Demonstrating quantum advantage for combinatorial optimization requires more than standalone algorithmic results; it calls for end-to-end case studies that integrate problem modelling, quantum execution, and classical refinement into practical workflows. This paper presents a sandbox platform for experimenting with hybrid quantum-classical workflows in graph optimization, enabling the systematic study of end-to-end optimization pipelines. Using our platform, we investigate three classically intractable and mutually reducible graph problems -- Minimum Vertex Cover, Maximum Independent Set, and Maximum Clique -- by transforming them into an unconstrained problem and solving the resulting instances with QAOA on IBM platforms. Our workflow combines classical pre-processing to reduce instance size, quantum optimization on the reduced problem, and classical postprocessing to map quantum outputs to high-quality feasible solutions, thereby avoiding direct constraint encoding in the quantum circuit. We evaluate the approach on synthetic graphs, benchmark instances, and real-world networks, and report hardware experiments on IBM Quantum System One at PINQ2 in Bromont, Quebec, powered by IBM's 156-qubit Heron r2 processor on graphs up to 128 vertices, with circuits involving up to 128 qubits and 13,555 two-qubit gates. The results illustrate how sandbox-style end-to-end experimentation can expose bottlenecks, clarify the role of classical-quantum workload partitioning, and provide domain experts and practitioners with a practical guide for interpreting quantum optimization outputs and assessing quantum utility on the road to quantum advantage in combinatorial optimization.
Demonstrating quantum advantage for combinatorial optimization requires more than standalone algorithmic results; it calls for end-to-end case studies that integrate problem modelling, quantum execution, and classical refinement into practical workflows. This paper presents a sandbox platform for experimenting with hybrid quantum-classical workflows in graph optimization, enabling the systematic study of end-to-end optimization pipelines. Using our platform, we investigate three classically intractable and mutually reducible graph problems -- Minimum Vertex Cover, Maximum Independent Set, and Maximum Clique -- by transforming them into an unconstrained problem and solving the resulting instances with QAOA on IBM platforms. Our workflow combines classical pre-processing to reduce instance size, quantum optimization on the reduced problem, and classical postprocessing to map quantum outputs to high-quality feasible solutions, thereby avoiding direct constraint encoding in the quantum circuit. We evaluate the approach on synthetic graphs, benchmark instances, and real-world networks, and report hardware experiments on IBM Quantum System One at PINQ2 in Bromont, Quebec, powered by IBM's 156-qubit Heron r2 processor on graphs up to 128 vertices, with circuits involving up to 128 qubits and 13,555 two-qubit gates. The results illustrate how sandbox-style end-to-end experimentation can expose bottlenecks, clarify the role of classical-quantum workload partitioning, and provide domain experts and practitioners with a practical guide for interpreting quantum optimization outputs and assessing quantum utility on the road to quantum advantage in combinatorial optimization.
We perform an end-to-end benchmark of a hybrid sequential quantum computing (HSQC) solver for higher-order unconstrained binary optimization (HUBO), executed on IBM Heron r3 quantum processors to evaluate the potential of current quantum hardware for combinatorial optimization with sub-second end-to-end runtimes. All reported runtimes include the complete pipeline--from preprocessing to QPU execution and postprocessing--under strict wall-clock accounting. Across 20 benchmark instances, a single hybrid attempt produces high-quality solutions in less than one second, matching the ground-state energy in 14 cases. At the same runtime, CPU-based solvers, including simulated annealing, memetic tabu search, and EasySolve, do not reach the value obtained by HSQC, whereas an enhanced parallel tempering method and the GPU-accelerated solver ABS3 reach or surpass it. These results show that HSQC, executed on a single QPU, can achieve performance competitive with strong classical solvers running on 128 vCPUs or 8 NVIDIA A100 GPUs, while also providing a reproducible system-level benchmark for tracking progress as quantum hardware and hybrid sequential workflows improve.
Despite rapid progress in quantum optimization, broad real-hardware benchmarks comparing multiple algorithmic families across diverse combinatorial problems under a common protocol remain limited. We benchmark gate-based quantum optimization on four NP-hard problems: multi-dimensional knapsack (MDKP), maximum independent set (MIS), quadratic assignment (QAP), and market-share (MSP). We study VQE, CVaR-VQE, standard, multi-angle, and warm-start QAOA, together with qubit-efficient PCE and QRAO, on IBM Heron r1/r2 processors using resilience-level-2 mitigation. To our knowledge, this includes the first real-hardware QRAO results and the first multi-problem PCE hardware benchmark.
Across 247 method-instance combinations, we report transpiled circuit size, hardware outcomes, and an independent-error gate-count fidelity proxy, Fest. For MDKP and MIS, an empirical operating point near Fest≈0.1, corresponding to about 770 two-qubit gates at the median Heron-r2 CZ error rate, marks the onset of noise-dominated execution. QAP exposes a separate bottleneck: dense one-hot encodings and an exponentially sparse feasible manifold, with feasible fraction 10!/2100 at n=10; no tested hardware method produces a feasible assignment. Compiled QAOA-family circuits are generally noise dominated, and a matched uniform-random control shows that most feasible low-fidelity outcomes fall within the random range, apart from one finite-sample MIS warm-start exception. A SWAP-aware, fractional-gate, Nighthawk-topology compilation counterfactual reduces two-qubit counts but leaves all circuits below Fest=10−3. These conclusions apply to the tested implementations rather than QAOA in general. Qubit-efficient methods extend runnable instance sizes, but only within the empirical fidelity budget.
Quantum annealing is a proposed combinatorial optimization technique meant to exploit quantum mechanical effects such as tunneling and entanglement. Real-world quantum annealing-based solvers require a combination of annealing and classical pre- and post-processing; at this early stage, little is known about how to partition and optimize the processing. This article presents an experimental case study of quantum annealing and some of the factors involved in real-world solvers, using a 504-qubit D-Wave Two machine and the graph isomorphism problem. To illustrate the role of classical pre-processing, a compact Hamiltonian is presented that enables a reduced Ising model for each problem instance. On random N-vertex graphs, the median number of variables is reduced from N(2) to fewer than N log2 N and solvable graph sizes increase from N = 5 to N = 13. Additionally, error correction via classical post-processing majority voting is evaluated. While the solution times are not competitive with classical approaches to graph isomorphism, the enhanced solver ultimately classified correctly every problem that was mapped to the processor and demonstrated clear advantages over the baseline approach. The results shed some light on the nature of real-world quantum annealing and the associated hybrid classical-quantum solvers.
Near-term quantum approaches to combinatorial optimization are limited by qubit counts, circuit fidelity, sampling cost, and the difficulty of encoding constraints, while machine learning is increasingly used to configure and control quantum optimization workflows. We call such workflows adaptive: decisions conventionally fixed in advance, from formulation and penalties to shot budgets, backends, and whether to invoke a quantum processor at all, are made by learned policies that respond to the instance, the progress of the solve, or the hardware. This review examines three paradigms, AI for quantum optimization, quantum for AI-driven optimization, and AI-quantum co-optimization, and organizes the literature by the decision being learned rather than by application. A structured search identified a 119-paper corpus, of which 67 papers form an evidence core coded in detail. The evidence is considerably stronger for AI-assisted quantum optimization than for the reverse direction: learning already reduces quantum evaluations, improves initialization, supports decomposition and penalty control, and mitigates noise, whereas evidence that quantum computation improves learned optimizers remains largely confined to small-scale simulation. Experimental controls are thin: of 52 studies in the three paradigms, 22 include no classical baseline, the quantum contribution is fully isolated in 10 of 24 where an ablation applies, and the median largest experiment uses 16 qubits. We introduce an M0-M5 evidence hierarchy, from simulation to practical quantum advantage, and find no broadly convincing result at the highest level. We argue that scaling is increasingly a systems problem: the question is not only whether a problem fits on a quantum processor, but how classical and quantum resources should be allocated across the optimization process.
This dissertation addresses fundamental challenges in solving constrained combinatorial optimization problems using hybrid quantum-classical computing approaches. Quantum computing shows promise for tackling computationally intractable problems, current approaches face significant limitations. The path forward in computing involves combining quantum and classical computing approaches. Quantum processing units (QPUs) will play a key role in accelerating algorithms in optimization, simulation, and machine learning applications beyond what classical computers can achieve on their own, similar to how graphical processing units (GPUs) have become integral to general-purpose computing. The current generation of quantum devices is limited by noise and circuit depth constraints, making it challenging to gain practical utility. Hybrid quantum-classical approaches, which combine quantum computing resources with classical computing resources, are essential for addressing these limitations on near-term devices. In the area of combinatorial optimization, current practical state-of-the-art hybrid approaches are variational in nature, relying on parameterized quantum circuits to encode solutions and classical optimization techniques to find optimal parameters. While the ultimate goal of solving computationally intractable problems is to find provably optimal solutions, practical constraints of real-world scenarios often necessitate focusing on efficiently obtaining high-quality, near-optimal solutions. The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical approach for tackling these challenging problems that are encoded using quadratic and higher-order unconstrained binary optimization problems (QUBO and HUBO). QAOA alternates between two operators: a problem-specific cost Hamiltonian that encodes the optimization objective, and a mixer Hamiltonian that explores the solution space. The algorithm's parameters are iteratively optimized using classical methods to maximize the probability of sampling high-quality solutions. However, these methods often struggle with solution feasibility, especially for problems where the solutions must satisfy specific constraints. Penalty-based encodings require careful parameter tuning and risk producing infeasible solutions, while feasibility-preserving methods demand deeper quantum circuits that are challenging to implement on near-term quantum devices. In this dissertation, we present novel strategies that reduce or omit dependency on penalty parameters while maintaining solution quality, effectively capturing both optimal and near-optimal solutions, and balancing the trade-offs between circuit depth and solution feasibility. We present SCOOP, a novel QAOA-based framework for solving constrained optimization problems. SCOOP transforms a constrained problem into an unconstrained counterpart, forming SCOOP problem twins. QAOA operates on the unconstrained twin to identify potential optimal and near-optimal solutions. Effective classical post-processing reduces the solution set to the constrained problem space. Our SCOOP approach is solution-enhanced, objective-function-compatible, and scalable. We demonstrate the effectiveness of the SCOOP framework on selected NP-hard problems, some of which can be encoded quadratically (such as Minimum Vertex Cover and Maximum Independent Set), and some that can be encoded using higher-order terms (such as Minimum Dominating Set, Minimum Maximal Matching, and Minimum Set Cover). We also apply our framework to constrained combinatorial optimization problems that can be solved in polynomial time, such as Minimum Edge Cover and Maximum Matching. Experimental results on full statevector simulators and tensor network simulators show that SCOOP significantly improves the probability of finding high-quality feasible solutions across various problem classes, on problems that can be encoded quadratically as well as problems that can be encoded using higher-order terms. We also demonstrate the effectiveness of SCOOP on real quantum hardware, specifically IBM Quantum Heron R2 processors with 156 qubits, showing that SCOOP can be applied to practical problems with current quantum hardware limitations. This work makes significant contributions toward achieving quantum utility by enabling systematic investigation of constrained optimization problems on quantum devices. While current quantum computing research primarily focuses on unconstrained problems like MaxCut, real-world applications typically involve constraints. SCOOP provides a scalable framework for encoding constrained problems, extending quantum computing's practical utility beyond simple unconstrained cases. By eliminating dependency on penalty parameters and providing natural problem formulations, SCOOP makes it feasible to explore quantum solutions for constrained problems where classical exact methods become intractable. This capability is crucial for identifying promising avenues for demonstrating quantum utility across diverse constrained optimization problems, advancing the practical applicability of quantum computing approaches.
Emerging quantum processors provide an opportunity to explore new approaches for solving traditional problems in the post Moore’s law supercomputing era. However, the limited number of qubits makes it infeasible to tackle massive real-world datasets directly in the near future, leading to new challenges in utilizing these quantum processors for practical purposes. Hybrid quantum-classical algorithms that leverage both quantum and classical types of devices are considered as one of the main strategies to apply quantum computing to large-scale problems. In this article, we advocate the use of multilevel frameworks for combinatorial optimization as a promising general paradigm for designing hybrid quantum-classical algorithms. To demonstrate this approach, we apply this method to two well-known combinatorial optimization problems, namely, the Graph Partitioning Problem, and the Community Detection Problem. We develop hybrid multilevel solvers with quantum local search on D-Wave’s quantum annealer and IBM’s gate-model based quantum processor. We carry out experiments on graphs that are orders of magnitude larger than the current quantum hardware size, and we observe results comparable to state-of-the-art solvers in terms of quality of the solution. Reproducibility : Our code and data are available at Reference [1].