  Quantum approximated optimization algorithm (QAOA) has shown promise for
solving combinatorial optimization problems by providing quantum speedup on
near-term gate-based quantum computing systems. However, QAOA faces challenges
in optimizing variational parameters for high-dimensional problems due to the
large number of qubits required and the complexity of deep circuits, which
limit its scalability for real-world applications. In this study, we propose a
distributed QAOA (DQAOA), which leverages a high-performance computing-quantum
computing (HPC-QC) integrated system. DQAOA leverages distributed computing
strategies to decompose a large job into smaller tasks, which are then
processed on the HPC-QC system. The global solution is iteratively updated by
aggregating sub-solutions obtained from DQAOA, allowing convergence toward the
optimal solution. We demonstrate that DQAOA can handle considerably large-scale
optimization problems (e.g., 1,000-bit problem) achieving high accuracy (~99%)
and short time-to-solution (~276 s). To apply this algorithm to material
science, we further develop an active learning algorithm integrated with our
DQAOA (AL-DQAOA), which involves machine learning, DQAOA, and active data
production in an iterative loop. We successfully optimize photonic structures
using AL-DQAOA, indicating that solving real-world optimization problems using
gate-based quantum computing is feasible with our strategies. We expect the
proposed DQAOA to be applicable to a wide range of optimization problems and
AL-DQAOA to find broader applications in material design.
