Quantum Algorithms for the Real-Time Control Loop: HHL Equilibrium Reconstruction, Disruption State-Estimation, and Edge-Turbulence Simulation with Resource-Honest Advantage Accounting
The real-time control loop of a compact spherical-tokamak breeder must reconstruct the equilibrium, watch for disruptions, and shape the plasma edge faster than a 9.66 plasma at =1.20 can move — a loop served today by classical GPU surrogates and a demonstrated in-winding quantum sensor. This paper asks, and answers with equal weight on both sides, where a quantum computer (as distinct from a quantum sensor) could accelerate that loop. We formalise a stack of four control-loop quantum algorithms as rigorous problems: an HHL-class linear solver for live Grad–Shafranov equilibrium reconstruction; a quantum-kernel state-estimator for disruption and runaway-electron precursor detection inside the shattered-pellet mitigation window; a Hamiltonian simulation of edge and scrape-off-layer turbulence to feed a predictive detachment controller; and a quantum-machine-learning surrogate for live transport prediction. For each we derive the governing operator, its discretization, and its fault-tolerant complexity, and we place them against the classical frontier through an explicit advantage inequality. Alongside the designs we report a completed characterization: the quantum-ML classifier loses discriminating power as it scales — area under the receiver-operating-characteristic curve falls from 0.817 at four qubits to 0.691 at eight, the calibrated signature of exponential kernel concentration — and a fault-tolerant kernel evaluation is 9–11 orders of magnitude slower than the classical route for this problem. Fault-tolerant resource estimates for the solver-class algorithms (edge/scrape-off-layer kinetics: ~44 logical qubits and ~19 days of single-instance runtime; linear kinetics: 11–19 logical qubits) confirm that none clears the advantage bar today, and the equilibrium solve, costed in absolute terms, is on the order of 10^17 Toffoli-equivalent T-gates — eleven orders of magnitude beyond the sub-millisecond loop budget on any plausible magic-state throughput. The honest conclusion is stated once: there is no measured quantum advantage on these control problems today. The deliverable is a validated set of algorithm designs, a resource-honest map of where a fault-tolerant machine will and provably will not help, and a dated roadmap that keeps the classical loop authoritative until a quantum route clears an explicit latency-and-fidelity bar. All numbers trace to frozen entries in the Kronos Physics De-Risking Register by their serial identifiers. Key results (frozen anchors): iter = 1.6 2^n. Methods & codes: FreeGSNKE, FreeGS, CGYRO, Sauter, QAOA, VQE, ADAPT-VQE. Live verification: 9 gate validator(s) with live-recompute cards (5 estimated, 4 reproduced). 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.
The real-time control loop of a compact spherical-tokamak breeder must reconstruct the equilibrium, watch for disruptions, and shape the plasma edge faster than a 9.66 plasma at =1.20 can move — a loop served today by classical GPU surrogates and a demonstrated in-winding quantum sensor. This paper asks, and answers with equal weight on both sides, where a quantum computer (as distinct from a quantum sensor) could accelerate that loop. We formalise a stack of four control-loop quantum algorithms as rigorous problems: an HHL-class linear solver for live Grad–Shafranov equilibrium reconstruction; a quantum-kernel state-estimator for disruption and runaway-electron precursor detection inside the shattered-pellet mitigation window; a Hamiltonian simulation of edge and scrape-off-layer turbulence to feed a predictive detachment controller; and a quantum-machine-learning surrogate for live transport prediction. For each we derive the governing operator, its discretization, and its fault-tolerant complexity, and we place them against the classical frontier through an explicit advantage inequality. Alongside the designs we report a completed characterization: the quantum-ML classifier loses discriminating power as it scales — area under the receiver-operating-characteristic curve falls from 0.817 at four qubits to 0.691 at eight, the calibrated signature of exponential kernel concentration — and a fault-tolerant kernel evaluation is 9–11 orders of magnitude slower than the classical route for this problem. Fault-tolerant resource estimates for the solver-class algorithms (edge/scrape-off-layer kinetics: ~44 logical qubits and ~19 days of single-instance runtime; linear kinetics: 11–19 logical qubits) confirm that none clears the advantage bar today, and the equilibrium solve, costed in absolute terms, is on the order of 10^17 Toffoli-equivalent T-gates — eleven orders of magnitude beyond the sub-millisecond loop budget on any plausible magic-state throughput. The honest conclusion is stated once: there is no measured quantum advantage on these control problems today. The deliverable is a validated set of algorithm designs, a resource-honest map of where a fault-tolerant machine will and provably will not help, and a dated roadmap that keeps the classical loop authoritative until a quantum route clears an explicit latency-and-fidelity bar. All numbers trace to frozen entries in the Kronos Physics De-Risking Register by their serial identifiers. Key results (frozen anchors): iter = 1.6 2^n. Methods & codes: FreeGSNKE, FreeGS, CGYRO, Sauter, QAOA, VQE, ADAPT-VQE. Live verification: 9 gate validator(s) with live-recompute cards (5 estimated, 4 reproduced). 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.
Quantum computing and quantum sensing are routinely proposed as accelerants for fusion; a serious design program should test those proposals quantitatively rather than assume them. We deliver a resource-honest map of where the quantum toolbox helps a compact spherical-tokamak breeder and its tandem-mirror burner, and where it does not, formalised as three coupled estimation problems and computed on a single GPU-HPC campaign. On the machine-learning front we build a quantum-kernel transport surrogate and measure its classifier area-under-curve (AUC) falling with encoding width — 0.817 at four qubits to 0.691 at eight — the signature of exponential kernel concentration, for which we give the variance bound and the induced shot-cost scaling; at a realistic 37-logical-qubit problem the quantum Gram evaluation is 9–11 orders of magnitude slower than its classical counterpart. On the sensing front we derive the entanglement-advantage condition from the quantum Fisher information and specify the in-cryostat magnetometry chain for the breeder centrepost (peak field 16.84 T, kept formally distinct from the burner plug at 26.49 T): an NMR/EPR probe as the absolute monitor to ≤1 ppm, a SQUID and an NV-diamond ensemble (1 pT Hz^-1/2) for the 1 mT quench precursor, with the entanglement knee located at a normalised signal ≈0.14. Every Kronos in-bore diagnostic sits below the knee, where entanglement buys only +0.00077 AUC; above it (deep error-field mapping) a Heisenberg-limited protocol carries the standard-quantum-limit sensitivity from 5.7×10^-7 to 5.7×10^-8 T at 100 probes. On the fault-tolerant front we recompute the resource ledger from first formulas — the Grover iteration count 1.62^n and the surface-code footprint 2d^2 physical qubits per logical qubit — across 50 rows with zero discrepancy, and date the single fusion-relevant near-term niche: linear Vlasov kinetics at 11–19 logical qubits around 2029±2, with nonlinear five-dimensional gyrokinetics beyond 2035 and fleet-scale Grover search never practical (1.8×10^18 iterations). Three further workloads keep the verdict resource-honest: full six-dimensional kinetic Hamiltonian simulation at ~44 logical qubits and ~19 days, fault-tolerant phase estimation of the exact REBCO/alloy ground state at ~238 logical qubits and ~10^9 Toffoli gates, and a Harrow–Hassidim–Lloyd equilibrium linear solve at ~10^17 T-gates — all calibration-only. Two of the six quantum-track items are already twin-testable classically: a materials variational eigensolver reproduces a model ground state to 3.9×10^-6 Ha and reaches chemical accuracy on 3d alloy fragments with 4–14× shallower circuits than unitary coupled cluster (no advantage claimed), and a tensor-network kinetic solver compresses the distribution function with exponential rank convergence (2.2×10^-10 at bond rank 16, 0.19× storage). The honest bottom line is that nothing on the design critical path waits on quantum hardware — and establishing that, with the numbers behind it, is the result. Key results (frozen anchors): iter = 1.8 ×10^18; iter = 1.6 2^n. Methods & codes: CGYRO, GENE, QAOA, VQE, ADAPT-VQE. Live verification: 11 gate validator(s) with live-recompute cards (5 reproduced, 5 estimated, 1 revised). 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.
Quantum computing and quantum sensing are routinely proposed as accelerants for fusion; a serious design program should test those proposals quantitatively rather than assume them. We deliver a resource-honest map of where the quantum toolbox helps a compact spherical-tokamak breeder and its tandem-mirror burner, and where it does not, formalised as three coupled estimation problems and computed on a single GPU-HPC campaign. On the machine-learning front we build a quantum-kernel transport surrogate and measure its classifier area-under-curve (AUC) falling with encoding width — 0.817 at four qubits to 0.691 at eight — the signature of exponential kernel concentration, for which we give the variance bound and the induced shot-cost scaling; at a realistic 37-logical-qubit problem the quantum Gram evaluation is 9–11 orders of magnitude slower than its classical counterpart. On the sensing front we derive the entanglement-advantage condition from the quantum Fisher information and specify the in-cryostat magnetometry chain for the breeder centrepost (peak field 16.84 T, kept formally distinct from the burner plug at 26.49 T): an NMR/EPR probe as the absolute monitor to ≤1 ppm, a SQUID and an NV-diamond ensemble (1 pT Hz^-1/2) for the 1 mT quench precursor, with the entanglement knee located at a normalised signal ≈0.14. Every Kronos in-bore diagnostic sits below the knee, where entanglement buys only +0.00077 AUC; above it (deep error-field mapping) a Heisenberg-limited protocol carries the standard-quantum-limit sensitivity from 5.7×10^-7 to 5.7×10^-8 T at 100 probes. On the fault-tolerant front we recompute the resource ledger from first formulas — the Grover iteration count 1.62^n and the surface-code footprint 2d^2 physical qubits per logical qubit — across 50 rows with zero discrepancy, and date the single fusion-relevant near-term niche: linear Vlasov kinetics at 11–19 logical qubits around 2029±2, with nonlinear five-dimensional gyrokinetics beyond 2035 and fleet-scale Grover search never practical (1.8×10^18 iterations). Three further workloads keep the verdict resource-honest: full six-dimensional kinetic Hamiltonian simulation at ~44 logical qubits and ~19 days, fault-tolerant phase estimation of the exact REBCO/alloy ground state at ~238 logical qubits and ~10^9 Toffoli gates, and a Harrow–Hassidim–Lloyd equilibrium linear solve at ~10^17 T-gates — all calibration-only. Two of the six quantum-track items are already twin-testable classically: a materials variational eigensolver reproduces a model ground state to 3.9×10^-6 Ha and reaches chemical accuracy on 3d alloy fragments with 4–14× shallower circuits than unitary coupled cluster (no advantage claimed), and a tensor-network kinetic solver compresses the distribution function with exponential rank convergence (2.2×10^-10 at bond rank 16, 0.19× storage). The honest bottom line is that nothing on the design critical path waits on quantum hardware — and establishing that, with the numbers behind it, is the result.Key results (frozen anchors): y_i = 0 eq; hbar = 2 ×28.03; iter = 1.8 ×10^18; iter = 1.6 2^n.Methods & codes: CGYRO, GENE, QAOA, VQE, ADAPT-VQE.Live verification: 11 gate validator(s) with live-recompute cards (5 reproduced, 5 estimated, 1 revised). See the Verification section and data/verification.csv.Related: Paper page · De-risking register · 3D model · Learn more about KronosPart 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.
Quantum computing is routinely invoked for fusion without a disciplined account of what is reachable now and what is a fault-tolerant-horizon problem. We supply that account for a specific machine — the Kronos Hyperion compact spherical-tokamak 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, elongation κ=2.0, major radius R_0=1.20 m, aspect ratio A=2.5, centrepost peak field 16.84 T) and its distinct tandem-mirror burner — through two grounded demonstrations, a resource frontier, and a pre-registered honest negative. First, a genuinely kinetic collisionless phenomenon, Landau damping, is reproduced on a 5-qubit circuit: casting the linearized Vlasov–Poisson system in an asymmetrically-weighted Fourier–Hermite basis and folding in the Poisson closure yields a Hermitian generator whose unitary evolution is a legitimate Hamiltonian-simulation target, and the circuit reproduces the analytic Landau rate to 1–7% across three wavenumbers, landing the complex dispersion at /=1.4157-0.1534 i against the textbook 1.4156-0.1533 i. Second, an adaptive variational quantum eigensolver (ADAPT-VQE) reaches chemical accuracy (better than 1.6 mHa) on the transition-metal fragments of the candidate low-activation refractory alloys, four orders of magnitude below the ~300mHa spread of density-functional exchange–correlation choices, with a fixed-ansatz depth 4–14× shallower than unitary coupled cluster. A third strand is a genuinely measured quantum capability: an in-winding nitrogen-vacancy (NV) diamond magnetometer resolves a ~100 superconducting-quench step at a per-shot signal-to-noise ratio of ~1.5×10^9 (sensitivity 0.189/Hz, bandwidth 125), giving single-shot quench-precursor detection — but its headline femto/pico-tesla sensitivity is forfeited in the reactor vessel, a second pre-registered negative we keep rather than hide. The honest boundaries are explicit: under a representative noise model the field-energy observable decoheres to the maximally-mixed floor 1/32 and the damping signature is lost, so present-day noisy hardware shows no advantage; and while the linear collisionless kinetic operator is an early-fault-tolerant target (11–19 logical qubits, 10^5–10^6 non-Clifford gates), the nonlinear five-dimensional gyrokinetic problem that actually sets confinement remains a fault-tolerant-horizon target (~10^2 logical qubits, 10^9–10^13 non-Clifford gates), and the Harrow–Hassidim–Lloyd equilibrium linear solve — the single most-invoked ``quantum inner kernel'' — costs ~1.8×10^17 T-gates once read-in and read-out are counted, a boundary and not a route this decade. Against the classical frontier — nonlinear CGYRO gyrokinetics and a quantum-inspired matrix-product-state solver that runs 21.2× faster at bond dimension 7, is the only feasible method past bond dimension 10, and drives its error to 2.2×10^-10 at bond dimension 16 — the classical tool is cheaper at every scale demonstrated here. The contribution is a set of validated encodings, a rigorous numerical formulation, and an honest, dated resource map, not a claim of quantum advantage. Key results (frozen anchors): k = 0.30 0.50. Methods & codes: CGYRO, QAOA, VQE, ADAPT-VQE. Live verification: 8 gate validator(s) with live-recompute cards (5 reproduced, 3 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. Previous version: DOI 10.5281/zenodo.22132180. Please cite this version.
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.
Quantum computing is routinely invoked for fusion without a disciplined account of what is reachable now and what is a fault-tolerant-horizon problem. We supply that account for a specific machine — the Kronos Hyperion compact spherical-tokamak breeder (design point I_p=9.66, Q=3.076, P_ fus=85.04, negative triangularity δ=-0.30, B_0=8, elongation κ=2.0, major radius R_0=1.20, aspect ratio A=2.5, centrepost peak field 16.84) and its distinct tandem-mirror burner — through two grounded demonstrations, a resource frontier, and a pre-registered honest negative. First, a genuinely kinetic collisionless phenomenon, Landau damping, is reproduced on a 5-qubit circuit: casting the linearized Vlasov–Poisson system in an asymmetrically-weighted Fourier–Hermite basis and folding in the Poisson closure yields a Hermitian generator whose unitary evolution is a legitimate Hamiltonian-simulation target, and the circuit reproduces the analytic Landau rate to 1–7% across three wavenumbers, landing the complex dispersion at /=1.4157-0.1534 i against the textbook 1.4156-0.1533 i. Second, an adaptive variational quantum eigensolver (ADAPT-VQE) reaches chemical accuracy (better than 1.6 mHa) on the transition-metal fragments of the candidate low-activation refractory alloys, four orders of magnitude below the ~300mHa spread of density-functional exchange–correlation choices, with a fixed-ansatz depth 4–14× shallower than unitary coupled cluster. A third strand is a genuinely measured quantum capability: an in-winding nitrogen-vacancy (NV) diamond magnetometer resolves a ~100 superconducting-quench step at a per-shot signal-to-noise ratio of ~1.5×10^9 (sensitivity 0.189/Hz, bandwidth 125), giving single-shot quench-precursor detection — but its headline femto/pico-tesla sensitivity is forfeited in the reactor vessel, a second pre-registered negative we keep rather than hide. The honest boundaries are explicit: under a representative noise model the field-energy observable decoheres to the maximally-mixed floor 1/32 and the damping signature is lost, so present-day noisy hardware shows no advantage; and while the linear collisionless kinetic operator is an early-fault-tolerant target (11–19 logical qubits, 10^5–10^6 non-Clifford gates), the nonlinear five-dimensional gyrokinetic problem that actually sets confinement remains a fault-tolerant-horizon target (~10^2 logical qubits, 10^9–10^13 non-Clifford gates), and the Harrow–Hassidim–Lloyd equilibrium linear solve — the single most-invoked ``quantum inner kernel'' — costs ~1.8×10^17 T-gates once read-in and read-out are counted, a boundary and not a route this decade. Against the classical frontier — nonlinear CGYRO gyrokinetics and a quantum-inspired matrix-product-state solver that runs 21.2× faster at bond dimension 7, is the only feasible method past bond dimension 10, and drives its error to 2.2×10^-10 at bond dimension 16 — the classical tool is cheaper at every scale demonstrated here. The contribution is a set of validated encodings, a rigorous numerical formulation, and an honest, dated resource map, not a claim of quantum advantage.Key results (frozen anchors): k = 0.30 0.50; mathrmSNR = 1.5 ×10^9.Methods & codes: CGYRO, QAOA, VQE, ADAPT-VQE.Live verification: 8 gate validator(s) with live-recompute cards (5 reproduced, 3 estimated). See the Verification section and data/verification.csv.Related: Paper page · De-risking register · 3D model · Learn more about KronosPart 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.Note (REPLACE): This record replaces and supersedes DOI 10.5281/zenodo.22132180; please cite this version.
Quantum computing, a prominent non-Von Neumann paradigm beyond Moore's law, can offer superpolynomial speedups for certain problems. Yet its advantages in efficiency for tasks like machine learning remain under investigation, and quantum noise complicates resource estimations and classical comparisons. We provide a detailed estimation of space, time, and energy resources for fault-tolerant superconducting devices running the Harrow-Hassidim-Lloyd (HHL) algorithm, a quantum linear system solver relevant to linear algebra and machine learning. Excluding memory and data transfer, possible quantum advantages over the classical conjugate gradient method could emerge at N≈233∼248 or even lower, requiring O(105) physical qubits, O(1012∼1013) Joules, and O(106) seconds under surface code fault-tolerance with three types of magic state distillation (15-1, 116-12, 225-1). Key parameters include condition number, sparsity, and precision κ,s≈O(10∼100), ε∼0.01, and physical error 10−5. Our resource estimator adjusts N,κ,s,ε, providing a map of quantum-classical boundaries and revealing where a practical quantum advantage may arise. Our work quantitatively determine how advanced a fault-tolerant quantum computer should be to achieve possible, significant benefits on problems related to real-world.