From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz

Many challenging scheduling, planning, and resource allocation problems come with real-world input data and hard problem constraints, and reduce to optimizing a cost function over a combinatorially defined feasible set, such as colorings of a graph. Toward tackling such problems with quantum computers using quantum approximate optimization algorithms, we present novel efficient quantum alternating operator ansatz (QAOA) constructions for optimization problems over proper colorings of chordal graphs. As our primary application, we consider the flight-gate assignment problem , where flights are assigned to airport gates as to minimize the total transit time of all passengers, and feasible assignments correspond to proper graph colorings of a conflict graph derived instancewise from the input data. We leverage ideas from classical algorithms and graph theory to show our constructions have the desirable properties of restricting quantum state evolution to the feasible subspace, and satisfying a particular reachability condition for most problem parameter regimes. Using classical preprocessing we show that we can always find and construct a suitable initial quantum (superposition) state efficiently. We show our constructions in detail, including explicit decompositions to a universal set of basic quantum gates, which we use to bound the required resource scaling as low-degree polynomials of the input parameters. In particular, we derive novel QAOA mixing operators and show that their implementation cost is commensurate with that of the QAOA phase operator for flight-gate assignment. A number of quantum circuit diagrams are included such that our constructions may be used as a template toward development and implementation of quantum gate-model approaches for a wider variety of potentially impactful real-world applications.

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 quantum approximate optimization algorithm (QAOA) is a leading iterative variational quantum algorithm for heuristically solving combinatorial optimization problems. A large portion of the computational effort in QAOA is spent by the optimization steps, which require many executions of the quantum circuit. Therefore, there is active research focusing on finding better initial circuit parameters, which would reduce the number of required iterations and hence the overall execution time. While existing methods for parameter initialization have shown great success, they often offer a single set of parameters for all problem instances. We propose a practical method that uses a simple, fully connected neural network that leverages previous executions of QAOA to find better initialization parameters tailored to a new given problem instance. We benchmark state-of-the-art initialization methods for solving the MaxCut problem of Erdős-Rényi graphs using QAOA and show that our method is consistently the fastest to converge while also yielding the best final result. Furthermore, the parameters predicted by the neural network are shown to match very well with the fully optimized parameters, to the extent that no iterative steps are required, thereby effectively realizing an iteration-free QAOA scheme.

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.

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.

We propose and study Th-QAOA (pronounced Threshold QAOA), a variation of the Quantum Alternating Operator Ansatz (QAOA) that replaces the standard phase separator operator, which encodes the objective function, with a threshold function that returns a value $1$ for solutions with an objective value above the threshold and a $0$ otherwise. We vary the threshold value to arrive at a quantum optimization algorithm. We focus on a combination with the Grover Mixer operator; the resulting GM-Th-QAOA can be viewed as a generalization of Grover's quantum search algorithm and its minimum/maximum finding cousin to approximate optimization. Our main findings include: (i) we provide intuitive arguments and show empirically that the optimum parameter values of GM-Th-QAOA (angles and threshold value) can be found with $O(\log(p) \times \log M)$ iterations of the classical outer loop, where $p$ is the number of QAOA rounds and $M$ is an upper bound on the solution value (often the number of vertices or edges in an input graph), thus eliminating the notorious outer-loop parameter finding issue of other QAOA algorithms; (ii) GM-Th-QAOA can be simulated classically with little effort up to 100 qubits through a set of tricks that cut down memory requirements; (iii) somewhat surprisingly, GM-Th-QAOA outperforms non-thresholded GM-QAOA in terms of approximation ratios achieved. This third result holds across a range of optimization problems (MaxCut, Max k-VertexCover, Max k-DensestSubgraph, MaxBisection) and various experimental design parameters, such as different input edge densities and constraint sizes.

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.

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.

In recent years, quantum computers and algorithms have made significant progress indicating the prospective importance of quantum computing (QC). Especially combinatorial optimization has gained a lot of attention as an application field for near-term quantum computers, both by using gate-based QC via the Quantum Approximate Optimization Algorithm and by quantum annealing using the Ising model. However, demonstrating an advantage over classical methods in real-world applications remains an active area of research. In this work, we investigate the robustness verification of ReLU networks, which involves solving many-variable mixed-integer programs (MIPs), as a practical application. Classically, complete verification techniques struggle with large networks as the combinatorial space grows exponentially, implying that realistic networks are difficult to be verified by classical methods. To alleviate this issue, we propose to use QC for neural network verification and introduce a hybrid quantum procedure to compute provable certificates. By applying Benders decomposition, we split the MIP into a quadratic unconstrained binary optimization and a linear program which are solved by quantum and classical computers, respectively. We further improve existing hybrid methods based on the Benders decomposition by reducing the overall number of iterations and placing a limit on the maximum number of qubits required. We show that, in a simulated environment, our certificate is sound, and provides bounds on the minimum number of qubits necessary to approximate the problem. Finally, we evaluate our method within simulations and on quantum hardware.

The quantum approximate optimization algorithm (QAOA) is considered to be one of the most promising approaches towards using near-term quantum computers for practical application. In its original form, the algorithm applies two different Hamiltonians, called the mixer and the cost Hamiltonian, in alternation with the goal being to approach the ground state of the cost Hamiltonian. Recently, it has been suggested that one might use such a set-up as a parametric quantum circuit with possibly some other goal than reaching ground states. From this perspective, a recent work (Lloyd, arXiv:1812.11075) argued that for one-dimensional local cost Hamiltonians, composed of nearest neighbour ZZ terms, this set-up is quantum computationally universal and provides a universal gate set, i.e. all unitaries can be reached up to arbitrary precision. In the present paper, we complement this work by giving a complete proof and the precise conditions under which such a one-dimensional QAOA might produce a universal gate set. We further generalize this type of gate-set universality for certain cost Hamiltonians with ZZ and ZZZ terms arranged according to the adjacency structure of certain graphs and hypergraphs.

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 $p\ensuremath{\ge}11$.

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.

Automatic optical inspection (AOI) systems for drones are increasingly becoming core tools for intelligent operation and maintenance of industrial infrastructure. However, their traditional implementation paradigms face fundamental technical bottlenecks in communication security, collaborative control, and harsh environment perception. This paper aims to provide a systematic review and analysis of research progress in enabling nextgeneration UAV AOI systems through emerging quantum technologies, including quantum communication, quantum computing, and quantum sensing. The article first provides a comprehensive overview of the unique potential and current engineering challenges faced by technologies such as quantum key distribution (QKD), quantum approximate optimization algorithms (QAOA), and quantum imaging in addressing the limitations of classical systems. Furthermore, this analysis delved into the cutting-edge concept of a “quantum-classical hybrid architecture” that integrates quantum modules with mature classical systems, analyzing its integration mechanisms and performance gains across three dimensions: communication, computation, and sensing. The review indicates that while this fusion paradigm can theoretically significantly enhance system security, optimize efficiency, and improve robustness, its practical deployment remains constrained by the environmental sensitivity of quantum devices, high costs, and complex system integration. Finally, this paper offers a forward-looking discussion on future development directions in hardware miniaturization, algorithmic collaboration frameworks, and standardization research within this field, providing a framework for researchers to comprehensively understand the opportunities and challenges in this interdisciplinary domain.

Quantum computing is poised to transform computational paradigms across science and industry. As the field evolves, it can benefit from established classical methodologies, including promising paradigms such as Transfer of Knowledge (ToK). This work serves as a brief, self‐contained reference for ToK, unifying its core principles under a single formal framework. We introduce a joint notation that consolidates and extends prior work in Transfer Learning and Transfer Optimisation, bridging traditionally separate research lines and enabling a common language for knowledge reuse. Building on this foundation, we classify existing ToK strategies and principles into a structured taxonomy that helps researchers position their methods within a broader conceptual map. We then extend key transfer protocols to quantum computing, introducing two novel use cases—reverse annealing and multitasking Quantum Approximate Optimization Algorithm (QAOA)—alongside a sequential Variational Quantum Eigensolver (VQE) approach that supports and validates prior findings. These examples highlight ToK's potential to improve performance and generalisation in quantum algorithms. Finally, we outline challenges and opportunities for integrating ToK into quantum computing, emphasising its role in reducing resource demands and accelerating problem‐solving. This work lays the groundwork for future synergies between classical and quantum computing through a shared, transferable knowledge framework.