A Layerwise-Multi-Angle Approach to Fine-Tuning the Quantum Approximate Optimization Algorithm

Quantum computing promises new ways to address computational problems that become extremely difficult as their size increases. Among these, combinatorial optimization is particularly relevant: selecting the best solution from an enormous number of possible configurations appears in logistics, energy grids, scheduling, network modelling, resource allocation, and even some machine learning problems. Current quantum hardware, however, operates under significant constraints. Limited numbers of qubits and noise make deep, fault-tolerant algorithms difficult to execute. Variational Quantum Algorithms offer an alternative designed for this intermediate technological phase. They divide the computational effort between a parameterized quantum circuit and a classical optimizer: the quantum system evaluates candidate solutions, while the classical component progressively adjusts the circuit parameters. One of the most promising approaches in this family is the Quantum Approximate Optimization Algorithm (QAOA). Its effectiveness depends not only on the quantum circuit itself, but also on a difficult question: how should its parameters be optimized as the circuit becomes increasingly complex?

Standard QAOA constructs a quantum circuit from multiple layers. Each layer alternates a cost operator, representing the optimization problem, with a mixer operator that explores the space of possible solutions. Two sets of angles determine the behaviour of these operations and are iteratively adjusted through classical optimization. Increasing the number of layers can improve the ability of the circuit to approximate the optimal solution, but it also makes parameter optimization more challenging. Two existing approaches address this problem from opposite directions. A layerwise strategy progressively builds the circuit while fixing parameters that have already been optimized. This reduces the dimensionality of the optimization landscape, making the process easier to manage, but constrains the expressive power of the model. Multi-Angle QAOA takes almost the opposite approach. Instead of assigning a single parameter to an entire operator, it introduces individual parameters for different elements of the optimization problem. In the MaxCut case, for example, different angles can be associated with individual graph nodes and edges. This gives the quantum model considerably greater freedom, but also creates a much larger and more difficult parameter space to optimize.

The proposed solution combines these two ideas into a Layerwise Multi-Angle QAOA, or LMA-QAOA. The process begins with a Multi-Angle circuit containing a single layer. Its parameters are randomly initialized, the quantum circuit is executed, and its output is passed to a classical optimizer. This produces an optimized set of angles for the first layer. When a second layer is introduced, those previously optimized parameters are preserved. Only the parameters associated with the new layer are randomly initialized and optimized. The procedure can then be repeated as the quantum circuit grows. Instead of repeatedly optimizing every parameter from scratch, the algorithm therefore constructs its solution progressively. Each new layer adds expressive capacity while the knowledge acquired during previous optimization stages remains fixed. Conceptually, the quantum circuit learns one layer at a time: optimize what has just been added, preserve what already works, and then increase the complexity of the model.

An extended version, called eLMA-QAOA, adds another optimization stage. After the layerwise procedure has identified the parameters, all parameters can be released and jointly optimized again. This creates a two-stage strategy. Layerwise optimization first guides the circuit toward a promising region of the parameter space. A final unconstrained optimization can then refine the entire solution without keeping the parameters from previous layers fixed. The additional freedom can potentially improve the result, particularly for relatively shallow circuits, but it also increases computational requirements. The experiments show that this extra optimization does not systematically provide enough improvement to compensate for its substantially longer execution time when the number of layers grows.

The approach is evaluated using MaxCut, a well-known NP-hard combinatorial optimization problem. Given a graph, MaxCut attempts to divide its nodes into two groups so that the number of edges connecting nodes belonging to opposite groups is maximized. Different graph structures are considered, including complete graphs, Erdős–Rényi random graphs, complete binary trees, and random regular graphs. The experiments covered graphs containing 4, 6, 8, and 10 nodes and quantum circuits with 1, 2, 4, and 6 layers. Four approaches are compared under the same experimental setting: conventional QAOA, Multi-Angle QAOA, the layerwise Multi-Angle strategy, and its extended version. Performance is primarily evaluated through the approximation ratio, which measures how closely the solution obtained by the quantum algorithm approaches the actual optimum.

The numerical results show a clear difference between conventional QAOA and the more expressive variants. Across all experiments, standard QAOA obtained an average approximation ratio similar to Multi-Angle QAOA and LMA-QAOA. More importantly, LMA-QAOA produced the lowest standard deviation, indicating greater stability across the experiments. The performance index combining approximation quality, variability, and circuit depth is also highest for LMA-QAOA. It is approximately 3.3% higher than that of Multi-Angle QAOA and about 14.8% higher than conventional QAOA. The results therefore suggest that the layerwise strategy can retain the expressive advantages of Multi-Angle QAOA while making parameter optimization more controlled and producing more stable solutions.

The experiments also highlight challenges that remain unresolved. Current quantum hardware restricts practical applications to relatively small problems, noise can degrade computation, graph structure strongly affects performance, and results depend on the selected classical optimizer and its configuration. Scalability to substantially deeper circuits also remains uncertain. Nevertheless, the layerwise Multi-Angle approach demonstrates an important principle for variational quantum computing: increasing expressive power does not necessarily require optimizing an increasingly large parameter space all at once. By progressively expanding the quantum circuit while preserving previously optimized parameters, complex quantum models can be trained through a sequence of smaller optimization problems. The result is a balance between the flexibility of highly parameterized quantum circuits and the tractability of layerwise optimization, a direction that could become increasingly important as variational quantum algorithms are applied to larger and more demanding optimization problems.