Improving Quantum Approximate Optimization by Noise-Directed Adaptive Remapping

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.

We compare the performance of the Quantum Approximate Optimization Algorithm (QAOA) with state-of-the-art classical solvers Gurobi and MQLib to solve the MaxCut problem on 3-regular graphs. We identify the minimum noiseless sampling frequency and depth p required for a quantum device to outperform classical algorithms. There is potential for quantum advantage on hundreds of qubits and moderate depth with a sampling frequency of 10 kHz. We observe, however, that classical heuristic solvers are capable of producing high-quality approximate solutions in linear time complexity. In order to match this quality for large graph sizes N, a quantum device must support depth p > 11. Additionally, multi-shot QAOA is not efficient on large graphs, indicating that QAOA p ≤ 11 does not scale with N. These results limit achieving quantum advantage for QAOA MaxCut on 3-regular graphs. Other problems, such as different graphs, weighted MaxCut, and 3-SAT, may be better suited for achieving quantum advantage on near-term quantum devices.

The Max-Cut Problem (MCP) is a classic problem in the field of combinatorial optimization and has important applications in various domains, including statistical physics and image processing. However, except for some special cases, the Max-Cut problem remains 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 Max-Cut problem. 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 Max-Cut problem is a critical challenge in demonstrating the advantages of quantum computing in the NISQ era.<br>In this paper, based on the standard QAOA algorithm, we introduce Pauli Y rotation gates into the target Hamiltonian circuit for the Max-Cut problem. 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 Max-Cut problem.<br>We conduct extensive numerical simulations using the MindSpore Quantum platform, comparing 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 demonstrate that the RY-layer-assisted QAOA achieves a higher approximation ratio across all graph types, particularly in regular and random graphs, where traditional QAOA variants struggle. Moreover, the proposed method exhibits strong robustness as the graph size increases, maintaining high performance even for larger graphs. Importantly, the RY-layer-assisted QAOA requires fewer CNOT gates and has a lower circuit depth compared to standard QAOA and its variants, making it more suitable for NISQ devices with limited coherence times and high error rates.<br>In conclusion, the RY-layer-assisted QAOA offers a promising approach to solving Max-Cut problems in the NISQ era. By improving the approximation ratio while reducing circuit complexity, this method demonstrates significant potential for practical quantum computing applications, paving the way for more efficient and reliable quantum optimization algorithms.

Feature selection is of great importance in Machine Learning, where it can be used to reduce the dimensionality of classification, ranking and prediction problems. The removal of redundant and noisy features can improve both the accuracy and scalability of the trained models. However, feature selection is a computationally expensive task with a solution space that grows combinatorically. In this work, we consider in particular a quadratic feature selection problem that can be tackled with the Quantum Approximate Optimization Algorithm (QAOA), already employed in combinatorial optimization. First we represent the feature selection problem with the QUBO formulation, which is then mapped to an Ising spin Hamiltonian. Then we apply QAOA with the goal of finding the ground state of this Hamiltonian, which corresponds to the optimal selection of features. In our experiments, we consider seven different real-world datasets with dimensionality up to 21 and run QAOA on both a quantum simulator and, for small datasets, the 7-qubit IBM (ibm-perth) quantum computer. We use the set of selected features to train a classification model and evaluate its accuracy. Our analysis shows that it is possible to tackle the feature selection problem with QAOA and that currently available quantum devices can be used effectively. Future studies could test a wider range of classification models as well as improve the effectiveness of QAOA by exploring better performing optimizers for its classical step.

The Quantum Approximate Optimization Algorithm (QAOA) is a variational quantum algorithm for Near-term Intermediate-Scale Quantum computers (NISQ) providing approximate solutions for combinatorial optimization problems. The QAOA utilizes a quantum-classical loop, consisting of a quantum ansatz and a classical optimizer, to minimize some cost function, computed on the quantum device. This paper presents an investigation into the impact of realistic noise on the classical optimizer and the determination of optimal circuit depth for the Quantum Approximate Optimization Algorithm (QAOA) in the presence of noise. We find that, while there is no significant difference in the performance of classical optimizers in a state vector simulation, the Adam and AMSGrad optimizers perform best in the presence of shot noise. Under the conditions of real noise, the SPSA optimizer, along with ADAM and AMSGrad, emerge as the top performers. The study also reveals that the quality of solutions to some 5 qubit minimum vertex cover problems increases for up to around six layers in the QAOA circuit, after which it begins to decline. This analysis shows that increasing the number of layers in the QAOA in an attempt to increase accuracy may not work well in a noisy device.

Motivated by the recent advancement of quantum processors, we investigate quantum approximate optimization algorithm (QAOA) to employ quasi-maximum-likelihood (ML) decoding of classical channel codes. QAOA is a hybrid quantum-classical variational algorithm, which is advantageous for the near-term noisy intermediate-scale quantum (NISQ) devices, where the fidelity of quantum gates is limited by noise and de-coherence. We first describe how to construct Ising Hamiltonian model to realize quasi-ML decoding with QAOA. For level-1 QAOA, we derive the systematic way to generate theoretical expressions of cost expectation for arbitrary binary linear codes. Focusing on [7], [4] Hamming code as an example, we analyze the impact of the degree distribution in associated generator matrix on the quantum decoding performance. The excellent performance of higher-level QAOA decoding is verified when Pauli rotation angles are optimized through meta-heuristic variational quantum eigensolver (VQE). Furthermore, we demonstrate the QAOA decoding performance in a real quantum device.

Recently, Hadfield et al. proposed the quantum alternating operator ansatz algorithm (QAOA+), an extension of the quantum approximate optimization algorithm (QAOA), to solve constrained combinatorial optimization problems (CCOPs). Compared with QAOA, QAOA+ enables the search for optimal solutions within a feasible solution space by encoding problem constraints into the mixer Hamiltonian, thereby reducing the search space and eliminating the possibility of yielding infeasible solutions. However, QAOA+ may incur high overall gate costs when the mixer is applied to all qubits in each layer, and each mixer is costly to implement. To address this challenge, an adaptive mixer allocation strategy is tailored for QAOA+. The resulting algorithm, which integrates this strategy into the original QAOA+ framework, is referred to as AMA‐QAOA+. Unlike QAOA+, AMA‐QAOA+ adaptively applies the mixer to a subset of qubits in each layer of the mixer unitary operator based on an evaluation function. The performance of AMA‐QAOA+ is evaluated on the maximum independent set problem. Numerical simulation results show that, under the same number of optimization runs, AMA‐QAOA+ achieves better solution quality than QAOA+, with the optimal approximation ratio improved by on Erdős–Rényi random graphs and on 3‐regular graphs. Moreover, AMA‐QAOA+ significantly reduces the CNOT gate consumption, requiring only and of the CNOT gates used by QAOA+ on Erdős–Rényi and 3‐regular random graphs, respectively. These results demonstrate that AMA‐QAOA+ enhances solution quality and computational efficiency, enabling the design of more compact and resource‐efficient quantum circuits.

The quantum approximate optimization algorithm (QAOA) promises to solve classically intractable computational problems in the area of combinatorial optimization. A growing amount of evidence suggests that the originally proposed form of the QAOA ansatz is not optimal, however. To address this problem, we propose an alternative ansatz, which we call QAOA+, that augments the traditional p=1 QAOA ansatz with an additional multiparameter problem-independent layer. The QAOA+ ansatz allows obtaining higher approximation ratios than p=1 QAOA while keeping the circuit depth below that of p = 2 QAOA, as benchmarked on the MaxCut problem for random regular graphs. We additionally show that the proposed QAOA+ ansatz, while using a larger number of trainable classical parameters than with the standard QAOA, in most cases outperforms alternative multiangle QAOA ansätze for fixed number of independent parameters.

The quantum approximate optimization algorithm (QAOA) is a method of approximately solving combinatorial optimization problems. While QAOA is developed to solve a broad class of combinatorial optimization problems, it is not clear which classes of problems are best suited for it. One factor in demonstrating quantum advantage is the relationship between a problem instance and the circuit depth required to implement the QAOA method. As errors in noisy intermediate-scale quantum (NISQ) devices increase exponentially with circuit depth, identifying lower bounds on circuit depth can provide insights into when quantum advantage could be feasible. Here, we identify how the structure of problem instances can be used to identify lower bounds for circuit depth for each iteration of QAOA and examine the relationship between problem structure and the circuit depth for a variety of combinatorial optimization problems including MaxCut and MaxIndSet. Specifically, we show how to derive a graph, G, that describes a general combinatorial optimization problem and show that the depth of circuit is at least the chromatic index of G. By looking at the scaling of circuit depth, we argue that MaxCut, MaxIndSet, and some instances of vertex covering and Boolean satisfiability problems are suitable for QAOA approaches while knapsack and traveling salesperson problems are not.

Quantum approximate optimization algorithm (QAOA) is a promising quantum-classical hybrid technique to solve NP-hard problems in near-term gate-based noisy quantum devices. In QAOA, the gate parameters of a parameterized quantum circuit (PQC) are varied by a classical optimizer to generate a quantum state with a significant support to the optimal solution. The existing analysis fails to consider nonidealities in the qubit quality i.e., short lifetime and imperfect gate operations in a realistic quantum hardware. In this article, we study the impact of various noise sources on the performance of QAOA both in simulation and on a real quantum computer from IBM. Our analysis indicates that QAOA performance is noise-sensitive (especially higher-depth QAOA instances). Therefore, the optimal number of stages (p-value) for any QAOA instance is limited by the noise in the target hardware as opposed to the current perception that QAOA will provide monotonically better performance with higher-depth. We show that the two-qubit gate error has to be decreased by more than 75% of the current state-of-the-art levels to attain a performance within 10% of the maximum value for the lowest-depth QAOA.

Although quantum approximate optimization algorithm (QAOA) has demonstrated its quantum supremacy, its performance on Noisy Intermediate-Scale Quantum (NISQ) devices would be influenced by complicated noises, e.g. quantum colored noises. To evaluate the performance of QAOA under these noises, this paper presents a framework for running QAOA on non-Markovian quantum systems which are represented by an augmented system model. In this model, a non-Markovian environment carrying quantum colored noises is modelled as an ancillary system driven by quantum white noises which is directly coupled to the corresponding principal system; i.e. the computational unit for the algorithm. With this model, we mathematically formulate QAOA as piecewise Hamiltonian control of the augmented system, where we also optimize the control depth to fit into the circuit depth of current quantum devices. For efficient simulation of QAOA in non-Markovian quantum systems, a boosted algorithm using quantum trajectory is further presented. Finally, we show that non-Markovianity can be utilized as a quantum resource to achieve a relatively good performance of QAOA, which is characterized by our proposed exploration rate.

Quantum approximate optimization algorithm (QAOA) is a quantum–classical hybrid algorithm proposed with the goal of approximately solving combinatorial optimization problems such as the MAX-CUT problem. It has been considered a potential candidate for achieving quantum advantage in the noisy intermediate-scale quantum era and has been extensively studied. However, the performance limitations of low-level QAOA have also been demonstrated across various instances. In this work, we first analytically prove the performance limitations of the standard level-1 QAOA in solving the MAX-CUT problem on bipartite graphs. To this end, we derive an upper bound for the approximation ratio based on the average degree of bipartite graphs. Second, we demonstrate that recursive QAOA (RQAOA), which recursively reduces graph size using QAOA as a subroutine, outperforms the standard level-1 QAOA. However, the performance of RQAOA exhibits limitations as the graph size increases. Finally, we show that RQAOA with a restricted parameter regime can fully address these limitations. Surprisingly, this modified RQAOA always finds the exact maximum cut for any bipartite graphs and even for a more general graph with parity-signed weights.

Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate algorithm for solving combinatorial optimization problems on quantum computers. However, in many cases QAOA requires computationally intensive parameter optimization. The challenge of parameter optimization is particularly acute in the case of weighted problems, for which the eigenvalues of the phase operator are non-integer and the QAOA energy landscape is not periodic. In this work, we develop parameter setting heuristics for QAOA applied to a general class of weighted problems. First, we derive optimal parameters for QAOA with depth p=1 applied to the weighted MaxCut problem under different assumptions on the weights. In particular, we rigorously prove the conventional wisdom that in the average case the first local optimum near zero gives globally-optimal QAOA parameters. Second, for p≥1 we prove that the QAOA energy landscape for weighted MaxCut approaches that for the unweighted case under a simple rescaling of parameters. Therefore, we can use parameters previously obtained for unweighted MaxCut for weighted problems. Finally, we prove that for p=1 the QAOA objective sharply concentrates around its expectation, which means that our parameter setting rules hold with high probability for a random weighted instance. We numerically validate this approach on general weighted graphs and show that on average the QAOA energy with the proposed fixed parameters is only 1.1 percentage points away from that with optimized parameters. Third, we propose a general heuristic rescaling scheme inspired by the analytical results for weighted MaxCut and demonstrate its effectiveness using QAOA with the XY Hamming-weight-preserving mixer applied to the portfolio optimization problem. Our heuristic improves the convergence of local optimizers, reducing the number of iterations by 7.4x on average.

Recent results on the Quantum Approximate Optimization Algorithm (QAOA) have cast pessimism on its potential to exhibit practical quantum speedups. For instance, QAOA’s locality limits its performance on tasks such as coloring bipartite graphs—which is easy for classical methods. Motivated by these limitations, the Recursive QAOA was introduced to overcome the locality and symmetry of QAOA. Despite being more powerful than QAOA, RQAOA is fully classically simulable at level-1 depth (p = 1). We report results on RQAOA in this classically simulable regime, benchmarked on random Quantum Unconstrainted Binary Optimization (QUBO) problems with up to 100 variables. We find that RQAOA generally matches the performance of classical simulated annealing and significantly outperforms ordinary QAOA.

The quantum approximate optimization algorithm (QAOA) generates an approximate solution to combinatorial optimization problems using a variational ansatz circuit defined by parameterized layers of quantum evolution. In theory, the approximation improves with increasing ansatz depth but gate noise and circuit complexity undermine performance in practice. Here, we investigate a multi-angle ansatz for QAOA that reduces circuit depth and improves the approximation ratio by increasing the number of classical parameters. Even though the number of parameters increases, our results indicate that good parameters can be found in polynomial time for a test dataset we consider. This new ansatz gives a 33% increase in the approximation ratio for an infinite family of MaxCut instances over QAOA. The optimal performance is lower bounded by the conventional ansatz, and we present empirical results for graphs on eight vertices that one layer of the multi-angle anstaz is comparable to three layers of the traditional ansatz on MaxCut problems. Similarly, multi-angle QAOA yields a higher approximation ratio than QAOA at the same depth on a collection of MaxCut instances on fifty and one-hundred vertex graphs. Many of the optimized parameters are found to be zero, so their associated gates can be removed from the circuit, further decreasing the circuit depth. These results indicate that multi-angle QAOA requires shallower circuits to solve problems than QAOA, making it more viable for near-term intermediate-scale quantum devices.