Dynamic depth quantum approximate optimization algorithm for solving constrained shortest path problem

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{p=1}$$\end{document} and progressing up to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{p=10}$$\end{document}, 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{p = 10}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{p = 15.}$$\end{document} In particular, while standard QAOA at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{p=15}$$\end{document} 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.