The Quantum Approximate Optimization Algorithm (QAOA) is a promising method for solving combinatorial optimization problems on near-term quantum devices. However, as the cost Hamiltonian becomes more complex, QAOA suffers from barren plateaus and optimization instability. In this work, we propose Cascade QAOA, a novel variant that splits the cost Hamiltonian into two clause-aware sub-Hamiltonians and utilizes an intermediate mixer to enable structured mixing. Clause partitioning is performed using betweenness centrality, prioritizing structurally important edges in early optimization stages. Experimental results on MaxCut problems using Erdős-Rényi graphs show that Cascade QAOA consistently outperforms vanilla QAOA across all tested settings. Notably, the approximation ratio improved by up to 5.2 % at moderate edge densities, where clause complexity benefits most from intermediate mixing. These findings suggest that Hamiltonian decomposition combined with mixer insertion is an effective strategy for enhancing QAOA performance without increasing the number of control gates, making it suitable for practical execution on Noisy Intermediate-Scale Quantum (NISQ)-era quantum hardware.
The Quantum Approximate Optimization Algorithm (QAOA) is a promising method for solving combinatorial optimization problems on near-term quantum devices. However, as the cost Hamiltonian becomes more complex, QAOA suffers from barren plateaus and optimization instability. In this work, we propose Cascade QAOA, a novel variant that splits the cost Hamiltonian into two clause-aware sub-Hamiltonians and utilizes an intermediate mixer to enable structured mixing. Clause partitioning is performed using betweenness centrality, prioritizing structurally important edges in early optimization stages. Experimental results on MaxCut problems using Erdős-Rényi graphs show that Cascade QAOA consistently outperforms vanilla QAOA across all tested settings. Notably, the approximation ratio improved by up to 5.2 % at moderate edge densities, where clause complexity benefits most from intermediate mixing. These findings suggest that Hamiltonian decomposition combined with mixer insertion is an effective strategy for enhancing QAOA performance without increasing the number of control gates, making it suitable for practical execution on Noisy Intermediate-Scale Quantum (NISQ)-era quantum hardware.
The quantum approximate optimization algorithm (QAOA) has numerous promising applications in solving the combinatorial optimization problems on near-term Noisy Intermediate Scalable Quantum (NISQ) devices. QAOA has a quantum-classical hybrid structure. Its quantum part consists of a parameterized alternating operator ansatz, and its classical part comprises an optimization algorithm, which optimizes the parameters to maximize the expectation value of the problem Hamiltonian. This expectation value depends highly on the parameters, this implies that a set of good parameters leads to an accurate solution. However, at large circuit depth of QAOA, it is difficult to achieve global optimization due to the multiple occurrences of local minima or maxima. In this paper, we propose a parameters fixing strategy which gives high approximation ratio on average, even at large circuit depths, by initializing QAOA with the optimal parameters obtained from the previous depths. We test our strategy on the Max-cut problem of certain classes of graphs such as the 3-regular graphs and the Erdös-Rényi graphs.
The Quantum Approximate Optimization Algorithm (QAOA) is a well-known hybrid quantum-classical algorithm for combinatorial optimization problems. Improving QAOA involves enhancing its approximation ratio while addressing practical constraints of Noisy Intermediate Scale Quantum (NISQ) devices, such as minimizing the number of two-qubit gates and reducing circuit depth. Although existing research has optimized designs for phase and mixer operators to improve performance, challenges remain, particularly concerning the excessive use of two-qubit gates in the construction of phase operators. To address these issues, we introduce a Dynamic Adaptive Phase Operator (DAPO) algorithm, which dynamically constructs phase operators based on the output of previous layers and neighborhood search approach, optimizing the problem Hamiltonian more efficiently. By using solutions generated by QAOA itself to simplify the problem Hamiltonian at each layer, the algorithm captures the problem's structural properties more effectively, progressively steering the solution closer to the optimal target. Experimental results on MaxCut and NAE3SAT problems show that DAPO achieves higher approximation ratios and significantly reduces two-qubit RZZ gates, especially in dense graphs. Compared to vanilla QAOA, DAPO uses only 66% of RZZ gates at the same depth while delivering better results, demonstrating its potential for efficient combinatorial optimization in the NISQ era.
Constructing an optimal mixer for Quantum Approximate Optimization Algorithm (QAOA) Hamiltonian is crucial for enhancing the performance of QAOA in solving combinatorial optimization problems. We present a systematic methodology for constructing the QAOA tailored mixer Hamiltonian, ensuring alignment with the inherent symmetries of classical optimization problem objectives. The key to our approach is to identify an operator that commutes with the action of the group of symmetries on the QAOA underlying Hilbert space and meets the essential technical criteria for effective mixer Hamiltonian functionality. We offer a construction method specifically tailored to the symmetric group Sd, prevalent in a variety of combinatorial optimization problems. By rigorously validating the required properties, providing a concrete formula and corresponding quantum circuit for implementation, we establish the viability of the proposed mixer Hamiltonian. Furthermore, we demonstrate that the classical mixer B commutes only with a subgroup of Sd of significantly smaller order than the group itself, enhancing the efficiency of the proposed approach. To evaluate the effectiveness of our methodology, we compare two QAOA variants utilizing different mixer Hamiltonians: conventional B=∑Xi and the newly proposed HM in edge coloring and graph partitioning problems across various graphs. We observe statistically significant differences in mean values, with the new variant consistently demonstrating superior performance across multiple independent simulations. Additionally, we analyze the phenomenon of poor performance in alternative warm-start QAOA variants, providing a conceptual explanation supported by recent literature findings.
Abstract The quantum approximate optimisation algorithm (QAOA) is a hybrid quantum–classical algorithm used to approximately solve combinatorial optimisation problems. It involves multiple iterations of a parameterised ansatz that consists of a problem and mixer Hamiltonian, with the parameters being classically optimised. While QAOA can be implemented on near-term quantum hardware, physical limitations such as gate noise, restricted qubit connectivity, and state-preparation-and-measurement (SPAM) errors can limit circuit depth and decrease performance. To address these limitations, this work introduces the eXpressive QAOA (XQAOA), an overparameterised variant of QAOA that assigns more classical parameters to the ansatz to improve its performance at low depths. XQAOA also introduces an additional Pauli-Y component in the mixer Hamiltonian, allowing the mixer to implement arbitrary unitary transformations on each qubit. To benchmark the performance of XQAOA at unit depth, we derive its closed-form expression for the MaxCut problem and compare it to QAOA, Multi-Angle QAOA (MA-QAOA) (Herrman et al 2022 Sci. Rep. 12 6781), a classical-relaxed algorithm, and the state-of-the-art Goemans–Williamson algorithm on a set of unweighted regular graphs with 128 and 256 nodes for degrees ranging from 3 to 10. Our results indicate that at unit depth, XQAOA has benign loss landscapes with local minima concentrated near the global optimum, allowing it to consistently outperform QAOA, MA-QAOA, and the classical-relaxed algorithm on all graph instances and the Goemans–Williamson algorithm on graph instances with degrees greater than 4. Small-scale simulations also reveal that unit-depth XQAOA invariably surpasses both QAOA and MA-QAOA on all tested depths up to five. Additionally, we find an infinite family of graphs for which XQAOA solves MaxCut exactly and analytically show that for some graphs in this family, special cases of XQAOA are capable of achieving a much larger approximation ratio than QAOA. Overall, XQAOA is a more viable choice for variational quantum optimisation on near-term quantum devices, offering competitive performance at low depths.
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), which is a variational quantum algorithm, aims to give sub-optimal solutions of combinatorial optimization problems. It is widely believed that QAOA has the potential to demonstrate application-level quantum advantages in the noisy intermediate-scale quantum(NISQ) processors with shallow circuit depth. Since the core of QAOA is the computation of expectation values of the problem Hamiltonian, an important practical question is whether we can find an efficient classical algorithm to solve quantum mean value in the case of general shallow quantum circuits. Here, we present a novel graph decomposition based classical algorithm that scales linearly with the number of qubits for the shallow QAOA circuits in most optimization problems except for complete graph case. Numerical tests in Max-cut, graph coloring and Sherrington-Kirkpatrick model problems, compared to the state-of-the-art method, shows orders of magnitude performance improvement. Our results are not only important for the exploration of quantum advantages with QAOA, but also useful for the benchmarking of NISQ processors.