Dynamic depth quantum approximate optimization algorithm for solving constrained shortest path problem
Rakesh Kumar Saini, Nora Mohamed, Saif M. Al-Kuwari, Ahmed Farouk · Quantum Machine Intelligence · 2026
Abstract The Quantum Approximate Optimization Algorithm (QAOA) has emerged as a promising approach for solving NP-hard combinatorial optimization problems on noisy intermediate-scale quantum (NISQ) hardware. However, its performance critically depends on the choice of circuit depth–a parameter that must be specified a priori without clear guidance. In this paper, we introduce a variant of QAOA, called the Dynamic Depth Quantum Approximate Optimization Algorithm (DDQAOA), that addresses the challenge of pre-selecting a fixed circuit depth. Our method adaptively expands circuit depth, starting from $$\varvec{p=1}$$ and progressing up to $$\varvec{p=10}$$ , by transferring learned parameters to deeper circuits based on convergence criteria. We tested this approach on 100 instances of the Constrained Shortest Path Problem (CSPP) at 10-qubit and 16-qubit scales, and on 20 additional instances at the 22-qubit scale. DDQAOA achieved competitive approximation ratios and success probabilities with substantially fewer CNOT evaluations than standard QAOA at $$\varvec{p = 10}$$ and $$\varvec{p = 15.}$$ In particular, while standard QAOA at $$\varvec{p=15}$$ achieved results close to our approach, it used 217%, 159.3%, and 315% more CNOT gates for 10-qubit, 16-qubit, and 22-qubit instances, respectively. These results demonstrate that convergence-driven DDQAOA, building on the INTERP parameter-transfer protocol (Zhou et al. 2020), offers a practical route to applying QAOA to constrained NP-hard problems on near-term devices.