Quantum Optimization and Reinforcement Learning for Real-Time Plasma Control: QUBO/QAOA Actuator Scheduling and Quantum RL for Shape Control, without Claimed Advantage
Real-time control of a compact spherical-tokamak breeder poses, at several points, a combinatorial-optimization problem inside a control cycle: which actuators to schedule, how to lay out a discrete blanket, and — the control-relevant case — how to sequence poloidal- and toroidal-field coil currents under hard constraints while tracking a shape and vertical-position target. Because these problems are naturally binary and quadratic, they are the canonical near-term target for quantum optimization, and it is a fair, testable question whether a quantum processor solves them faster or better than a classical one. This paper answers that question with a formal treatment and a pre-registered result. We derive compact quadratic-unconstrained-binary-optimization (QUBO) encodings for all three problems from first principles — pulse cadence as a number-partitioning Ising model, blanket layout as a penalty-embedded assignment, and coil-current/actuator scheduling as a fixed-point-encoded quadratic tracking objective built on the free-boundary equilibrium shape-response Jacobian — and map each to two fault-tolerant primitives, a depth-three quantum approximate optimization ansatz (QAOA, p=3) and a D\"urr–Hyer Grover minimum-finder. At the frozen breeder design point (I_p=9.66 MA, Q=3.076, P_fus=85.04 MW, negative triangularity δ=-0.30, B_0=8 T, κ=2.0, =1.20 m, aspect ratio A=2.5, centrepost peak field 16.84 T) the design-scale instances need 23–61 logical qubits and 5×10^3–5.5×10^7 T-gates, verified against the Fowler surface-code and D\"urr–Hyer formulas across 50 resource rows with zero discrepancies. On the question of whether quantum hardware helps, we report an explicit negative: classical simulated annealing matches or beats QAOA on all three breeder QUBOs out to n=120 variables; the exact-to-heuristic crossover sits at n=12–16 (exact optima in 0.03–292 s of wall time); the QAOA approximation ratio plateaus at 0.90–0.95 and the circuits become classically unverifiable past n≈24; and a fleet-scale (n=120) Grover formulation would need 1.8×10^18 oracle calls and 1.2×10^23 T-gates — a resource boundary, not a route. We therefore place variational quantum reinforcement learning for shape and vertical-position control on a dated, benchmark-gated schedule: a quantum policy is admitted only if it meets or beats the classical controller at equal wall-clock with a control-barrier-certified safety filter held inviolate. All quantum and classical runs were carried out on the NVIDIA GPU HPC campaign. The deliverable is the honest, resource-grounded map of where quantum computing does — and does not — pay for real-time fusion control. Key results (frozen anchors): N_T = 1.2 ×10^23. Methods & codes: FreeGSNKE, FreeGS, Sauter, QAOA, VQE, ADAPT-VQE. Live verification: 4 gate validator(s) with live-recompute cards (2 reproduced, 1 revised, 1 estimated). See the Verification section and data/verification.csv. Related: Paper page · De-risking register · 3D model · Learn more about Kronos Part of the 2026 Kronos publication series; independently re-run and stamped in the Kronos de-risking register (DOI 10.5281/zenodo.22645689). All numerical values are frozen design-point anchors; see the register. Public research artifact. No proprietary, financial, or supply-chain information is included.
Real-time control of a compact spherical-tokamak breeder poses, at several points, a combinatorial-optimization problem inside a control cycle: which actuators to schedule, how to lay out a discrete blanket, and — the control-relevant case — how to sequence poloidal- and toroidal-field coil currents under hard constraints while tracking a shape and vertical-position target. Because these problems are naturally binary and quadratic, they are the canonical near-term target for quantum optimization, and it is a fair, testable question whether a quantum processor solves them faster or better than a classical one. This paper answers that question with a formal treatment and a pre-registered result. We derive compact quadratic-unconstrained-binary-optimization (QUBO) encodings for all three problems from first principles — pulse cadence as a number-partitioning Ising model, blanket layout as a penalty-embedded assignment, and coil-current/actuator scheduling as a fixed-point-encoded quadratic tracking objective built on the free-boundary equilibrium shape-response Jacobian — and map each to two fault-tolerant primitives, a depth-three quantum approximate optimization ansatz (QAOA, p=3) and a D"urr–Hyer Grover minimum-finder. At the frozen breeder design point (I_p=9.66 MA, Q=3.076, P_fus=85.04 MW, negative triangularity δ=-0.30, B_0=8 T, κ=2.0, =1.20 m, aspect ratio A=2.5, centrepost peak field 16.84 T) the design-scale instances need 23–61 logical qubits and 5×10^3–5.5×10^7 T-gates, verified against the Fowler surface-code and D"urr–Hyer formulas across 50 resource rows with zero discrepancies. On the question of whether quantum hardware helps, we report an explicit negative: classical simulated annealing matches or beats QAOA on all three breeder QUBOs out to n=120 variables; the exact-to-heuristic crossover sits at n=12–16 (exact optima in 0.03–292 s of wall time); the QAOA approximation ratio plateaus at 0.90–0.95 and the circuits become classically unverifiable past n≈24; and a fleet-scale (n=120) Grover formulation would need 1.8×10^18 oracle calls and 1.2×10^23 T-gates — a resource boundary, not a route. We therefore place variational quantum reinforcement learning for shape and vertical-position control on a dated, benchmark-gated schedule: a quantum policy is admitted only if it meets or beats the classical controller at equal wall-clock with a control-barrier-certified safety filter held inviolate. All quantum and classical runs were carried out on the NVIDIA GPU HPC campaign. The deliverable is the honest, resource-grounded map of where quantum computing does — and does not — pay for real-time fusion control. Key results (frozen anchors): N_T = 1.2 ×10^23. Methods & codes: FreeGSNKE, FreeGS, Sauter, QAOA, VQE, ADAPT-VQE. Live verification: 4 gate validator(s) with live-recompute cards (2 reproduced, 1 revised, 1 estimated). See the Verification section and data/verification.csv. Related: Paper page · De-risking register · 3D model · Learn more about Kronos Part of the 2026 Kronos publication series; independently re-run and stamped in the Kronos de-risking register (DOI 10.5281/zenodo.22645689). All numerical values are frozen design-point anchors; see the register. Public research artifact. No proprietary, financial, or supply-chain information is included.
Large-scale combinatorial scheduling problems -- job-shop scheduling, flow-shop scheduling, and vehicle routing among them -- remain NP-hard at practically relevant scale, and quantum computing has entered this space along two related tracks: genuinely quantum solvers, principally quantum annealers and gate-based devices running the quantum approximate optimization algorithm (QAOA), and quantum-inspired classical solvers, such as coherent Ising machines and digital annealers, that borrow the same Ising/QUBO problem structure without requiring quantum hardware. This article synthesizes both tracks as applied specifically to scheduling. We trace the theoretical foundations from quantum and simulated annealing through the QUBO formulation techniques that make scheduling problems solver-ready, then review direct applications to job-shop, flow-shop, and vehicle-routing problems, reporting the concrete, verified numbers the underlying studies provide. The consistent finding is that quantum hardware remains constrained by qubit connectivity, embedding overhead, and shallow achievable circuit depth to small benchmark instances, while quantum-inspired classical solvers, unconstrained by qubit count, already reach scheduling-relevant problem sizes with reported performance competitive with or exceeding classical heuristics. We close by identifying benchmarking rigor, hardware connectivity, and the near-total absence of work on dynamic, online scheduling as the field's clearest open problems.
Rotational irrigation scheduling in water-scarce Central Asia is a densely coupled combinatorial problem: soil-moisture memory links each irrigation decision to all later days within a zone, field adjacency couples zones on overlapping window days, and rigid canal rotations quantize water delivery in time. We formulate it as a Quadratic Unconstrained Binary Optimization (QUBO) by linearizing the root-zone water balance, so the quadratic crop-stress objective generates the physical couplings as 2-local Ising interactions with no higher-order terms; only the water-budget constraint requires an artificial all-to-all penalty, which we certify with an instance-adaptive weight bound an order of magnitude tighter than generic prescriptions. Every instance is built from observed data for a cotton district in Khorezm, Uzbekistan: NASA POWER meteorology, FAO-56 Penman--Monteith evapotranspiration, SoilGrids~2.0 hydraulics, measured capillary fluxes, and documented canal-rotation windows that enter as qubit-count reductions. We benchmark four tiers -- exact solvers, matched-budget heuristics, ideal-statevector quantum approximate optimization algorithm (QAOA), and noise-model plus IBM Heron execution -- and add a scaling study on soil instances up to 584 variables. Exact branch-and-bound proves optimality in seconds through 150 variables, and heuristic degradation at fixed evaluation budget is repaired by scaling the budget, so no classical scalability wall appears at deployment-relevant sizes, and none is claimed. On hardware, the informative signal is optimum-sampling enrichment over uniform sampling. We claim no quantum advantage; we deliver a physically grounded, data-complete, reproducible encoding of a societally critical scheduling problem for the quantum-utility era.
This thesis is devoted to four modern quantum optimisation algorithms and their applications. In the first chapter, we provide a complete Quantum Approximate Optimisation Algorithm (QAOA) pipeline for the discrete global minimum-variance portfolio (DGMVP) problem. We introduce an ansatz with efficient initial-state preparation, new hard mixing operators satisfying the constraints of the DGMVP problem, and scalable cost operators utilising binary encoding of the qubits. Benchmarking the method with the most effective optimisation strategies---dual annealing and unfrozen layerwise optimisation---indicates a scaling advantage over classical methods in finding the global optimum of the DGMVP problem. In the second chapter, we show an end-to-end quantum/quantum-inspired Goemans-Williamson semidefinite programming (SDP) method for relaxing Quadratic Unconstrained Binary Optimisation (QUBO) problems and extracting their rounding energy. Our method identifies a class of QUBO instances that is effectively solvable in polylogarithmic time relative to the problem dimension, under the assumptions of Pauli sparsification of the QUBO cost matrix, reducibility of the diagonal constraints, and efficient evaluation of the corresponding Gibbs-state expectation values. Further, we identify the commuting 1D Hamiltonian and the Kronecker graph as practically important instances. Our numerical experiments show that our method solves the SDP and QUBO with up to 250 variables for 1D Hamiltonian instances and achieves results competitive with the best classical solver for Kronecker graphs, giving strong evidence of the power of our algorithm. In the third chapter, we improve the current QAOA method with an iterative warm-start initial-state preparation method. We implement the t-th order statistic state as the warm-start initial state generated from each QAOA optimisation loop. We achieve an improved approximation ratio when solving the 3-regular MaxCut problem, and find that the worst-case approximation ratio converges to that of the best classical algorithm for p=1 QAOA. Additionally, our method achieves more favourable scaling than standalone QAOA, single-string warm-started QAOA, and a classical constrained-sampling approach when solving the DGMVP problem. In the fourth chapter, we provide a hard-constrained Grover adaptive search (GAS) method for fixed-sum quadratic binary optimisation problems, and an extension of the alternating direction method of multipliers (ADMM) framework for fixed-sum quartic binary optimisation. Our method guarantees at least a quadratic speed-up over the classical method, and exponentially reduces the gate count for solving fixed-sum binary optimisation when compared with the soft-constrained GAS in the fault-tolerant quantum computing regime. In summary, our contributions include efficient algorithms and novel frameworks that substantially improve performance and broaden the applicability of these quantum optimisation methods. The proposed methods exhibit favourable scaling behaviour and indicate potential quantum advantages in solving challenging optimisation problems, thereby motivating further research at the interface of quantum computing and optimisation.
The Quantum Approximate Optimisation Algorithm (QAOA) has emerged as a leading candidate for near-term quantum advantage in combinatorial optimisation, but most evaluations remain confined to synthetic benchmarks such as random MaxCut instances. Translating QAOA to real-world problem instances -- with irregular graph structures, heterogeneous constraint types, and practical scale requirements -- introduces additional challenges including QUBO formulation overhead, hardware embedding costs, and the performance gap between QAOA on structured real instances vs. random benchmarks. This paper presents the Real-World QAOA (RW-QAOA) framework, a systematic methodology for applying QAOA to four real-world application domains: supply chain network optimisation, financial portfolio rebalancing, telecommunications frequency assignment, and traffic signal timing optimisation. RW-QAOA introduces three methodological contributions: a domain-adaptive QUBO formulator (DAQF) encoding real-world constraints as penalised QUBO terms with penalty coefficient auto-tuning; a warm-start initialisation strategy (WSIS) using classical heuristic solutions to seed QAOA parameter search; and a layered noise adaptation protocol (LNAP) adjusting circuit depth based on real-time device calibration data. RW-QAOA is evaluated on real problem instances from industry partners across all four domains on IBM Quantum Eagle (127-qubit). RW-QAOA achieves solution quality within 4.2% of classical exact solvers across all four domains, outperforming classical greedy heuristics by 12.4% (SD = 3.8%) on average. WSIS reduces QAOA parameter convergence iterations by 48.4%. LNAP improves approximation ratio by 8.4pp vs. fixed-depth QAOA under device noise variation.
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.
Large-scale combinatorial scheduling problems -- job-shop scheduling, flow-shop scheduling, and vehicle routing among them-- remain NP-hard at practically relevant scale, and quantum computing has entered this space along two related tracks:genuinely quantum solvers, principally quantum annealers and gate-based devices running the quantum approximateoptimization algorithm (QAOA), and quantum-inspired classical solvers, such as coherent Ising machines and digitalannealers, that borrow the same Ising/QUBO problem structure without requiring quantum hardware. This article synthesizesboth tracks as applied specifically to scheduling. We trace the theoretical foundations from quantum and simulated annealingthrough the QUBO formulation techniques that make scheduling problems solver-ready, then review direct applications tojob-shop, flow-shop, and vehicle-routing problems, reporting the concrete, verified numbers the underlying studies provide. Theconsistent finding is that quantum hardware remains constrained by qubit connectivity, embedding overhead, and shallowachievable circuit depth to small benchmark instances, while quantum-inspired classical solvers, unconstrained by qubitcount, already reach scheduling-relevant problem sizes with reported performance competitive with or exceeding classicalheuristics. We close by identifying benchmarking rigor, hardware connectivity, and the near-total absence of work on dynamic,online scheduling as the field's clearest open problems.