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Parameterized quantum circuit defined by a vector θ = (θ 1, …, θ m ) of m parameters, here for m = 19. (a) For a number p of circuit layers the block of gates inside the dashed rectangle is repeated p times. Here we show the circuit for p = 1. The F-VQE algorithm samples the entire circuit to evaluate a global observable indicated by the blue rectangle. (b) and (c) Highlighted qubits and gates constitute the causal cone that HE-ITE uses to evaluate two-local observables on two neighboring qubits and two non-neighboring qubits, respectively.
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Current gate-based quantum computers have the potential to provide a computational advantage if algorithms use quantum hardware efficiently. To make combinatorial optimization more efficient, we introduce the Filtering Variational Quantum Eigensolver (F-VQE) which utilizes filtering operators to achieve faster and more reliable convergence to the o...
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Citations
... Additionally, the multi-angle QAOA demonstrated superior approximation ratios compared to QAOA at equivalent depths across a set of MaxCut instances on fifty and one-hundred vertex graphs. Amaro et al. [62] introduced the filtering variational quantum eigensolver which utilized filtering operators to attain accelerated and more dependable convergence toward the optimal solution. Through numerical evaluations using randomly weighted MaxCut instances, they demonstrated the superior performance of filtering variational quantum eigensolver compared to both the original VQE and QAOA. ...
Quantum computing is a new discipline combining quantum mechanics and computer science, which is expected to solve technical problems that are difficult for classical computers to solve efficiently. At present, quantum algorithms and hardware continue to develop at a high speed, but due to the serious constraints of quantum devices, such as the limited numbers of qubits and circuit depth, the fault-tolerant quantum computing will not be available in the near future. Variational quantum algorithms(VQAs) using classical optimizers to train parameterized quantum circuits have emerged as the main strategy to address these constraints. However, VQAs still have many challenges, such as trainability, hardware noise, expressibility and entangling capability. The fundamental concepts and applications of VQAs are reviewed. Then, strategies are introduced to overcome the challenges of VQAs and the importance of further researching VQAs is highlighted.
... Among them, the quantum approximate optimization algorithm (QAOA) [8] is a variational algorithm transforming the QUBO problem into finding the ground-state energy of a spin system, employing an ansatz alternatively implementing two unitaries. Another algorithm is based on a variational simulation of a filtering operation that amplifies the amplitudes of low energy eigenstates [9]. By applying the filtering operation, the input quantum state can evolve to the ground state with a much more shallow circuit than QAOA. ...
Quadratically Constrained Quadratic Programs (QCQPs) are an important class of optimization problems with diverse real-world applications. In this work, we propose a variational quantum algorithm for general QCQPs. By encoding the variables in the amplitude of a quantum state, the requirement for the qubit number scales logarithmically with the dimension of the variables, which makes our algorithm suitable for current quantum devices. Using the primal-dual interior-point method in classical optimization, we can deal with general quadratic constraints. Our numerical experiments on typical QCQP problems, including Max-Cut and optimal power flow problems, demonstrate better performance of our hybrid algorithm over classical counterparts.
... Firstly, the original QAOA ansatz is inefficient and hardware-unfriendly for dense models. Recent studies have introduced new modified ansätze for QAOA that are more hardware-friendly [27,28]. However, the effectiveness of these ansätze for fully connected models has yet to be thoroughly investigated. ...
The formation of energy communities is pivotal for advancing decentralized and sustainable energy management. Within this context, Coalition Structure Generation (CSG) emerges as a promising framework. The complexity of CSG grows rapidly with the number of agents, making classical solvers impractical for even moderate sizes (number of agents>30). Therefore, the development of advanced computational methods is essential. Motivated by this challenge, this study conducts a benchmark comparing classical solvers with quantum annealing on Dwave hardware and the Quantum Approximation Optimization Algorithm (QAOA) on both simulator and IBMQ hardware to address energy community formation. Our classical solvers include Tabu search, simulated annealing, and an exact classical solver. Our findings reveal that Dwave surpasses QAOA on hardware in terms of solution quality. Remarkably, QAOA demonstrates comparable runtime scaling with Dwave, albeit with a significantly larger prefactor. Notably, Dwave exhibits competitive performance compared to the classical solvers, achieving solutions of equal quality with more favorable runtime scaling.
... As a result, considerable effort has been spent developing methods for these VQAs, and studying how different approaches affect performance across different classes of problems. For example, ore sophisticated variants have been proposed, such as filtered VQE, warm-start QAOA, and recursive QAOA [15]- [17]. ...
... Firstly, we introduce the concept of variational time evolution into the flow of VQE methods [18] as a way to simulate the parameter optimization process and propose the concept of Proposed dynamic VQE (D-VQE). Secondly, with the introduction of causal cones [19], we use new judgment rules to construct dynamically tunable PQC ansatz. Thirdly, we use the simulated dynamics strategy to address the difficulty of solving for thermodynamic averages in Boltzmann machine learning [20]. ...
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... As quantum processors control delicate quantum states, they are necessarily complex and physical access to high-performance systems is limited. Cloud-based approaches, where users can remotely access quantum servers, are likely to be the working model in the near term and beyond; many users already perform computations on commercially available devices for state-of-the-art research [1][2][3][4][5]. ...
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... Other QAOAs. There are plenty of promising QAOA and VQE-inspired algorithms in the literature [15,[46][47][48][49][50][51][52][53][54][55]. However, this work focuses on ADAPT-QAOAs [20]-mainly due to their relatively shallow ansatz circuits. ...
The quantum approximate optimization algorithm (QAOA) is an appealing proposal to solve NP problems on noisy intermediate-scale quantum (NISQ) hardware. Making NISQ implementations of the QAOA resilient to noise requires short ansatz circuits with as few controlled-not (cnot) gates as possible. Here we present the dynamic adaptive quantum approximate optimization algorithm (Dynamic-ADAPT-QAOA). Our algorithm significantly reduces the circuit depth and the cnot count of standard ADAPT-QAOA, a leading proposal for near-term implementations of the QAOA. Throughout our algorithm, the decision to apply cnot-intensive operations is made dynamically, based on algorithmic benefits. Using density-matrix simulations, we benchmark the noise resilience of ADAPT-QAOA and Dynamic-ADAPT-QAOA. We compute the gate-error probability pgate★ below which these algorithms provide, on average, more accurate solutions than the classical, polynomial-time approximation algorithm by Goemans and Williamson. For small systems with six to ten qubits, we show that pgate★>10−3 for Dynamic-ADAPT-QAOA. Compared to standard ADAPT-QAOA, this constitutes an order-of-magnitude improvement in noise resilience. This improvement should make Dynamic-ADAPT-QAOA viable for implementations on superconducting NISQ hardware, even in the absence of error mitigation.
... In order to solve the QUBO with the gate-based approach of quantum computing, we use the QAOA-and the VQEansatz and compare them. This was inspired by [11]. In addition to using the out-of-the-box QAOA algorithm of IBM, we compare the results with a customised solution scheme for QAOA. ...
The problem of selecting an appropriate number of features in supervised learning problems is investigated. Starting with common methods in machine learning, the feature selection task is treated as a quadratic unconstrained optimisation problem (QUBO), which can be tackled with classical numerical methods as well as within a quantum computing framework. The different results in small problem instances are compared. According to the results of the authors’ study, whether the QUBO method outperforms other feature selection methods depends on the data set. In an extension to a larger data set with 27 features, the authors compare the convergence behaviour of the QUBO methods via quantum computing with classical stochastic optimisation methods. Due to persisting error rates, the classical stochastic optimisation methods are still superior.
... Since its introduction by Farhi et al. [1] in 2014, the quantum approximate optimization algorithm (QAOA) has been explored in the quantum computing literature as one of the most promising heuristics for achieving quantum advantage on near-term devices [2,3]. This is only one example of a larger class of variational quantum optimization algorithms, which attempt to produce good solutions to combinatorial optimization problems by sampling a parametrized quantum circuit [4][5][6][7][8]. In the absence of full quantum error correction [9], the required circuits must be sufficiently shallow to withstand noise, yet expressive enough to find states with high overlap onto the ground state. ...
We embed 1-layer QAOA circuits into the larger class of parametrized instantaneous quantum polynomial circuits to produce an improved variational quantum algorithm for solving combinatorial optimization problems. The use of analytic expressions to find optimal parameters classically makes our protocol robust against barren plateaus and hardware noise. The average overlap with the ground state scales as 2−0.31(2)N with the number of qubits N for random Sherrington-Kirkpatrick (SK) Hamiltonians of up to 29 qubits, a polynomial improvement over 1-layer QAOA. Additionally, we observe that performing variational imaginary time evolution on the manifold approximates low-temperature pseudo-Boltzmann states. Our protocol outperforms 1-layer QAOA on the recently released Quantinuum H2 trapped-ion quantum hardware and emulator, where we obtain an average approximation ratio of 0.985 across 312 random SK instances of 7 to 32 qubits, from which almost 44% are solved optimally using only 4 to 1208 shots per instance.
... In this section, we examine how the WS-QAOA performance varies with choice of approximate solutions {x 0 i } . For that purpose, we numerically study the MAX-CUT problem on weighted 3-regular (w3R) graphs 13,26 . In w3R graphs, each vertex is connected to three others chosen at random, and each edge has weight w ij randomly set from [0, 1). ...
Quantum approximate optimization algorithm (QAOA) is a promising hybrid quantum-classical algorithm to solve combinatorial optimization problems in the era of noisy intermediate-scale quantum computers. Recently it has been revealed that warm-start approaches can improve the performance of QAOA, where approximate solutions are obtained by classical algorithms in advance and incorporated into the initial state and/or unitary ansatz. In this work, we study in detail how the accuracy of approximate solutions affects the performance of the warm-start QAOA (WS-QAOA). We numerically find that in typical MAX-CUT problems, WS-QAOA achieves higher fidelity (probability that exact solutions are observed) and approximation ratio than QAOA as the Hamming distance of approximate solutions to the exact ones becomes smaller. We reveal that this could be quantitatively attributed to the initial state of the ansatz. We also solve MAX-CUT problems by WS-QAOA with approximate solutions obtained via QAOA, having higher fidelity and approximation ratio than QAOA especially when the circuit is relatively shallow. We believe that our study may deepen understanding of the performance of WS-QAOA and also provide a guide as to the necessary quality of approximate solutions.
