An Adaptive Mixer Allocation Algorithm for the Quantum Alternating Operator Ansatz

Recently, Hadfield et al. proposed the quantum alternating operator ansatz algorithm (QAOA+), an extension of the quantum approximate optimization algorithm (QAOA), to solve constrained combinatorial optimization problems (CCOPs). Compared with QAOA, QAOA+ enables the search for optimal solutions within a feasible solution space by encoding problem constraints into the mixer Hamiltonian, thereby reducing the search space and eliminating the possibility of yielding infeasible solutions. However, QAOA+ may incur high overall gate costs when the mixer is applied to all qubits in each layer, and each mixer is costly to implement. To address this challenge, an adaptive mixer allocation strategy is tailored for QAOA+. The resulting algorithm, which integrates this strategy into the original QAOA+ framework, is referred to as AMA-QAOA+. Unlike QAOA+, AMA-QAOA+ adaptively applies the mixer to a subset of qubits in each layer of the mixer unitary operator based on an evaluation function. The performance of AMA-QAOA+ is evaluated on the maximum independent set problem. Numerical simulation results show that, under the same number of optimization runs, AMA-QAOA+ achieves better solution quality than QAOA+, with the optimal approximation ratio improved by $5.30%$ on ER random graphs and $5.41%$ on 3-regular graphs. Moreover, AMA-QAOA+ significantly reduces the CNOT gate consumption, requiring only $15.30%$ and $25.18%$ of the CNOT gates used by QAOA+ on ER and 3-regular random graphs, respectively. These results demonstrate that AMA-QAOA+ enhances solution quality and computational efficiency, enabling the design of more compact and resource-efficient quantum circuits.

💡 Research Summary

The paper addresses a critical bottleneck in the quantum alternating operator ansatz with constraints (QAOA+), namely the high gate overhead caused by applying the mixer Hamiltonian to all qubits in every layer. To mitigate this, the authors introduce an Adaptive Mixer Allocation strategy, yielding the AMA‑QAOA+ algorithm. The key idea is to apply the mixer only to a carefully chosen subset of qubits in each layer, based on a composite evaluation function that incorporates both the average gradient of the cost function with respect to the parameters and the average expectation value obtained from multiple random parameter samples. This dual‑criterion approach aims to capture both local sensitivity and global performance, avoiding the pitfalls of single‑criterion adaptive methods.

Algorithmically, AMA‑QAOA+ starts from a shallow circuit (p = 1) and iteratively expands it. At each expansion step, a predefined pool of possible mixer operators is examined. For each candidate, the evaluation function is computed; the top‑k operators (where k is a hyper‑parameter) are appended to the circuit without immediate re‑optimization. After a batch of operators has been added, the entire parameter set is optimized using a classical optimizer (e.g., COBYLA or Adam). This batch‑wise update reduces the number of optimizer calls compared with earlier adaptive VQE/QAOA schemes that re‑optimize after every single operator addition.

The authors evaluate AMA‑QAOA+ on the Maximum Independent Set (MIS) problem, a representative constrained combinatorial optimization task. Numerical simulations are performed on Erdős–Rényi (ER) random graphs (edge probability 0.5) and 3‑regular graphs, with vertex counts ranging from 8 to 14. For each graph instance, 100 independent optimization runs are executed. The baselines include the original QAOA+, an Adaptive‑QAOA+ that uses only gradient information, and a random‑subset mixer approach (PNU).

Results show three consistent improvements. First, the approximation ratio (AR) of AMA‑QAOA+ exceeds that of QAOA+ by about 5.30 % on ER graphs and 5.41 % on 3‑regular graphs, indicating higher solution quality under the same number of optimization runs. Second, the CNOT gate count is dramatically reduced: AMA‑QAOA+ uses only 15.30 % of the CNOTs required by QAOA+ on ER graphs and 25.18 % on 3‑regular graphs, corresponding to reductions of 84.70 % and 74.82 % respectively. Third, the number of optimizer iterations is lowered by 35 %–49 % compared with Adaptive‑QAOA+, reflecting a more efficient search in the parameter space.

Ablation studies confirm that the combined use of gradient and expectation information in the evaluation function outperforms either metric alone, and that batch‑wise operator insertion yields fewer optimizer calls without sacrificing performance. The authors also discuss limitations: the evaluation function requires sampling over random parameters, which may become costly for larger problem sizes, and the performance depends on the size and composition of the operator pool. Future work is suggested in the directions of meta‑learning‑driven evaluation metrics, dynamic pool expansion, and experimental validation on noisy intermediate‑scale quantum (NISQ) hardware. Extending the adaptive mixer allocation concept to other constrained problems such as maximum clique or graph coloring is identified as a promising avenue.

In summary, AMA‑QAOA+ offers a practical pathway to reduce quantum circuit depth and gate count while simultaneously improving solution quality for constrained combinatorial optimization, thereby bringing QAOA+ closer to feasibility on near‑term quantum devices.