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Maximum weighted independent set and quantum alternating operator ansatz

Zain H. Saleem

Published May 12, 2019• 0 citations
OpenAlex

Abstract

We study the maximum weighted independent set problem of graph theory using the quantum approximate optimization algorithm. We perform simulations on the Rigetti Forrest simulator and analyze the dependence of the algorithm on the depth of the circuit, intial states and the wieghts of the vertices. We also numerically obtain an estimate of the expected approximation ratio of the algorithm for the circuit depths of p=1 and p=2 for a degree 2 graph.

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Max-independent set and the quantum alternating operator ansatz

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The maximum-independent set (MIS) problem of graph theory using the quantum alternating operator ansatz is studied. We perform simulations on the Rigetti Forest simulator for the square ring, [Formula: see text], and [Formula: see text] graphs and analyze the dependence of the algorithm on the depth of the circuit and initial states. The probability distribution of observation of the feasible states representing maximum-independent sets is observed to be asymmetric for the MIS problem, which is unlike the Max-Cut problem where the probability distribution of feasible states is symmetric. For asymmetric graphs, it is shown that the algorithm clearly favors the independent set with the larger number of elements even for finite circuit depth. We also compare the approximation ratios for the algorithm when we choose different initial states for the square ring graph and show that it is dependent on the choice of the initial state.

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Parameters Fixing Strategy for Quantum Approximate Optimization Algorithm

Xinwei Lee, Yoshiyuki Saito, Dongsheng Cai +1 more

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.

arXiv:2108.05288•IEEE International Conference on Quantum Computing and Engineering (QCE)•Oct 1, 2021•34 citations
We study the maximum weighted independent set problem of graph theory using the quantum approximate optimization algorithm. We perform simulations on the Rigetti Forrest simulator and analyze the dependence of the algorithm on the depth of the circuit, intial states and the wieghts of the vertices. We also numerically obtain an estimate of the expected approximation ratio of the algorithm for the circuit depths of p=1 and p=2 for a degree 2 graph.
The maximum-independent set (MIS) problem of graph theory using the quantum alternating operator ansatz is studied. We perform simulations on the Rigetti Forest simulator for the square ring, [Formula: see text], and [Formula: see text] graphs and analyze the dependence of the algorithm on the depth of the circuit and initial states. The probability distribution of observation of the feasible states representing maximum-independent sets is observed to be asymmetric for the MIS problem, which is unlike the Max-Cut problem where the probability distribution of feasible states is symmetric. For asymmetric graphs, it is shown that the algorithm clearly favors the independent set with the larger number of elements even for finite circuit depth. We also compare the approximation ratios for the algorithm when we choose different initial states for the square ring graph and show that it is dependent on the choice of the initial state.
Abstract The quantum alternating operator ansatz algorithm (QAOA+) is widely used for constrained combinatorial optimization problems (CCOPs) due to its ability to construct feasible solution spaces. In this paper, we propose a progressive quantum algorithm (PQA) to reduce qubit requirements for QAOA+ in solving the maximum independent set (MIS) problem. PQA iteratively constructs a subgraph likely to include the MIS solution of the original graph and solves the problem on it to approximate the global solution. Specifically, PQA starts with a small-scale subgraph and progressively expands its graph size utilizing heuristic expansion strategies. After each expansion, PQA solves the MIS problem on the newly generated subgraph using QAOA+. In each run, PQA repeats the expansion and solving process until a predefined stopping condition is reached. Simulation results show that PQA achieves an approximation ratio of 0.95 using only 5.57% (2.17%) of the qubits and 17.59% (6.43%) of the runtime compared with directly solving the original problem with QAOA+ on Erdös–Rényi (3-regular) graphs, highlighting the efficiency and scalability of PQA.
We investigate a hybrid quantum-classical algorithm for solving the Maximum Independent Set (MIS) problem on regular graphs, combining the Quantum Approximate Optimization Algorithm (QAOA) with a minimal degree classical greedy algorithm. The method leverages pre-computed QAOA angles, derived from depth-ppp QAOA circuits on regular trees, to compute local expectation values and inform sequential greedy decisions that progressively build an independent set. This hybrid approach maintains shallow quantum circuit and avoids instance-specific parameter training, making it well-suited for implementation on current quantum hardware: we have implemented the algorithm on a 20 qubit IQM superconducting device to find independent sets in graphs with thousands of nodes. We perform tensor network simulations to evaluate the performance of the algorithm beyond the reach of current quantum hardware and compare to established classical heuristics. Our results show that even at low depth (p=4p=4p=4), the quantum-enhanced greedy method significantly outperforms purely classical greedy baselines as well as more sophisticated approximation algorithms. The modular structure of the algorithm and relatively low quantum resource requirements make it a compelling candidate for scalable, hybrid optimization in the NISQ era and beyond.
Hadfield et al. proposed a novel Quantum Alternating Operator Ansatz algorithm (QAOA+), and this algorithm has wide applications in solving constrained combinatorial optimization problems (CCOPs) because of the advantages of QAOA+ ansatz in constructing a feasible solution space. In this paper, we propose a Progressive Quantum Algorithm (PQA) with QAOA+ ansatz to solve the Maximum Independent Set (MIS) problem using fewer qubits. The core idea of PQA is to construct a subgraph that is likely to contain the MIS solution of the target graph and then solve the MIS problem on this subgraph to obtain an approximate solution. To construct such a subgraph, PQA starts with a small-scale initial subgraph and progressively expands its graph size utilizing heuristic expansion strategies. After each expansion, PQA solves the MIS problem on the newly generated subgraph. In each run, PQA repeats the expansion and solving process until a predefined stopping condition is reached. Simulation results demonstrate that to achieve an approximation ratio of 0.95, PQA requires only 5.565%5.565\%5.565% (2.170%2.170\%2.170%) of the qubits and 17.59%17.59\%17.59% (6.430%6.430\%6.430%) of the runtime compared with directly solving the original problem using QAOA+ on Erdős-Rényi (3-regular) graphs, highlighting the efficiency of PQA.
In recent years, quantum computing has shown promise in solving certain problems more efficiently than classical methods. The Quantum Approximate Optimization Algorithm (QAOA) has emerged as a leading candidate for achieving quantum advantage on noisy intermediate-scale quantum (NISQ) devices. This study explores the application of QAOA to the Maximum Weighted Independent Set (MWIS) problem, with particular focus on circuit transpilation strategies and parameter optimization methods. Using IBM Qiskit, we systematically examine factors affecting QAOA performance, including different transpilation optimization levels and classical optimizer selection. Our results demonstrate that optimization level 3 reduced circuit depth by 32% compared to level 0 for p=3 circuits, while maintaining solution quality. Among classical optimizers, COBYLA achieved an 81.24% approximation ratio with only 54.8 function evaluations at p=2, outperforming both grid search and SPSA methods. Execution on IBM’s real quantum hardware resulted in performance degradation ranging from 16% at p=1 to 34% at p=3 compared to ideal simulations, highlighting the compounding impact of noise and gate errors as circuit depth increases.These findings provide practical guidance for quantum algorithm implementation on NISQ devices, demonstrating that appropriate selection of transpilation techniques and optimization methods can substantially improve quantum resource efficiency while maintaining solution quality.
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.