Iteration-Free quantum approximate optimization algorithm using neural networks

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 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.

The parity mapping provides a geometrically local encoding of the Quantum Approximate Optimization Algorithm (QAOA), at the expense of having a quadratic qubit overhead for all-to-all connected problems. In this work, we benchmark the parity-encoded QAOA on spin-glass models. We address open questions in the scaling of this algorithm. In particular, we show that for fixed number of parity-encoded QAOA layers, the performance or the output energy, vanishes towards zero (the value achieved by random guessing) with problem size N as N−1/2. Our results suggest that the parity-encoded QAOA does not have a promising scaling compared to the standard version of QAOA. We perform tensor-network calculations to confirm our results, and comment on the concentration of optimal QAOA parameters over problem instances.

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 performance. We introduce the initialization of QAOA parameters based on the Trotterized quantum annealing (TQA) protocol, parameterized by the Trotter time step. We find that the TQA initialization allows to circumvent the issue of false minima for a broad range of time steps, yielding the same performance as the best result out of an exponentially scaling number of random initializations. Moreover, we demonstrate that the optimal value of the time step coincides with the point of proliferation of Trotter errors in quantum annealing. Our results suggest practical ways of initializing QAOA protocols on near-term quantum devices and reveals new connections between QAOA and quantum annealing.

Despite its popularity, several empirical and theoretical studies suggest that the quantum approximate optimization algorithm (QAOA) has persistent issues in providing a substantial practical advantage. Numerical results for few qubits and shallow circuits are, at best, ambiguous, and the well-studied barren plateau phenomenon draws a rather sobering picture for deeper circuits. However, as more and more sophisticated strategies are proposed to circumvent barren plateaus, it stands to reason which issues are actually fundamental and which merely constitute – admittedly difficult – engineering tasks. By shifting the scope from the usually considered parameter landscape to the quantum state space's geometry we can distinguish between problems that are fundamentally difficult to solve, independently of the parameterization, and those for which there could at least exist a favorable parameterization. Here, we find clear evidence for a 'no free lunch'-behavior of QAOA on a general optimization task with no further structure; individual cases have, however, to be analyzed more carefully. Based on our analysis, we propose and justify a performance indicator for the deep-circuit QAOA that can be accessed by solely evaluating statistical properties of the classical objective function. We further discuss the various favorable properties a generic QAOA instance has in the asymptotic regime of infinitely many gates, and elaborate on the immanent drawbacks of finite circuits. We provide several numerical examples of a deep-circuit QAOA method based on local search strategies and find that – in alignment with our performance indicator – some special function classes, like QUBOs, indeed admit a favorable optimization landscape.

The next few years will be exciting as prototype universal quantum processors emerge, enabling the implementation of a wider variety of algorithms. Of particular interest are quantum heuristics, which require experimentation on quantum hardware for their evaluation and which have the potential to significantly expand the breadth of applications for which quantum computers have an established advantage. A leading candidate is Farhi et al.’s quantum approximate optimization algorithm, which alternates between applying a cost function based Hamiltonian and a mixing Hamiltonian. Here, we extend this framework to allow alternation between more general families of operators. The essence of this extension, the quantum alternating operator ansatz, is the consideration of general parameterized families of unitaries rather than only those corresponding to the time evolution under a fixed local Hamiltonian for a time specified by the parameter. This ansatz supports the representation of a larger, and potentially more useful, set of states than the original formulation, with potential long-term impact on a broad array of application areas. For cases that call for mixing only within a desired subspace, refocusing on unitaries rather than Hamiltonians enables more efficiently implementable mixers than was possible in the original framework. Such mixers are particularly useful for optimization problems with hard constraints that must always be satisfied, defining a feasible subspace, and soft constraints whose violation we wish to minimize. More efficient implementation enables earlier experimental exploration of an alternating operator approach, in the spirit of the quantum approximate optimization algorithm, to a wide variety of approximate optimization, exact optimization, and sampling problems. In addition to introducing the quantum alternating operator ansatz, we lay out design criteria for mixing operators, detail mappings for eight problems, and provide a compendium with brief descriptions of mappings for a diverse array of problems.

Quantum Approximate Optimization Algorithm (QAOA) is a hybrid algorithm whose\ncontrol parameters are classically optimized. In addition to the variational\nparameters, the right choice of hyperparameter is crucial for improving the\nperformance of any optimization model. Control depth, or the number of\nvariational parameters, is considered as the most important hyperparameter for\nQAOA. In this paper we investigate the control depth selection with an\nautomatic algorithm based on proximal gradient descent. The performances of the\nautomatic algorithm are demonstrated on 7-node and 10-node Max-Cut problems,\nwhich show that the control depth can be significantly reduced during the\niteration while achieving an sufficient level of optimization accuracy. With\ntheoretical convergence guarantee, the proposed algorithm can be used as an\nefficient tool for choosing the appropriate control depth as a replacement of\nrandom search or empirical rules. Moreover, the reduction of control depth will\ninduce a significant reduction in the number of quantum gates in circuit, which\nimproves the applicability of QAOA on Noisy Intermediate-scale Quantum (NISQ)\ndevices.\n

The quantum approximate optimization algorithm (QAOA) is a near-term combinatorial optimization algorithm suitable for noisy quantum devices. However, little is known about performance guarantees for $p>2$. A recent work computing MaxCut performance guarantees for 3-regular graphs conjectures that any $d$-regular graph evaluated at particular fixed angles has an approximation ratio greater than some worst-case guarantee. In this work, we provide numerical evidence for this fixed angle conjecture for $p<12$. We compute and provide these angles via numerical optimization and tensor networks. These fixed angles serve for an optimization-free version of QAOA and have universally good performance on any 3-regular graph. Heuristic evidence is presented for the fixed angle conjecture on graph ensembles, which suggests that these fixed angles are ``close'' to global optimum. Under the fixed angle conjecture, QAOA has a larger performance guarantee than the Goemans Williamson algorithm on 3-regular graphs for $p\ensuremath{\ge}11$.

The classical distortion models similarly to temporal data analysis provide a way to predict the trends in the output of the algorithms and discard the anomalies in the output which may impact the accuracy of the method. In this paper, we introduce a quantum distortion modeling framework that enables variational quantum algorithms to operate effectively in the presence of errors without traditional error correction. Drawing inspirations from classical distortion-tolerant computing, we describe different distortions measures and formulation which can be used in variational quantum algorithms. In particular, we develop mathematical foundations for distortion metrics including energy progression, parameter stability, and state fidelity distortions, and demonstrate their applicability to variational quantum eigensolver, quantum approximate optimization algorithm, and quantum power method. We believe this paper will provide a milestone for distortion-aware quantum computing which can expand the practical applicability of the pre-fault-tolerant era devices to problems where approximate solutions provide sufficient value, representing a paradigm shift from exact error elimination to managed accuracy degradation.

Current hardware limitations restrict the potential when solving quadratic unconstrained binary optimization (QUBO) problems via the quantum approximate optimization algorithm (QAOA) or quantum annealing (QA). Thus, we consider training neural networks in this context. We first discuss QUBO problems that originate from translated instances of the traveling salesman problem (TSP): Analyzing this representation via autoencoders shows that there is way more information included than necessary to solve the original TSP. Then we show that neural networks can be used to solve TSP instances from both QUBO input and autoencoders' hidden state representation. We finally generalize the approach and successfully train neural networks to solve arbitrary QUBO problems, sketching means to use neuromorphic hardware as a simulator or an additional co-processor for quantum computing.

Quantum Approximate Optimization Algorithms (QAOA) promise efficient solutions to classically intractable combinatorial optimization problems by harnessing shallowdepth quantum circuits. Yet, their performance and scalability often hinge on effective parameter optimization, which remains nontrivial due to rugged energy landscapes and hardware noise. In this work, we introduce a quantum meta-learning framework that combines quantum neural networks, specifically Quantum Long Short-Term Memory (QLSTM) architectures, with QAOA. By training the QLSTM optimizer on smaller graph instances, our approach rapidly generalizes to larger, more complex problems, substantially reducing the number of iterations required for convergence. Through comprehensive benchmarks on Max-Cut and Sherrington-Kirkpatrick model instances, we demonstrate that QLSTM-based optimizers converge faster and achieve higher approximation ratios compared to classical baselines, thereby offering a robust pathway toward scalable quantum optimization in the NISQ era.

Abstract: This paper explores hybrid quantum-classical optimization algorithms implemented across multiple quantum software development kits (SDKs): Qiskit and Cirq,. We evaluate the Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA) in terms of convergence rate, final cost value, and fidelity, using both simulators and real quantum hardware backends. Our results offer comparative insights into SDK performance and propose a standardized benchmarking approach for future quantum algorithm research. Keywords: Hybrid Quantum Algorithms, Qiskit, Cirq, VQE, QAOA, Quantum SDKs, Quantum Optimization, NISQ

Quantum Artificial Intelligence (Quantum AI) represents a rapidly developing interdisciplinary field at the intersection of quantum computing and machine learning (ML). It holds the promise of unlocking unprecedented computational capabilities for complex optimization tasks, large-scale data processing, and advanced pattern recognition. In this research, we provide a comprehensive examination of two principal quantum algorithms—the Quantum Approximate Optimization Algorithm (QAOA) and the Variational Quantum Eigensolver (VQE)—applied to classical ML challenges. Using a hybrid simulation framework that integrates TensorFlow, scikit-learn, Qiskit, and Cirq, we extensively benchmark quantum-enhanced approaches against conventional methods on both combinatorial optimization and image classification tasks. Our findings indicate that while noise and qubit limitations remain critical barriers, quantum-enhanced models can achieve competitive, and sometimes superior, performance compared to purely classical solutions. We elaborate on the practical implications of these results, discuss hardware and algorithmic constraints, and propose future research directions focusing on error mitigation, scalability, and quantum-native ML models. These insights pave the way for a new computational paradigm, in which quantum resources are harnessed to address previously intractable ML problems.

This paper explores the potential of quantum computing on a WCET(Worst-Case Execution Time (of a program).)-related combinatorial optimization problem applied to a set of several polynomial special cases. We consider the maximization problem of determining the most expensive path in a control flow graph. In these graphs, vertices represent blocks of code whose execution times are fixed and known in advance. We port the considered optimization problem to the quantum framework by expressing it as a QUBO. We then experimentally compare the performances in solving the problem of classic Simulated Annealing (SA), Quantum Annealing (QA), and Quantum Approximate Optimization Algorithm (QAOA). Our experiments suggest that QA represents a fast equivalent of simulated annealing. Indeed, we measured the approximation ratio on the results of QA and SA, showing that their performances are comparable, at least on our set of simplified problems.KeywordsWCETsQuantum computingQUBOCombinatorial optimizationQuantum annealingQAOA

Abstract: This chapter explores the transformative potential of quantum algorithms in solving complex optimization problems across industries such as logistics, finance, and material science. Quantum algorithms, including Quantum Annealing, Quantum Approximate Optimization Algorithm (QAOA), and Variational Quantum Eigensolver (VQE), offer unprecedented computational capabilities for addressing challenges that are impractical for classical algorithms. In logistics, quantum algorithms optimize supply chains, vehicle routing, and warehouse operations, significantly reducing costs and improving efficiency. In finance, quantum optimization enables better portfolio management, risk assessment, and high-frequency trading strategies, providing faster, more accurate solutions. In material science, quantum algorithms accelerate the discovery and design of new materials, such as superconductors and catalysts, by simulating molecular interactions more efficiently. The chapter discusses real-world implementations, current technical challenges, and the future potential of quantum algorithms to revolutionize industries through faster and more scalable optimization solutions. Keywords: Quantum algorithms, Quantum Annealing, QAOA, VQE, optimization, logistics, finance, material science, portfolio optimization, risk management, supply chain, molecular simulation, quantum computing, quantum optimization, quantum chemistry, high-frequency trading.