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.
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.
Near-optimal solutions to Maximum Independent Set on dense graphs sit in local optima that greedy correction and maximality extension cannot escape. 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. 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 the superposition recovers the certified MIS. 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. 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.
A novel quantum algorithm named the Quantum Approximate Shift-Phase Algorithm (QASPA) is proposed for the approximate solution of the Max-Cut problem. The Max-Cut problem consists in partitioning the set of vertices into two subsets in such a way that the total weight of the edges connecting vertices from different subsets is maximized. This problem is known to be NP-complete and represents a combinatorial challenge. Classical solution algorithms require excessive computational time, rendering them inefficient for medium and large graphs. Approximate solution methods offer guaranteed approximation ratios but still face significant limitations in terms of accuracy and performance on larger graphs. The proposed algorithm features a simple scheme and a minimal number of optimized parameters. The input graph is represented by an adjacency matrix, after which the edge weights are linearly normalized to fixed phase angles. The quantum circuit begins by applying a Hadamard transform to each qubit, thereby creating an equal superposition of all possible partitions. Subsequently, for each pair of adjacent vertices, a sequence comprising a controlled NOT gate, a single-qubit rotation around the Y-axis by an angle proportional to the edge weight, and a second controlled NOT gate is applied. This encodes the phase information about the edge weights into the quantum state of the system. After circuit execution, the qubits are measured in the standard computational basis, and the resulting probability distribution allows the selection of the most probable partition as an approximate solution to the problem. Experimental studies conducted on quantum processor simulators have demonstrated that the proposed algorithm achieves accuracy comparable to the Variational Quantum Approximate Optimization Algorithm (QAOA) while significantly reducing computation time due to the absence of iterative parameter optimization. Moreover, as the graph size increases, the algorithm's runtime grows considerably slower compared to classical brute-force approaches and QAOA, confirming its potential for solving medium-sized Max-Cut problems.
Large-scale combinatorial optimization is a challenge for near-term quantum computing because dense Quadratic Unconstrained Binary Optimization (QUBO) formulations yield interaction graphs that exceed the limits of NISQ processors. This work introduces Hybrid Quantum Neighborhood Selection (HQNS), a hybrid framework mitigating this via stochastic frontier decomposition. Instead of encoding all N variables into a monolithic circuit, HQNS selects a compact frontier of F << N active variables per stage, freezing the rest into reduced QUBO coefficients. A multi-stage crawling procedure rotates these frontiers, letting local quantum subproblems refine a global solution. We evaluate HQNS on the Maximum Diversity Subset Selection Problem (MDSSP) across six scales, N up to 1000. Circuit burden is reduced from the dense QAOA requirement of O(N^2) two-qubit terms per layer to O(F^2) per stage, with total complexity governed by the number of stages and classical overhead. Benchmarks show that HQNS achieves competitive solution quality relative to parallel simulated annealing (SA) while maintaining bounded circuit width and stable QPU time. In the N=1000 benchmark over ten executions, HQNS preserves 99.9908% of the mean diversity score of an 11-restart parallel SA baseline, while reducing wall-clock time by 65.03%, peak CPU usage by 55.97%, and peak memory by 35.21%. Ablation shows performance depends on frontier size, warm-starts, CVaR filtering, and stochastic rotation. These results demonstrate that structured frontier decomposition makes variational optimization executable for dense QUBO instances unsuitable for direct QAOA on present hardware.
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.
Abstract We port the Generative Quantum Eigensolver + Quantum-Selected Configuration Interaction (GQE+QSCI) architecture from quantum chemistry to combinatorial graph optimization. Our framework, GQO+QSS (Generative Quantum Optimization + Quantum-Selected Search), follows the same principle: ML generates circuit programs, quantum samplers produce bitstrings, classical recovery repairs and expands them into a restricted subspace, and a classical solver finds the optimum within that subspace. We formalize the key quantity as the recovery-basin mass pr(qθ,G) = Prx∼qθ [x ∈ Ar(G)], where Ar(G) = {x : Er(R(x))∩X∗(G) ̸= ∅} is the preimage of the optimal set under repair R and r-flip expansion Er. Under B independent shots, success probability becomes Psuccess = 1 − (1 − pr)B. The quantum-benefit question reduces to: does pr(qquantum,G) > pr(qclassical,G)? We test this on MaximumWeighted Independent Set (MWIS) with a preregistered, frozen protocol. All 150 ground truths are certified with CBC + brute-force verification. Exact recovery-basin masks are enumerated for all n ≤ 20 instances. Nine samplers are compared on a held-out test set: uniform, greedy, simulated annealing, tabu search, multi-start local search, Metropolis, warm-start QAOA, oracle QAOA, and a matched learned classical policy. Key findings: 1. Tabu search dominates at all sizes (pr = 0.82–0.94), followed by simulated annealing. All quantum samplers and the learned classical policy lose to both at every problem size under the frozen test protocol. 2. Oracle QAOA loses to SA under fair evaluation: With random QAOA configurations (not per-instance optimized), oracle QAOA achieves pr = 0.014 at n = 28 vs SA’s 0.088. The earlier positive result (pr = 0.185 vs 0.130) required per-instance best-of-50 configuration selection—an oracle upper bound, not a deployable method. 3. Learned classical policy matches oracle QAOA: A matched 140K-parameter classical policy achieves pr = 0.005–0.101, comparable to oracle QAOA’s 0.014–0.113. The quantum circuit’s proposal distribution provides no structural advantage over a matched learned classical model in this setting. 4. Recovery-basin masks reveal the phase structure: At r = 2, uniform-sampling Precover drops from 0.975 (n = 16) to 0.775 (n = 20) at B = 200, identifying the regime where sampler quality matters. 5. Hardware robustness without advantage: 136 cores on Piast-Q show 93.4% recovery consistency between QPU and simulator, but no basin-mass advantage. This is a falsification study: Under rigorous, preregistered evaluation with matched learned baselines, shallow QAOA-derived proposal distributions do not outperform matched classical proposal models, and both lose to strong classical baselines (SA, tabu). The earlier positive oracle result was an artifact of per-instance configuration selection. We claim no quantum advantage.