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.
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.
Combinatorial optimization lies at the heart of numerous real-world applications. For a broad category of optimization problems, quantum computing is expected to exhibit quantum speed-up over classic computing. Among various quantum algorithms, the Quantum Approximate Optimization Algorithm (QAOA), as one of variational quantum algorithms, shows promise on demonstrating quantum advantage on noisy intermediate-scale quantum (NISQ) hardware. However, with increasing problem size, the circuit depth demanded by original QAOA scales rapidly and quickly surpasses the threshold at which meaningful results can be obtained. To address this challenge, in this work, we propose a variant of QAOA (termed linear chain QAOA) and demonstrate its advantages over original QAOA on paradigmatic MaxCut problems. In original QAOA, each graph edge is encoded with one entangling gate. In our ansatz, we locate a linear chain from the original MaxCut graph and place entangling gates sequentially along this chain. This linear-chain ansatz is featured by shallow quantum circuits and with the low execution time that scales independently of the problem size. Leveraging this ansatz, we demonstrate an approximation ratio of 0.78 (without post-processing) on non-hardware-native random regular MaxCut instances with 100 vertices in a digital quantum processor using 100 qubits. Our findings offer new insights into the design of hardware-efficient ansatz and point toward a promising route for tackling large-scale combinatorial optimization problems on NISQ devices.
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 max-cut problem (MCP) is a classic problem in the field of combinatorial optimization and has important applications in various fields, including statistical physics and image processing. However, except for some special cases, the MCP still encounters a non-deterministic polynomial complete problem (an NP-complete problem), and there is currently no known efficient classical algorithm that can solve it in polynomial time. The quantum approximate optimization algorithm (QAOA), as a pivotal algorithm in the noisy intermediate-scale quantum (NISQ) computing era, has shown significant potential for solving the MCP. However, due to the lack of quantum error correction, the reliability of computations in NISQ systems sharply declines as the circuit depth of the algorithm increases. Therefore, designing an efficient, shallow-depth, and low-complexity QAOA for the MCP is a critical challenge in demonstrating the advantages of quantum computing in the NISQ era.In this paper, according to the standard QAOA algorithm, we introduce Pauli Y rotation gates into the target Hamiltonian circuit for the MCP. By enhancing the flexibility of quantum trial functions and improving the efficiency of Hilbert space exploration within a single iteration, we significantly improve the performance of QAOA on the MCP.We conduct extensive numerical simulations using the MindSpore quantum platform, and compare the proposed RY-layer-assisted QAOA with standard QAOA and its existing variants, including MA-QAOA and QAOA+. The experiments are performed on various graph types, including complete graphs, 3-regular graphs, 4-regular graphs, and random graphs with edge probabilities between 0.3 and 0.5. Our results show that the RY-layer-assisted QAOA achieves higher approximation ratios in all graph types, particularly in regular and random graphs, where traditional QAOA variants are difficult to implement. Moreover, the proposed method exhibits strong robustness as the graph size increases, and can maintain high performance even for larger graphs. Importantly, the RY-layer-assisted QAOA requires fewer CNOT gates and has a lower circuit depth than the standard QAOA and its variants, making it more suitable for NISQ devices with limited coherence times and high error rates.In conclusion, the RY-layer-assisted QAOA provides a promising approach for solving MCP in the NISQ era. By improving the approximation ratio while reducing circuit complexity, this method demonstrates significant potential for practical quantum computing applications, thus paving the way for developing more efficient and reliable quantum optimization algorithms.
Quantum approximate optimization algorithm (QAOA) is one of the popular quantum algorithms that are used to solve combinatorial optimization problems via approximations. QAOA is able to be evaluated on both physical and virtual quantum computers simulated by classical computers, with virtual ones being favored for their noise-free feature and availability. Nevertheless, performing QAOA on virtual quantum computers suffers from a slow simulation speed for solving combinatorial optimization problems which require large-scale quantum circuit simulation (QCS). In this paper, we propose techniques to accelerate QCS for QAOA using mathematical optimizations to compress quantum operations, incorporating efficient bitwise operations to further lower the computational complexity, and leveraging different levels of parallelisms from modern multi-core processors, with a study case to show the effectiveness on solving max-cut problems. The experimental results reveal substantial performance improvements, surpassing a state-of-the-art simulator, QuEST, by a factor of 17 on a virtual quantum computer running on a 16-core, 32-thread AMD Ryzen 9 5950X processor. We believe that this work opens up new possibilities for accelerating various QAOA applications.
Quantum computer systems require robust software integration to support quantum algorithm development towards addressing practical challenges in many important problem areas including logistics, chemistry, and AI/ML. Researchers predict quantum algorithms will achieve computational advantage over classical algorithms for these specific problem areas in the Noisy Intermediate Scale Quantum (NISQ) era [1]. We consider one such algorithm, the Quantum Approximate Optimization Algorithm (QAOA), a variational quantum algorithm for solving combinatorial optimization problems [2]. QAOA promises to be a practical alternative to inefficient classical algorithms for large problem sizes, but quantum hardware capabilities lag behind the requirements for practical application. We show that the Superstaq quantum compiler can support performance improvements for QAOA by exploiting optimization opportunities only revealed at levels lower than logic gates, i.e., at the level of pulses, quantum hardware instructions.
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.