We study variational quantum algorithms for the Maximum Independent Set (MIS) problem on benchmark graphs of 64, 99, and 180 vertices. The Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA) are compared across SPSA and COBYLA optimizers at multiple circuit depths. A preprocessing pipeline comprising spectral graph reordering (via the Fiedler vector) and distance-based sparsification reduces circuit depth while preserving energy fidelity. Classical post-processing via history-guided bitstring correction and stepwise maximality extension recovers the exact MIS across all instances. With CVaR optimization, VQE with SPSArecovers up to 6 distinct MIS per run for the 64-node instance and up to 10 distinct MIS per run for the 99-node instance, sampling broadly from the optimal solution population. Repeated runs with different SPSA trajectories collectively enumerate a larger fraction of all MIS for each instance. For the 180-node instance, where standard approaches stall at size 14 (MIS is 15), we introduce ancilla-assisted superposition initialization: ancilla qubits prepare a uniform superposition over classically-found near-optimal solutions, and an excitation-preserving ansatz evolves this state while conserving Hamming weight. This novel construction enables quantum-parallel variational search over multiple seeds simultaneously, discovering the exact MIS where single-seed methods fail. The 180-qubit simulation represents, to our knowledge, the largest scale at which gate-based variational algorithms have solved MIS to optimality. Hardware validation on IBM Quantum hardware ibm_marrakesh confirms that converged simulator parameters transfer effectively to noisy quantum execution.
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Corrections in citations and update on bond dimensions
We study variational quantum algorithms for the Maximum Independent Set (MIS) problem on benchmark graphs of 64, 99, and 180 vertices. The Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA) are compared across SPSA and COBYLA optimizers at multiple circuit depths. A preprocessing pipeline comprising spectral graph reordering (via the Fiedler vector) and distance-based sparsification reduces circuit depth while preserving energy fidelity. Classical post-processing via history-guided bitstring correction and stepwise maximality extension recovers the exact MIS across all instances. With CVaR optimization, VQE with SPSArecovers up to 6 distinct MIS per run for the 64-node instance and up to 10 distinct MIS per run for the 99-node instance, sampling broadly from the optimal solution population. Repeated runs with different SPSA trajectories collectively enumerate a larger fraction of all MIS for each instance. For the 180-node instance, where standard approaches stall at size 14 (MIS is 15), we introduce ancilla-assisted superposition initialization: ancilla qubits prepare a uniform superposition over classically-found near-optimal solutions, and an excitation-preserving ansatz evolves this state while conserving Hamming weight. This novel construction enables quantum-parallel variational search over multiple seeds simultaneously, discovering the exact MIS where single-seed methods fail. The 180-qubit simulation represents, to our knowledge, the largest scale at which gate-based variational algorithms have solved MIS to optimality. Hardware validation on IBM Quantum hardware ibm_marrakesh confirms that converged simulator parameters transfer effectively to noisy quantum execution.
Abstract Near-optimal solutions to Maximum Independent Set on dense graphs sit in local optima that greedy correction and maximality extension cannot escape, even with repeated restarts. We encode near-optimal seeds as a uniform quantum superposition on ancilla qubits and evolve them under an excitation-preserving variational ansatz that holds the search inside the feasible Hamming-weight subspace. A preprocessing stage of spectral reordering and distance-based sparsification, together with history-guided post-processing of the sampled bitstrings, takes the method to 200 nodes. The ansatz entangles the seed branches and the bond dimension of the simulated state grows with circuit depth, which places deeper circuits outside the reach of exact matrix product state simulation at the bond dimensions available to us. Larger instances therefore need quantum hardware. Measured on the data register alone, the superposition behaves as a classical mixture over the seeds, so coherence between the branches has to be created and then looked for. We do this with a CRZ phase layer and post-selection on the ancilla qubits, which brings the branches into interference and exposes the cross terms. The idea is to see if interference between near-optimal seeds widens the range of independent sets the circuit returns. We measure the cross terms rather than assume them and place an upper limit on their size, and we trace that limit to the distance between the vertices that distinguish near-optimal solutions once the graph is reordered. Instances that fit within the qubit count of the hardware available to us need none of the ancilla machinery. Standard VQE with this pipeline recovers the certified optimum for instances up to 125 nodes, and these run on ibm_marrakesh with parameters transferred from noiseless simulation. The ancilla construction is introduced for the sizes past that point. On a 180-node hard instance where standard VQE and our classical post-processing stall at size 14, the superposition recovers the certified maximum of 15, and an ablation over single seeds shows the superposition is responsible. Five 200-node instances are solved to the certified optimum, and on the 400-node brock400-1 benchmark the method reaches size 25 against a certified optimum of 27, with the recovered size improving as depth increases, which points at circuit expressiveness as the bottleneck. These are the largest hard general-graph instances we know of where a gate-based variational algorithm optimises the full circuit directly.
We investigate a hybrid quantum-classical algorithm for solving the Maximum Independent Set (MIS) problem on regular graphs, combining the Quantum Approximate Optimization Algorithm (QAOA) with a minimal degree classical greedy algorithm. The method leverages pre-computed QAOA angles, derived from depth-p QAOA circuits on regular trees, to compute local expectation values and inform sequential greedy decisions that progressively build an independent set. This hybrid approach maintains shallow quantum circuit and avoids instance-specific parameter training, making it well-suited for implementation on current quantum hardware: we have implemented the algorithm on a 20 qubit IQM superconducting device to find independent sets in graphs with thousands of nodes. We perform tensor network simulations to evaluate the performance of the algorithm beyond the reach of current quantum hardware and compare to established classical heuristics. Our results show that even at low depth (p=4), the quantum-enhanced greedy method significantly outperforms purely classical greedy baselines as well as more sophisticated approximation algorithms. The modular structure of the algorithm and relatively low quantum resource requirements make it a compelling candidate for scalable, hybrid optimization in the NISQ era and beyond.
Quantum computing offers promising approaches to combinatorial optimization problems whose search spaces grow exponentially with problem size. The Closest String problem belongs to this category, which seeks a string that minimizes the maximum Hamming distance to any string in a given collection of reference strings. In this work, we present an end-to-end quantum implementation of the Closest String problem using the Non-Variational Quantum Walk Optimization Algorithm (NV-QWOA), evaluating its resource requirements and solution amplification capabilities. Our automated workflow converts arbitrary problem instances into quantum circuits expressed in one- or two-qubit gates that encode candidate solutions, evaluate the objective function, and apply NV-QWOA optimisation to increase the measurement probabilities of high-quality solutions. Analytical resource estimates yield a qubit complexity of O(nmlogk) and a circuit depth of O(pmlog2n+pnlogn), where n is the string length, m is the number of reference strings, k is the alphabet size, and p is the number of NV-QWOA iterations. We introduce a revised prescription for the phase-separation parameter, γ, that depends on problem size but requires no instance-specific optimization. This prescription mitigates excessive phase wrapping modulo 2π, which can weaken the relationship between objective values and the phases guiding the optimization dynamics. Across all simulated problem sizes, the algorithm consistently amplifies optimal and near-optimal candidates. This work provides an explicit gate-level implementation, an improved phase-scaling method, and a foundation for hardware experiments and benchmarking against classical algorithms.
Realizing quantum speedup for practically relevant, computationally hard problems is a central challenge in quantum information science. Using Rydberg atom arrays with up to 289 qubits in two spatial dimensions, we experimentally investigate quantum algorithms for solving the maximum independent set problem. We use a hardware-efficient encoding associated with Rydberg blockade, realize closed-loop optimization to test several variational algorithms, and subsequently apply them to systematically explore a class of graphs with programmable connectivity. We find that the problem hardness is controlled by the solution degeneracy and number of local minima, and we experimentally benchmark the quantum algorithm's performance against classical simulated annealing. On the hardest graphs, we observe a superlinear quantum speedup in finding exact solutions in the deep circuit regime and analyze its origins.
Abstract We introduce a novel hybrid quantum–classical variational optimization method for unconstrained binary combinatorial optimization problems on gate-model quantum computers, integrating a custom variational ansatz, staged feedback-based dual variational parameter update strategies, efficient parametric compilation, automated error suppression during hardware execution, and scalable O( n ) classical post-processing to correct for bitflip errors. Without this integrated approach, we show that standard circuit execution at scale produces output indistinguishable from random sampling, establishing the necessity of each pipeline component. We benchmark the method on IBM superconducting quantum computers for several optimization problems drawn from NP-hard problem classes, where the optimization is conducted on hardware with no use of classical simulation or prior knowledge of the solution. For Max-Cut on random regular graphs with topologies not matched to device connectivity, the method achieves approximation ratios (AR) of 100% for unweighted 3-regular graphs up to 156 nodes, weighted regular graphs up to 80 nodes, and weighted 7-regular graphs up 50 nodes. Applied to higher-order binary optimization, the method finds the ground state energy of 127- and 156-qubit spin-glass models matched to device topology with linear, quadratic, and cubic interaction terms, achieving ARs of at least 99.5% across all instances tested. The method consistently outperforms a classical local solver across all problems. Where published results on identical problem instances are available, our method demonstrates competitive or superior performance. These results demonstrate that an appropriately engineered variational approach enables gate-model quantum computers to produce high-quality solutions for binary optimization problems drawn from NP-hard problem classes at the 156-qubit scale, where naïve implementations are insufficient for good performance.
Variational quantum optimization provides a useful setting for examining how circuit structure, parameterization, and trainability interact in practice. Objectives: Among the most promising variational quantum algorithms, the Quantum Approximate Optimization Algorithm (QAOA) depends critically on circuit architecture, but greater circuit complexity does not necessarily lead to easier optimization or better solutions. Using the Maximum Cut (MaxCut) problem on graphs as a controlled setting, we investigate how variational behavior changes as problem-specific QAOA structure is progressively removed, with particular attention to representational capacity, optimization geometry, and finite-budget trainability. Methods: We analytically characterize a graph-agnostic product-state formulation of MaxCut and derive exact equivalences, redundancies, and local-landscape properties. These results motivate a controlled comparison among separable product ansätze, a representative graphagnostic entangling circuit, and graph-structured QAOA at several depths, including a parameter-count-matched control. The numerical benchmark comprises 95 small unweighted graph instances optimized under a common objective-evaluation budget. Results: The product-state formulation preserves the exact discrete MaxCut optimum, while its strict local maxima correspond to classical cuts that are stable under singlevertex flips. Redundant phase parameters increase optimization effort without enlarging the objective-relevant family. Under the prescribed optimization budget, the productstate ansatz attains higher average solution quality than the tested QAOA depths while requiring substantially fewer objective evaluations. The graph-agnostic entangling comparator provides no consistent advantage over the separable baseline, and a simple classical local-search method remains highly competitive. Conclusion: In the regime considered here, objective-relevant parameterization and optimization geometry can be as important as circuit richness.