficient to achieve exact solutions, at least for systems with up to 20 qubits.

Specifically, the optimal circuit depth, the minimum complexity needed to reliably find a solution, precisely matches the depth required for overparametrization. This suggests that, for this problem, simply adding more parameters to the quantum circuit guarantees finding the best possible solution, and that any less complex circuit is unlikely to succeed. Researchers even analytically determined the optimal depth for MAX-CUT on a specific type of graph, a ring of disagrees, confirming this scaling with the number of qubits. In contrast, MAX-2-SAT exhibits a different behavior. The research demonstrates that solutions to this problem can be found with circuits significantly less complex than those required for overparametrization.

This indicates that, for MAX-2-SAT, overparametrization is not a necessity; effective solutions can be achieved with fewer parameters, potentially offering advantages for implementation on current quantum hardware. This difference highlights that the relationship between overparametrization and solution quality is problem-dependent, and that a “one-size-fits-all” approach to circuit design may not be optimal. These results have important implications for the development of quantum algorithms. While overparametrization can be a powerful tool for achieving high accuracy, it is not always necessary, and may introduce unnecessary complexity. Understanding the interplay between circuit complexity and problem structure is crucial for designing efficient quantum algorithms that can be implemented on near-term quantum devices, and for establishing performance guarantees for QAOA in practical scenarios. The findings suggest that tailoring the circuit complexity to the specific problem at hand can lead to significant improvements in performance and resource utilization.

Overparameterization Guarantees Optimal QAOA Performance

This work investigates the impact of overparameterization on the performance of QAOA when applied to two challenging combinatorial problems, MAX-CUT and MAX-2-SAT. The research demonstrates that, for MAX-CUT instances, overparameterization provides a sufficient condition for obtaining accurate solutions, and, specifically for 2-regular graphs, is also a necessary condition, with the required circuit depth closely matching analytically determined optimal values. The success probability, measured as the fraction of optimization runs converging to the best solution, significantly increased when transitioning to overparameterized circuits, suggesting that the overparameterization depth offers a reasonable estimate of the optimal depth needed for high-probability solutions. In contrast, MAX-2-SAT instances exhibited markedly different behaviour, with a majority solved effectively even when using circuits with fewer parameters than expected, well within the underparametrized regime.

Approximately 85% of instances with a specific problem size were solved using circuits with depths as small as half the overparameterization depth. This suggests that, unlike MAX-CUT, QAOA can achieve good performance for MAX-2-SAT without requiring extensive circuit complexity. The authors acknowledge that further research is needed to fully understand these differing behaviours and to explore the potential of QAOA in both over and underparametrized regimes, particularly in the context of current noisy quantum devices.