Deep-Circuit QAOA

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.

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 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 $p\ensuremath{\ge}11$.

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.

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.

We present an analog version of the quantum approximate optimization algorithm suitable for current quantum annealers. The central idea of this algorithm is to optimize the schedule function, which defines the adiabatic evolution. It is achieved by choosing a suitable parametrization of the schedule function based on interpolation methods for a fixed time, with the potential to generate any function. This algorithm provides an approximate result of optimization problems that may be developed during the coherence time of current quantum annealers on their way toward quantum advantage.

The Quantum Approximate Optimization Algorithm (QAOA) is a promising hybrid quantum-classical approach for solving combinatorial optimization problems such as the NP-hard MaxCut on near-term quantum devices. However, QAOA suffers from frequent data exchange between the quantum circuit simulator and the classical optimizer, creating a sig-nificant performance bottleneck. To address these issues, this work presents HE-QAOA, a hardware-efficient FPGA-based accelerator that integrates quantum circuit evolution with the classical Broyden-Fletcher-Goldfarb-Shanno (BFGS) optimizer into a unified architecture, reducing data exchange overhead. The step-size adjustment strategy for BFGS optimizer is then developed to reduce multiplier overhead. Experimental results demonstrate that the proposed HE-QAOA achieves up to 303.6× and 289.7× speedup in quantum circuit evolution and BFGS optimization, respectively, while maintaining a 4.3% decrease in approximation ratio. The proposed FPGA-accelerated hardware architecture shows substantial practical value for enhancing the performance of quantum simulations.

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.

Limited by the decoherence of qubits as well as the errors of quantum gates, near-term superconducting quantum computers can only run low-depth quantum circuits to achieve acceptable fidelity. One possible way to overcome these limitations is to construct quantum circuits with additional high-fidelity expressive multi-qubit gates. Recently, a new three-qubit gate, denoted as Controlled-CPHASE-SWAP (CCZS), has been implemented through simultaneous Controlled-Z (CZ) gates. The CCZS gate takes less time than a single CZ gate and can be implemented at the coherence limit. However, how to use the CCZS gate in quantum circuit synthesis remains unexplored. In this paper, we construct the quantum fan-out/parity gates, the controlled-phase gate and the locally fully connected CZ gates with the CCZS gate, respectively. Furthermore, applications of the CCZS gate in quantum error correction, quantum Fourier transform and quantum approximate optimization algorithm are also proposed. We evaluate the performance of the CCZS gate in quantum circuit synthesis through simulation and explore its potential advantages over CZ gates.

Maritime transportation is essential for global trade but presents significant environmental challenges due to its greenhouse gas emissions. Existing studies have addressed these challenges through integrated routing and speed optimization frameworks, yet frequently lack explicit quantification of environmental impacts and exhibit limited scalability for large-scale ship routing operations. Conversely, existing quantum optimization research in vehicle routing predominantly targets land-based transportation scenarios, restricting its direct applicability to maritime logistics. Maritime logistics inherently involve distinct operational complexities, such as nonlinear interactions among speed, payload, fuel consumption, and numerous operational uncertainties. These combined limitations underscore the critical need for quantum optimization methods explicitly designed for green maritime supply chains. To bridge this gap, this paper proposes an efficient quantum-centric optimization framework that uses the quantum approximate optimization algorithm (QAOA) to jointly optimize ship routing and speed management within sustainable maritime supply chains. Specifically, we formulate an NP-hard cost minimization problem integrating critical maritime parameters, including fuel consumption, payload constraints, and operational speeds. We further develop a hybrid quantum-classical alternating optimization approach that iteratively addresses routing decisions through quantum computing techniques and optimizes ship speed using an analytical solution. Simulation results and real quantum hardware experiments demonstrate that our quantum-centric methodology achieves substantial cost reductions and highlights the potential for practical applicability in realistic maritime operations, significantly outperforming classical optimization benchmarks.