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

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, which increases 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, thus 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 to existing QAOA algorithms, making it a promising candidate for implementation on near-term quantum devices.

Figures (16)

• Figure 1: A single-layer quantum circuit for solving the MDS problem on a 4-vertex 3-regular graph, constructed according to Guerrero's QAOA algorithm guerrero2020solving. The qubit $q_k$ corresponds to vertex $v_k$ of the graph, and $\left | 0 \right \rangle_{aux}$ represents the auxiliary qubit. The blue dashed box represents the circuit part that implements all the $D_k(x)$ clauses, while the red dashed box indicates the circuit part that implements all the $T_k(x)$ clauses.
• Figure 2: Quantum circuits for (a) the single-qubit operator $e^{-i \theta \sigma_0^z /2}$, (b) the two-qubit operator $e^{-i \theta \sigma_0^z \sigma_1^z /2 }$, and (c) the three-qubit operator $e^{-i \theta \sigma_0^z \sigma_1^z \sigma_{2}^z /2}$. Similarly, a $k$-qubit operator $e^{-i \theta \sigma_0^z \sigma_1^z \cdots \sigma_{k-1}^z}$ can be implemented using $2(k-1)$ CNOT gates and a single-qubit $R_Z$ gate hadfield2021representation.
• Figure 3: Decomposition of a multi-OR-controlled phase gate with $n$ control qubits using $2n+2$ single-qubit gates and two $n$-controlled $X$ gates without auxiliary qubits. Here, $n=4$. It employs the relations $R_Z(\theta)R_Z(\theta)=R_Z(2\theta)$, $XR_Z(\theta)X=R_Z(-\theta)$ and $R_Z(\theta)R_Z(-\theta)=I$.
• Figure 4: Decomposition of an $n$-controlled $X$ gate ($C^{n}(X)$) using two $(n-1)$-controlled $X$ gates ($C^{n-1}(X)$), an $(n-1)$-controlled $\sqrt{X}$ gate ($C^{n-1}(\sqrt{X})$), and two single-controlled gates. Here, $\sqrt{X}$ is the square root of $X$, and $\sqrt{X}^\dagger$ is the Hermitian conjugate of $\sqrt{X}$. Adapted from Lemma 7.5 of Ref. barenco1995elementary.
• Figure 5: Decomposition of a $C^{n-1}(X)$ gate using two $m_1$-controlled $X$ gates ($C^{m_1}(X)$) and two $(m_2+1)$-controlled $X$ gates ($C^{m_2+1}(X)$), where $m_1 = \left \lceil \frac{n}{2} \right \rceil$ and $m_2 = n- m_1-1$. Adapted from Lemma 7.3 of Ref. barenco1995elementary.

Theorems & Definitions (3)

• Theorem 1
• proof
• Definition 1: Success Probability