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Journal of Physics Conference Series

Quantum approximate optimization and k-means algorithms for data clustering

Jirawat Saiphet, Sujin Suwanna, Thiparat Chotibut, Areeya Chantasri

Published Jan 1, 2021• 3 citations
Publisher

Abstract

Abstract Noisy intermediate-scale quantum (NISQ) devices are cutting-edge technology expected to demonstrate potential and advantages of quantum computing over classical computing. Its low number of qubits and imperfection from noises restrict running full-scale quantum algorithms on such devices; however, quantum advantages can still be obtained. To achieve quantum advantages from NISQ devices, the hybrid quantum-classical algorithms were introduced. Quantum approximate optimization algorithm (QAOA) is a variational hybrid algorithm, which utilizes a NISQ device as a sub-unit for specific tasks and performs most calculations on a classical computer. QAOA provides an approximate solution, with arbitrary precision as the number of operations increases, for optimization problems. In this work we investigate the possibility of applying QAOA to a clustering problem and compare its performance with the classical k-means algorithm. It turns out that the weights in graph connectivity can degrade the algorithm operation and make it more difficult to approximate the solution. We also benchmark the QAOA by comparing the approximated solutions with the exact one obtained from a classical clustering algorithm.

DOI
10.1088/1742-6596/1719/1/012100

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Abstract Noisy intermediate-scale quantum (NISQ) devices are cutting-edge technology expected to demonstrate potential and advantages of quantum computing over classical computing. Its low number of qubits and imperfection from noises restrict running full-scale quantum algorithms on such devices; however, quantum advantages can still be obtained. To achieve quantum advantages from NISQ devices, the hybrid quantum-classical algorithms were introduced. Quantum approximate optimization algorithm (QAOA) is a variational hybrid algorithm, which utilizes a NISQ device as a sub-unit for specific tasks and performs most calculations on a classical computer. QAOA provides an approximate solution, with arbitrary precision as the number of operations increases, for optimization problems. In this work we investigate the possibility of applying QAOA to a clustering problem and compare its performance with the classical k-means algorithm. It turns out that the weights in graph connectivity can degrade the algorithm operation and make it more difficult to approximate the solution. We also benchmark the QAOA by comparing the approximated solutions with the exact one obtained from a classical clustering algorithm.
Quantum computers are devices which allow the solution of problems unsolvable to their classical counterparts. As an error-corrected quantum computer is still a decade away the quantum computing community has dedicated much attention to developing algorithms for currently available Noisy Intermediate-Scale Quantum computers (NISQ). Thus far, within NISQ, optimization problems are one of the most commonly studied and are exclusively tackled with the Quantum Approximate Optimization Algorithm (QAOA). This algorithm predominantly computes graph partitions with a maximal separation of edges (MaxCut), but can also be modified to calculate other properties of graphs. Here, I present a novel quantum optimization algorithm which uses exponentially less qubits as compared to the QAOA while requiring a significantly reduced number of quantum operations to solve the MaxCut problem. Such an improved performance allowed me to partition a graphs 32 nodes on publicly available 5 qubit gate-based solid state quantum devices without any preprocessing such as division of the graph into smaller subgraphs. This results represent a 40% increase in graph size as compared to state-of-art experiments on solid state devices such as Google Sycamore. The obtained lower bound is 54.9% on the solution for actual hardware benchmarks and 77.6% on ideal quantum simulators. Furthermore, large-scale optimization problems represented by graphs of a 128 nodes are tackled with quantum simulators, again without any pre-division into smaller subproblems and a lower solution bound of 67.9% is achieved.
The quantum approximate optimization algorithm (QAOA) is one of the canonical algorithms designed to find approximate solutions for combinatorial optimization problems on current noisy intermediate-scale quantum (NISQ) devices. The primary focus of ongoing research is to exhibit its speed advantage over classical algorithms. However, the performance of QAOA is restricted at lower depths, while higher depths are limited by current experimental techniques. We propose an algorithm that amalgamates the capabilities of QAOA and the real-space renormalization group transformation. Numerical simulations indicate that our proposed algorithm can deliver precise solutions for specific randomly generated instances using QAOA at shallow depths, even at the lowest depth. This algorithm is particularly suitable for current NISQ devices, offering the potential to demonstrate a quantum advantage.
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.
This paper presents a qualitative analysis of hybrid quantum-classical computing approaches aimed at solving complex optimization problems using near-term quantum processors. Hybrid algorithms leverage the strengths of quantum and classical computing to tackle computationally intensive tasks often constrained by current quantum hardware limitations. Through an extensive literature review and synthesis of recent empirical studies, we benchmark various hybrid algorithms, such as the Quantum Approximate Optimization Algorithm (QAOA) and Variational Quantum Eigensolver (VQE), focusing on their performance, scalability, and practical applicability on noisy intermediate-scale quantum (NISQ) devices. The study highlights the advantages, challenges, and future prospects of integrating hybrid quantum-classical computation in optimization domains, providing a comprehensive framework and qualitative insights into benchmarking methodologies and critical performance metrics.
This study presents a rigorous empirical evaluation of the Quantum Approximate Optimization Algorithm (QAOA) within a hybrid quantum-classical framework for solving the NP-hard Max-Cut problem. Using Python and Qiskit Aer statevector simulations, we benchmark QAOA (depths p = 2 and p = 3) against established classical algorithms, including simulated annealing, the Goemans–Williamson algorithm, and greedy heuristics, across Erdős–Rényi and regular graphs with 4 to 20 nodes (100 instances per size). Results demonstrate that low-depth QAOA suffers from significant performance degradation, achieving a 0.467 approximation ratio at 20 nodes compared to 0.856 for Goemans–Williamson, alongside a 210× runtime overhead. Convergence analysis further reveals low optimality probabilities and high iteration costs, highlighting limitations of variational optimization in NISQ-era devices. This work provides a reproducible benchmarking framework and contributes to a realistic assessment of near-term quantum advantage.
Quantum technology enters in the new era of Noisy Intermediate-Scale Quantum (NISQ) with 50-100 qubits quantum computers already available. It is now important to compare quantum methods with the classical approach to solve optimization problems in order to determine if there is a potential speed-up and what quantum technologies will be able to perform. The Quantum Approximate optimization Algorithm (QAOA) is a hybrid quantum-classical variational algorithm designed to tackle combinatorial optimization problems. We provide the method to apply QAOA on a shortest path problem and a study of the performance of this quantum method by using heuristic strategies based on observed patterns in optimal parameters. First, we will present the state of the art of quantum hybrid algorithms and technologies. Then we will discuss the effect of the shape of the graph, the number of valid solutions, and the number of nodes of the graph we consider on the quality of the QAOA’s results and on the patterns in optimal variational parameters.