Quantum computing holds promise for outperforming classical computing in specialized applications such as optimization. With current Noisy Intermediate Scale Quantum (NISQ) devices, only variational quantum algorithms like the Quantum Alternating Operator Ansatz (QAOA) can be practically run. QAOA is effective for solving Quadratic Unconstrained Binary Optimization (QUBO) problems by approximating Quantum Annealing via Trotterization. Successful implementation on NISQ devices requires shallow circuits, influenced by the number of variables and the sparsity of the augmented interaction matrix. This paper investigates the necessary sparsity levels for augmented interaction matrices to ensure solvability with QAOA. By analyzing the Max-Cut problem with varying sparsity, we provide insights into how the Hamiltonian density affects the QAOA performance. Our findings highlight that, while denser matrices complicate the energy landscape, the performance of QAOA remains largely unaffected by sparsity variations. This study emphasizes the algorithm's robustness and potential for optimization tasks on near-term quantum devices, suggesting avenues for future research in enhancing QAOA for practical applications.
The classical homotopy optimization approach has the potential to deal with highly nonlinear landscape, such as the energy landscape of QAOA problems. Following this motivation, we introduce Hamiltonian-Oriented Homotopy QAOA (HOHo-QAOA), that is a heuristic method for combinatorial optimization using QAOA, based on classical homotopy optimization. The method consists of a homotopy map that produces an optimization problem for each value of interpolating parameter. Therefore, HOHo-QAOA decomposes the optimization of QAOA into several loops, each using a mixture of the mixer and the objective H…
We report on the energy-expectation-value landscapes produced by the single-layer ($p=1$) Quantum Approximate Optimization Algorithm (QAOA) when being used to solve Ising problems. The landscapes are obtained using an analytical formula that we derive. The formula allows us to predict the landscape for any given Ising problem instance and consequently predict the optimal QAOA parameters for heuristically solving that instance using the single-layer QAOA. We have validated our analytical formula by showing that it accurately reproduces the landscapes published in recent experimental reports. We…
Hybrid quantum-classical algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) are considered as one of the most encouraging approaches for taking advantage of near-term quantum computers in practical applications. Such algorithms are usually implemented in a variational form, combining a classical optimization method with a quantum machine to find good solutions to an optimization problem. The solution quality of QAOA depends to a high degree on the parameters chosen by the classical optimizer at each iteration. However, the solution landscape of those parameters is highly …
The quantum approximate optimization algorithm (QAOA) is a near-term hybrid algorithm intended to solve combinatorial optimization problems, such as MaxCut. QAOA can be made to mimic an adiabatic schedule, and in the $p\to\infty$ limit the final state is an exact maximal eigenstate in accordance with the adiabatic theorem. In this work, the connection between QAOA and adiabaticity is made explicit by inspecting the regime of $p$ large but finite. By connecting QAOA to counterdiabatic (CD) evolution, we construct CD-QAOA angles which mimic a counterdiabatic schedule by matching Trotter "error" …
Improving the efficiency of QAOA using efficient parameter transfer initialization and targeted-single-layer regularized optimization with minimal performance degradation
Department of Physics & Astrophysics, University of Delhi（德里大学物理与天体物理学系）
Physically motivated classical heuristic optimization algorithms such as simulated annealing (SA) treat the objective function as an energy landscape, and allow walkers to escape local minima. It has been argued that quantum properties such as tunneling may give quantum algorithms advantage in finding ground states of vast, rugged cost landscapes. Indeed, the Quantum Adiabatic Algorithm (QAO) and the recent Quantum Approximate Optimization Algorithm (QAOA) have shown promising results on various problem instances that are considered classically hard. Here, we argue that the type of control str…
The quantum approximate optimisation algorithm (QAOA) was partially inspired by digitising quantum annealing. Based on this inspiration, we develop techniques to use the additional flexibility of a universal gate-model quantum computer to mitigate fluctuation effects which are known to distort the search space within quantum annealing and lead to false minima. We find that even just the added ability to take Pauli X measurements allows us to modify the mixer angles to counteract these effects by scaling mixer terms in a way proportional to the diagonal elements of the Fubini-Study metric. We f…
We employ spectral analysis and compressed sensing to identify settings where a variational algorithm's cost function can be recovered purely classically or with minimal quantum computer access. We present theoretical and numerical evidence supporting the viability of sparse recovery techniques. To demonstrate this approach, we use basis pursuit denoising to efficiently recover simulated Quantum Approximate Optimization Algorithm (QAOA) instances of large system size from very few samples. Our results indicate that sparse recovery can enable a more efficient use and distribution of quantum res…
The quantum approximate optimization algorithm (QAOA) is a prospective near-term quantum algorithm due to its modest circuit depth and promising benchmarks. However, an external parameter optimization required in QAOA could become a performance bottleneck. This motivates studies of the optimization landscape and search for heuristic ways of parameter initialization. In this work we visualize the optimization landscape of the QAOA applied to the MaxCut problem on random graphs, demonstrating that random initialization of the QAOA is prone to converging to local minima with sub-optimal performan…
Farhi et al. recently proposed a class of quantum algorithms, the Quantum Approximate Optimization Algorithm (QAOA), for approximately solving combinatorial optimization problems. A level-p QAOA circuit consists of p steps; in each step a classical Hamiltonian, derived from the cost function, is applied followed by a mixing Hamiltonian. The 2p times for which these two Hamiltonians are applied are the parameters of the algorithm, which are to be optimized classically for the best performance. As p increases, parameter optimization becomes inefficient due to the curse of dimensionality. The suc…
Emergent Problem-Graph Alignment in RL-Discovered Entanglement Topologies for QAOA
Institute for Computer Science, LMU Munich（慕尼黑大学计算机科学研究所）；relAI – Konrad Zuse School of Excellence in Reliable AI, LMU Munich（relAI – 康拉德·楚泽可靠人工智能卓越学院，慕尼黑大学）；Department of Computer Science, University of Exeter（埃克塞特大学计算机系）
Quantum Approximate Optimization Algorithm (QAOA) is a hybrid classical-quantum algorithm to approximately solve NP optimization problems such as MAX-CUT. We describe a new application area of QAOA circuits: graph structure discovery. We omit the time-consuming parameter-optimization phase and utilize the dependence of QAOA energy on the graph structure for randomly or judiciously chosen parameters to learn about graphs. In the first part, Following up on Wang et. al. and Brandao et. al. we give explicit formulas. We show that the layer-one QAOA energy for the MAX-CUT problem for three regular…
SISSA（国际高等研究学院）；Universität Innsbruck（因斯布鲁克大学）；Bocconi University（博科尼大学）；International Centre for Theoretical Physics (ICTP)（国际理论物理中心）；CNR-IOM, Consiglio Nazionale delle Ricerche - Istituto Officina dei Materiali（意大利国家研究委员会-材料研究所）
We apply digitized Quantum Annealing (QA) and Quantum Approximate Optimization Algorithm (QAOA) to a paradigmatic task of supervised learning in artificial neural networks: the optimization of synaptic weights for the binary perceptron. At variance with the usual QAOA applications to MaxCut, or to quantum spin-chains ground state preparation, the classical Hamiltonian is characterized by highly non-local multi-spin interactions. Yet, we provide evidence for the existence of optimal smooth solutions for the QAOA parameters, which are transferable among typical instances of the same problem, and…
Energy Landscape Structure of Small Graph Isomorphism Under Variational Optimization
Calcutta Institute of Engineering and Management, Maulana Abul Kalam Azad University of Technology（加尔各答工程与管理学院，穆拉纳·阿布尔·卡拉姆·阿扎德技术大学）；University of Dhaka（达卡大学）；Institute of Theoretical and Applied Informatics, Polish Academy of Sciences（波兰科学院理论与应用信息研究所）；Silesian University of Technology（西里西亚理工大学）
Constrained Portfolio Optimization via Quantum Approximate Optimization Algorithm (QAOA) with XY-Mixers and Trotterized Initialization: A Hybrid Approach for Direct Indexing
SquareOne Capital（SquareOne资本）；School of Physics, Faculty of Sciences, Aristotle University of Thessaloniki（塞萨洛尼基亚里士多德大学理学院物理系）
Quantum variational algorithms have garnered significant interest recently, due to their feasibility of being implemented and tested on noisy intermediate scale quantum (NISQ) devices. We examine the robustness of the quantum approximate optimization algorithm (QAOA), which can be used to solve certain quantum control problems, state preparation problems, and combinatorial optimization problems. We demonstrate that the error of QAOA simulation can be significantly reduced by robust control optimization techniques, specifically, by sequential convex programming (SCP), to ensure error suppressio…
Constraint-Aware Quantum Optimization via Hamming Weight Operators
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China（电子科技大学基础与前沿科学学院）；Key Laboratory of Quantum Physics and Photonic Quantum Information, Ministry of Education, University of Electronic Science and Technology of China（电子科技大学量子物理与光子量子信息教育部重点实验室）；Center on Frontiers of Computing Studies, Peking University（北京大学前沿计算研究中心）；School of Computer Science, Peking University（北京大学计算机学院）
The Quantum Approximate Optimization Algorithm, QAOA, uses a shallow depth quantum circuit to produce a parameter dependent state. For a given combinatorial optimization problem instance, the quantum expectation of the associated cost function is the parameter dependent objective function of the QAOA. We demonstrate that if the parameters are fixed and the instance comes from a reasonable distribution then the objective function value is concentrated in the sense that typical instances have (nearly) the same value of the objective function. This applies not just for optimal parameters as the w…
CV-QAOA: Efficient Low-Depth Quantum Optimization of Continuous Variables
Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy, University of California, Los Angeles（加州大学洛杉矶分校曼尼·L·巴乌米克理论物理研究所）；Electrical and Computer Engineering Department, University of California, Los Angeles（加州大学洛杉矶分校电气与计算机工程系）；Department of Physics, Tsinghua University（清华大学物理系）；Institute of Advanced Study, Tsinghua University（清华大学高等研究院）
We demonstrate the application of the Google Sycamore superconducting qubit quantum processor to combinatorial optimization problems with the quantum approximate optimization algorithm (QAOA). Like past QAOA experiments, we study performance for problems defined on the (planar) connectivity graph of our hardware; however, we also apply the QAOA to the Sherrington-Kirkpatrick model and MaxCut, both high dimensional graph problems for which the QAOA requires significant compilation. Experimental scans of the QAOA energy landscape show good agreement with theory across even the largest instances …
Investigating layer-selective transfer learning of QAOA parameters for Max-Cut problem
Department of Engineering, University Pompeu Fabra（工程系，庞贝法布拉大学）；Department of Physics and Astronomy, University of Florence（物理与天文学系，佛罗伦萨大学）；QUANTIC, Barcelona Supercomputing Center（QUANTIC，巴塞罗那超级计算中心）；Institute for Cross-Disciplinary Physics and Complex Systems (IFISC) UIB-CSIC（跨学科物理与复杂系统研究所（IFISC） UIB-CSIC）；European Laboratory for Non-Linear Spectroscopy (LENS), University of Florence（非线性光谱欧洲实验室（LENS），佛罗伦萨大学）
When is global evolutionary search useful for variational quantum algorithms? A landscape-first study
VSB - Technical University of Ostrava（俄斯特拉发理工大学）；IT4Innovations National Supercomputing Center, VSB - Technical University of Ostrava（俄斯特拉发理工大学 IT4创新国家超算中心）；Klaipeda University（克莱佩达大学）
PACE-QAOA: Physics-Constrained Quantum Optimization for Qubit-Efficient Power System Islanding
Pennsylvania State University（宾夕法尼亚州立大学）；Rensselaer Polytechnic Institute（伦斯勒理工学院）；Kent State University（肯特州立大学）；Clemson University（克莱姆森大学）；University of Pretoria（比勒陀利亚大学）
Adaptive Differential Evolution and Multistart Search for Noisy QAOA Optimization
VSB–Technical University of Ostrava（俄斯特拉发科技大学）；IT4Innovations National Supercomputing Center, VSB–Technical University of Ostrava（俄斯特拉发科技大学 IT4创新国家超级计算中心）；Marine Research Institute, Klaipeda University（克莱佩达大学海洋研究所）；Indian Statistical Institute（印度统计研究所）；Università dell’Aquila（阿奎拉大学）
