Dynamic Depth Quantum Approximate Optimization Algorithm for Solving Constrained Shortest Path Problem

TL;DR

This work introduces Dynamic Depth QAOA (DDQAOA), an adaptive variant of QAOA that starts at $p=1$ and incrementally increases circuit depth based on convergence criteria, aided by a parameter transfer mechanism to warm-start deeper layers. The method is applied to the Constrained Shortest Path Problem (CSPP), an NP-hard problem, by formulating CSPP as a QUBO and Ising Hamiltonian and evaluating DDQAOA against fixed-depth QAOA at $p=3$, $5$, $10$, and $15$ on 100 CSPP instances at $10$- and $16$-qubit scales. Results show DDQAOA achieves higher approximation ratios and success probabilities with substantially lower or comparable CNOT gate costs, and exhibits robust, adiabatic-like parameter schedules with monotonic trends in $\gamma^*$ and $\beta^*$. These findings highlight DDQAOA as a practical, hardware-aware approach for solving combinatorial optimization on NISQ devices and motivate further validation on larger instances and real quantum hardware.

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 is critically dependent on the selection of the circuit depth a parameter that must be specified a priori without clear guidance. In this paper, we introduce a variant of QAOA called dynamic depth Quantum Approximate Optimization Algorithm (DDQAOA) that resolves the challenge of pre selecting a fixed circuit depth. Our method adaptively expands circuit depth, starting from p = 1 and progressing up to 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. Our DDQAOA achieved superior approximation ratios and success probabilities with fewer CNOT gate evaluations than the standard QAOA for p = 3, 5, 10, and 15. In particular, while standard QAOA at p = 15 achieved results close to our approach, it used 217% and 159.3% more CNOT gates for 10 qubit and 16 qubit instances, respectively. This demonstrates the performance and practical applicability of DDQAOA to solve combinatorial optimization problems on near term devices.

Figures (6)

• Figure 1: The dynamic depth quantum approximate algorithm (DDQAOA) quantum circuit schematic applied to 4-node CSPP instance. Each box in the circuit contains the cost Hamiltonian operator $e^{i\gamma H_C}$ and mixer operator $e^{i\beta H_M}$, $H_C$ corresponds to the problem. The algorithm iteratively optimize $(\gamma, \beta)$ at depth $p$ using a classical optimizer to maximize $\langle H_C(\gamma,\beta)\rangle$. Here, the DDQAOA starts with $p=1$. When the cost improvement between consecutive iterations falls below threshold $\epsilon$, the depth $p$ is increased by 1 and the optimized $p$ parameters are interpolated to $p+1$ parameters $(\gamma_1^*\ldots\gamma_{p+1}^*, \beta_1^*\ldots\beta_{p+1}^*)$. The solution shown in the right displays the optimal CSPP solution from node $0$ to node $3$, highlighted with a bold red line.
• Figure 2: The box plot shows the approximation ratio distributions across 100 problem instances achieved by DDQAOA and fixed-depth QAOAs. Black dashed lines indicate the optimal approximation ratio line. DDQAOA substantially exceeds all fixed-depth QAOA results and reached close to the optimal value in all 100 instances.
• Figure 3: It shows the convergence trajectories for (a), (b) approximation ratio, and (c), (d) success probability for 10 and 16 qubits. Vertical dashed lines indicate the increase in layer number in DDQAOA throughout the optimization, and the horizontal black dashed line indicates the optimal value for the approximation ratio and the baseline for random guessing success probability.
• Figure 4: The box plot shows the success probability distributions across 100 problem instances for DDQAOA and fixed-depth QAOAs. Black dashed lines indicate a random guessing baseline. DDQAOA substantially exceeds random performance as well as fixed-depth QAOA results.
• Figure 5: It illustrates the number of CNOT gates in the QAOA circuit at each optimization step. The fixed-depth QAOAs show a constant CNOT utilization throughout the optimization; the DDQAOA shows an increase in the number of CNOT gates, starting from a minimum and reaching equal to the number of CNOT gates utilized by $p=10$ QAOA.