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arXiv:2509.16653 · Chinese Physics B

Auxiliary-qubit-free quantum approximate optimization algorithm for the minimum dominating set problem

Guanghui Li, Xiaohui Ni, Junjian Su, Su‐Juan Qin, Fen-Zhuo Guo, Bingjie Xu, Wei Huang, Fei Gao

Published Mar 11, 2026• 0 citations
arXivHTMLPDF

Abstract

Abstract Quantum approximate optimization algorithm (QAOA) is a promising framework for solving combinatorial optimization problems on near-term quantum devices. One such problem is the minimum dominating set (MDS), which is known to be NP-hard. Existing QAOA algorithms for this problem typically require numerous auxiliary qubits, increasing circuit overhead and hardware requirements. In this paper, we propose an auxiliary-qubit-free QAOA algorithm based on Hamiltonian evolution (AQFH-QAOA) for the MDS problem. Unlike previous studies that require numerous auxiliary qubits, our algorithm eliminates the need for auxiliary qubits, thereby significantly reducing circuit overhead. In addition, we present an auxiliary-qubit-free optimized implementation of the previously proposed Guerrero’s QAOA algorithm (AQFG-QAOA) by utilizing gate decomposition techniques. Through a detailed analysis of gate complexity, we evaluate the applicability of these two algorithms. Numerical experiments demonstrate that our proposed algorithm achieves competitive solution quality compared with existing QAOA algorithms, making it a promising candidate for implementation on near-term quantum devices.

DOI
10.1088/1674-1056/ae5052

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Abstract Quantum approximate optimization algorithm (QAOA) is a promising framework for solving combinatorial optimization problems on near-term quantum devices. One such problem is the minimum dominating set (MDS), which is known to be NP-hard. Existing QAOA algorithms for this problem typically require numerous auxiliary qubits, increasing circuit overhead and hardware requirements. In this paper, we propose an auxiliary-qubit-free QAOA algorithm based on Hamiltonian evolution (AQFH-QAOA) for the MDS problem. Unlike previous studies that require numerous auxiliary qubits, our algorithm eliminates the need for auxiliary qubits, thereby significantly reducing circuit overhead. In addition, we present an auxiliary-qubit-free optimized implementation of the previously proposed Guerrero’s QAOA algorithm (AQFG-QAOA) by utilizing gate decomposition techniques. Through a detailed analysis of gate complexity, we evaluate the applicability of these two algorithms. Numerical experiments demonstrate that our proposed algorithm achieves competitive solution quality compared with existing QAOA algorithms, making it a promising candidate for implementation on near-term quantum devices.
Quantum Approximate Optimization Algorithm (QAOA) is a prospective candidate for providing quantum advantage in finding approximate solutions to optimization problems using near-term quantum devices. In this paper, the goal is to reduce the number of CNOT gates and the depth of quantum ansatz circuits for QAOA. First, we present a generalized QAOA formulation for any Hamiltonian that involves upto two-body interactions, as a graph. The circuit realization of the depth-p QAOA requires 2mp2mp2mp CNOT gates where the graph for the Hamiltonian has n vertices and m edges. Presently, a CNOT gate is one of the primary sources of error. We propose a graph-theoretic algorithm which can eliminate n−1n-1n−1 CNOTs from the QAOA ansatz while retaining functional equivalence with the original circuit. This improves upon previously proposed Depth First Search (DFS) method by restricting the increase in depth of the circuit, which was a drawback of the method, by ≃84.8%\simeq 84.8\%≃84.8% for Erdős-Renyi graphs. Finally, if the underlying connectivity (hardware coupling map) and the initial qubit placement for a hardware are known, then we present how to extend the greedy heuristic method in order to obtain a functionally equivalent circuit with ≃5%\simeq 5\%≃5%; fewer SWAP gates for Erdős-Renyi graphs.
The quantum approximate optimization algorithm (QAOA) is a promising method for solving certain classical combinatorial optimization problems on near-term quantum devices. When employing the QAOA to 3-SAT and Max-3-SAT problems, the quantum cost exhibits an easy-hard-easy or easy-hard pattern, respectively, as the clause density is changed. The quantum resources needed in the hard-region problems are out of reach for current noisy intermediate-scale quantum (NISQ) devices. We show by numerical simulations with up to 14 variables and analytical arguments that the adaptive-bias QAOA (ab-QAOA) greatly improves performance in the hard region of the 3-SAT problems and hard region of the Max-3-SAT problems. For similar accuracy, on average, ab-QAOA needs 3 levels for 10-variable 3-SAT problems as compared to 22 for QAOA. For 10-variable Max-3-SAT problems, the numbers are 7 levels and 62 levels. The improvement comes from a more targeted and more limited generation of entanglement during the evolution. We demonstrate that classical optimization is not strictly necessary in the ab-QAOA since local fields are used to guide the evolution. This leads us to propose an optimization-free ab-QAOA that can solve the hard-region 3-SAT and Max-3-SAT problems effectively with significantly fewer quantum gates as compared to the original ab-QAOA. Our work paves the way for realizing quantum advantages for optimization problems on NISQ devices.
The performance of the Quantum Approximate Optimization Algorithm (QAOA) on noisy intermediate-scale quantum (NISQ) devices is strongly limited by sparse qubit connectivity. When interactions required by QAOA Hamiltonians are not aligned to the hardware topology, transpilation introduces SWAP gates, increasing circuit depth and noise. We propose a SWAP-free QAOA framework based on modifying the cost Hamiltonian so that it can be implemented natively on the hardware. We formulate this as a mixed-integer semidefinite program (MISDP) that selects a hardware-compatible approximation of the original cost matrix and optimizes the allocation of logical variables to physical qubits. We prove that the associated decision problem is NP-complete and derive theoretical guarantees relating the MISDP objective to the loss in the original optimization problem through the Lovász number of the hardware graph. Since solving MISDPs is practical only for small instances, we introduce heuristics based on spectral properties of the problem matrix and hardware graph. Our experiments on a cardinality-constrained quadratic optimization model for index tracking show competitive performance against a baseline representing ideal QAOA under SWAP-induced noise. These results indicate that, on sparse NISQ architectures, a hardware-aware approximation of the objective may be more effective than an exact but heavily transpiled Hamiltonian implementation.
The quantum approximate optimization algorithm (QAOA) is an approach for near-term quantum computers to potentially demonstrate computational advantage in solving combinatorial optimization problems. However, the viability of the QAOA depends on how its performance and resource requirements scale with problem size and complexity for realistic hardware implementations. Here, we quantify scaling of the expected resource requirements by synthesizing optimized circuits for hardware architectures with varying levels of connectivity. Assuming noisy gate operations, we estimate the number of measurements needed to sample the output of the idealized QAOA circuit with high probability. We show the number of measurements, and hence total time to solution, grows exponentially in problem size and problem graph degree as well as depth of the QAOA ansatz, gate infidelities, and inverse hardware graph degree. These problems may be alleviated by increasing hardware connectivity or by recently proposed modifications to the QAOA that achieve higher performance with fewer circuit layers.
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.
The quantum approximate optimization algorithm (QAOA) is an appealing proposal to solve NP problems on noisy intermediate-scale quantum (NISQ) hardware. Making NISQ implementations of the QAOA resilient to noise requires short ansatz circuits with as few controlled- () gates as possible. Here we present the dynamic adaptive quantum approximate optimization algorithm (Dynamic-ADAPT-QAOA). Our algorithm significantly reduces the circuit depth and the count of standard ADAPT-QAOA, a leading proposal for near-term implementations of the QAOA. Throughout our algorithm, the decision to apply -intensive operations is made dynamically, based on algorithmic benefits. Using density-matrix simulations, we benchmark the noise resilience of ADAPT-QAOA and Dynamic-ADAPT-QAOA. We compute the gate-error probability p gate ★ below which these algorithms provide, on average, more accurate solutions than the classical, polynomial-time approximation algorithm by Goemans and Williamson. For small systems with six to ten qubits, we show that p gate ★ > 10 − 3 for Dynamic-ADAPT-QAOA. Compared to standard ADAPT-QAOA, this constitutes an order-of-magnitude improvement in noise resilience. This improvement should make Dynamic-ADAPT-QAOA viable for implementations on superconducting NISQ hardware, even in the absence of error mitigation. Published by the American Physical Society 2024