Breaking QAOA's Fixed Target Hamiltonian Barrier: A Fully Connected Quantum Boltzmann Machine via Bilevel Optimization

To overcome the limitations of classical partially connected Boltzmann machines and mainstream quantum Boltzmann machines (QBMs), this work extends the conventional circuit of the quantum approximate optimization algorithm (QAOA) to a bilevel optimization architecture and proposes a fully connected QBM. The inner-loop training simulates positive phase energy minimization based on the computational process of the conventional QAOA circuit, whereas the outer-loop training simulates negative phase contrastive divergence learning by optimizing the structural parameters of the target Hamiltonian. It is found that, first, the model exhibits superior performance using only a single layer p=1 in the QAOA circuit, with an average probability of 0.9559 in measuring the target quantum state under noiseless conditions. Second, the model exhibits notable noise robustness. Under the typical noise level of current mainstream commercial quantum computing devices, the average probability of measuring the target quantum state reaches 0.6047; when the noise rises to a more stringent level with doubled intensity, this probability remains at 0.3859. In both scenarios, the target quantum state maintains the highest measurement probability among all detected states, with a value several times higher than that of the second-ranked state. This indicates that the model retains strong robustness even when noise meets or exceeds the upper limit of current mainstream commercial quantum computing devices. Third, under a block-by-block learning strategy with p=1 and only 10 measurement shots, the model consistently generates the target "qubit" grid image regardless of noise interference, demonstrating strong robustness in image generation.