The Quantum Approximate Optimisation Algorithm (QAOA) is a widely studied quantum-classical iterative heuristic for combinatorial optimisation. While QAOA targets problems in complexity class NP, the classical optimisation procedure required in every iteration is itself known to be NP-hard. Still, advantage over classical approaches is suspected for certain scenarios, but nature and origin of its computational power are not yet satisfactorily understood. By introducing means of efficiently and accurately approximating the QAOA optimisation landscape from solution space structures, we derive a new algorithmic variant of unit-depth QAOA for two-level Hamiltonians (including all problems in NP): Instead of performing an iterative quantum-classical computation for each input instance, our non-iterative method is based on a quantum circuit that is instance-independent, but problem-specific. It matches or outperforms unit-depth QAOA for key combinatorial problems, despite reduced computational effort. Our approach is based on proving a long-standing conjecture regarding instance-independent structures in QAOA. By ensuring generality, we link existing empirical observations on QAOA parameter clustering to established approaches in theoretical computer science, and provide a sound foundation for understanding the link between structural properties of solution spaces and quantum optimisation.
The Quantum Approximate Optimisation Algorithm (QAOA) is a widely studied quantum-classical iterative heuristic for combinatorial optimisation. While QAOA targets problems in complexity class NP, the classical optimisation procedure required in every iteration is itself known to be NP-hard. Still, advantage over classical approaches is suspected for certain scenarios, but nature and origin of its computational power are not yet satisfactorily understood. By introducing means of efficiently and accurately approximating the QAOA optimisation landscape from solution space structures, we derive a new algorithmic variant of unit-depth QAOA for two-level Hamiltonians (including all problems in NP): Instead of performing an iterative quantum-classical computation for each input instance, our non-iterative method is based on a quantum circuit that is instance-independent, but problem-specific. It matches or outperforms unit-depth QAOA for key combinatorial problems, despite reduced computational effort. Our approach is based on proving a long-standing conjecture regarding instance-independent structures in QAOA. By ensuring generality, we link existing empirical observations on QAOA parameter clustering to established approaches in theoretical computer science, and provide a sound foundation for understanding the link between structural properties of solution spaces and quantum optimisation.
Structured variational quantum algorithms such as the Quantum Approximate Optimisation Algorithm (QAOA) have emerged as leading candidates for exploiting advantages of near-term quantum hardware. They interlace classical computation, in particular optimisation of variational parameters, with quantum-specific routines, and combine problem-specific advantages -- sometimes even provable -- with adaptability to the constraints of noisy, intermediate-scale quantum (NISQ) devices. While circuit depth can be parametrically increased and is known to improve performance in an ideal (noiseless) setting, on realistic hardware greater depth exacerbates noise: The overall quality of results depends critically on both, variational parameters and circuit depth. Although identifying optimal parameters is NP-hard, prior work has suggested that they may exhibit regular, predictable patterns for increasingly deep circuits and depending on the studied class of problems. In this work, we systematically investigate the role of classical parameters in QAOA performance through extensive numerical simulations and suggest a simple, yet effective heuristic scheme to find good parameters for low-depth circuits. Our results demonstrate that: (i) high-quality parameters often deviate substantially from expected patterns; (ii) QAOA performance becomes progressively less sensitive to specific parameter choices as depth increases; and (iii) iterative component-wise fixing performs on par with, and at shallow depth may even outperform, several established parameter-selection strategies. We identify conditions under which structured parameter patterns emerge, and when deviations from the patterns warrant further consideration. These insights for low-depth circuits may inform more robust pathways to harnessing QAOA in realistic quantum computing scenarios.
Quantum computing promises solutions to classically difficult and new-found problems through controlling the subtleties of quantum computing. The Quantum Approximate Optimisation Algorithm (QAOA) is a recently proposed quantum algorithm designed to tackle difficult combinatorial optimisation problems utilising both quantum and classical computation. The hybrid nature, generality and typically low gate-depth make it a strong candidate for near-term implementation in quantum computing. Finding the practical limits of the algorithm is currently an open problem. Until now, no tools to facilitate the design and validation of probabilistic quantum optimisation algorithms such as the QAOA on a non-trivial scale exist. Graph similarity is a long standing classically difficult problem withstanding decades of research from academia and industry. Determining the maximal edge overlap between all possible node label permutations is an NP-Complete task and provides an apt measure of graph similarity. We introduce a novel quantum optimisation simulation package facilitating investigation of all constituent components of the QAOA from desktop to cluster scale using graph similarity as an example. Our simulation provides flexibility and performance. We investigate eight classical optimisation methods each at six levels of decomposition. Moreover an encoding for permutation based problems such as graph similarity through edge overlap to the QAOA allows for significant quantum memory savings at the cost of additional operations. This compromise extends into the classical portion of the algorithm as the inclusion of infeasible solutions creates a challenging cost-function landscape. We present performance analysis of our simulation and of the QAOA setting a precedent for investigating and validating numerous other difficult problems to the QAOA as we move towards realising practical quantum computation.
Abstract The quantum approximate optimisation algorithm (QAOA) is a hybrid quantum–classical algorithm used to approximately solve combinatorial optimisation problems. It involves multiple iterations of a parameterised ansatz that consists of a problem and mixer Hamiltonian, with the parameters being classically optimised. While QAOA can be implemented on near-term quantum hardware, physical limitations such as gate noise, restricted qubit connectivity, and state-preparation-and-measurement (SPAM) errors can limit circuit depth and decrease performance. To address these limitations, this work introduces the eXpressive QAOA (XQAOA), an overparameterised variant of QAOA that assigns more classical parameters to the ansatz to improve its performance at low depths. XQAOA also introduces an additional Pauli-Y component in the mixer Hamiltonian, allowing the mixer to implement arbitrary unitary transformations on each qubit. To benchmark the performance of XQAOA at unit depth, we derive its closed-form expression for the MaxCut problem and compare it to QAOA, Multi-Angle QAOA (MA-QAOA) (Herrman et al 2022 Sci. Rep. 12 6781), a classical-relaxed algorithm, and the state-of-the-art Goemans–Williamson algorithm on a set of unweighted regular graphs with 128 and 256 nodes for degrees ranging from 3 to 10. Our results indicate that at unit depth, XQAOA has benign loss landscapes with local minima concentrated near the global optimum, allowing it to consistently outperform QAOA, MA-QAOA, and the classical-relaxed algorithm on all graph instances and the Goemans–Williamson algorithm on graph instances with degrees greater than 4. Small-scale simulations also reveal that unit-depth XQAOA invariably surpasses both QAOA and MA-QAOA on all tested depths up to five. Additionally, we find an infinite family of graphs for which XQAOA solves MaxCut exactly and analytically show that for some graphs in this family, special cases of XQAOA are capable of achieving a much larger approximation ratio than QAOA. Overall, XQAOA is a more viable choice for variational quantum optimisation on near-term quantum devices, offering competitive performance at low depths.
The transition from theoretical quantum computing to practical real-world applications heavily relies onhybrid quantum-classical architectures designed for NoisyIntermediate-Scale Quantum (NISQ) devices. Among these approaches, the Quantum Approximate Optimization Algorithm(QAOA) is a prominent strategy for addressing complex combinatorial optimization challenges, such as the NP-hard MaxCut problem. Instead of performing the entire computationon a quantum processor, QAOA utilizes a quantum circuit toevaluate a parameterized cost function, whereas a classical optimizer refines the parameters iteratively. This study presentsa rigorous empirical evaluation comparing the performanceof QAOA against established classical solvers on small-scaleMax-Cut instances. All experiments were implemented andexecuted using Python (Qiskit Aer) on a state-vector simulator.Using Erdős–Rényi random graphs and regular graphs with4 to 20 nodes, we benchmark QAOA at depths p = 2 andp = 3 against simulated annealing, the Goemans–WilliamsonSDP-based algorithm, and a linear-time greedy heuristic. Ourexperimental results demonstrate that low-depth QAOA implementations exhibit severe performance degradation, prohibitive computational overhead (O(2^n) scaling), and low convergencerates compared to mature classical algorithms. At 20 nodes,QAOA (p = 2) achieves only a 0.467 approximation ratio versus0.856 for Goemans–Williamson, with a 210× runtime slowdownover greedy heuristics
The quantum approximate optimisation algorithm (QAOA) and its variants are at the core of many scenarios that aim at combining the power of quantum computers (QC) and classical high-performance computing (HPC) appliances for combinatorial optimisation. Several obstacles challenge concrete benefits now and in the foreseeable future: Imperfections quickly degrade algorithmic performance below practical utility; over-heads arising from alternating between classical and quantum primitives can counter any advantage; and the choice of parameters or algorithmic variant can substantially influence runtime and result quality. Selecting the appropriate combination is a non-trivial issue, as it not only depends on end-user requirements, but also on details of the hardware and software stack. Appropriate automation can alleviate the burden of choosing optimal combinations for end-users: They should not be required to understand technicalities like detail differences between QAOA variants, required number of QAOA layers, or necessary measurement samples. Yet, they should receive the best possible satisfaction of their non-functional requirements, be it performance or other. We determine factors that influence approximation quality and temporal behaviour of four QAOA variants using comprehensive density-matrix-based numerical simulations targeting three widely studied optimisation problems. Our simulations consider ideal quantum computation, and a continuum of scenarios troubled by realistic imperfections. Our quantitative results, accompanied by a comprehensive reproduction package, show strong performance differences between QAOA variants that can be pinpointed to narrow and specific effects. We identify influential co-variables and relevant non-functional quality goals that, as we argue, mark the rele-vant ingredients for designing appropriate software engineering abstraction mechanisms and automated tool-chains for devising quantum solutions from higher-level problem specifications.
Establishing quantum advantage for variational quantum algorithms is an important direction in quantum computing. In this work, we apply the Quantum Approximate Optimisation Algorithm (QAOA) -- a popular variational quantum algorithm for general combinatorial optimisation problems -- to a variant of the satisfiability problem (SAT): Not-All-Equal SAT (NAE-SAT). We focus on regimes where the problems are known to have solutions with low probability and introduce a novel classical solver that outperforms existing solvers. Extensively benchmarking QAOA against this, we show that while the runtime of both solvers scales exponentially with the problem size, the scaling exponent for QAOA is smaller for large enough circuit depths. This implies a polynomial quantum speedup for solving NAE-SAT.