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 $N \approx 2^{33} \sim 2^{48}$ or even lower, requiring ${O}(10^5)$ physical qubits, ${O}(10^{12}\sim10^{13})$ Joules, and ${O}(10^6)$ 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 $κ, s\approx{O}(10\sim100)$, $ε\sim0.01$, 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.
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.
Quantum computers hold promise for solving problems intractable for classical computers, especially those with high time or space complexity. Practicalquantum advantagecan be said to exist for such problems when the end-to-end time for solving such a problem using a classical algorithm exceeds that required by a quantum algorithm. Reducing the power flow (PF) problem into a linear system of equations allows for the formulation of quantum PF (QPF) algorithms, which are based on solving methods for quantum linear systems such as the Harrow-Hassidim-Lloyd (HHL) algorithm. Speedup from using QPF algorithms is often claimed to be exponential when compared to classical PF solved by state-of-the-art algorithms. We investigate the potential for practical quantum advantage in solving QPF compared to classical methods on gate-based quantum computers. Notably, this paper does not present a new QPF solving algorithm but scrutinizes the end-to-end complexity of the QPF approach, providing a nuanced evaluation of the purported quantum speedup in this problem. Our analysis establishes a best-case bound for the HHL-based quantum power flow complexity, conclusively demonstrating that the HHL-based method has higher runtime complexity compared to the classical algorithm for solving the direct current power flow (DCPF) and fast decoupled load flow (FDLF) problem. Notably, our analysis and conclusions can be extended to any quantum linear system solver with rigorous performance guarantees, based on the known complexity lower bounds for this problem. Additionally, we establish that for potential practical quantum advantage (PQA) to exist it is necessary to consider DCPF-type problems with a very narrow range of condition number values and readout requirements.
The quantum algorithm by Harrow, Hassidim, and Lloyd solves a system of N linear equations and achieves exponential speedup over classical algorithms under certain conditions. The advantage to the algorithm is that log(N) rather than N registers are required. Given an N x N matrix A and vectors x and b, the quantum algorithm seeks to find x such that Ax = b. By representing vector b as a superposition of quantum states |b>, quantum phase estimation is used to find the corresponding eigenvalues of A. Applying the inverse Fourier transform, we solve for |x> such that |x> = A-1|b>. We model the algorithm using a quantum circuit diagram, with data qubits encoded using the Steane code for fault tolerant quantum phase estimation. Fresh ancilla for error correction are provided using an oracular pipelined ancilla architecture. We then analyze the classical and quantum resources needed for implementation. The significance of this case study is to examine how classical and quantum resources interact in implementing this algorithm. The issues raised in this analysis, such as fault tolerant phase estimation using pipelined ancilla, garbage collection, and the preparation of I/O registers to this architecture, will be explored in more detail in future research.
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.
While quantum computers promise to solve classically intractable problems, identifying the point at which fault-tolerant quantum computation outperforms the best classical algorithms for practical applications remains an outstanding challenge. Here we establish a concrete quantum-classical crossover for quantum many-body dynamics under realistic hardware conditions. We introduce a scalable fault-tolerant framework that combines coherent observable estimation with a space-time-efficient implementation of non-Clifford rotations, suppressing the residual logical errors that limit existing partially fault-tolerant approaches. A benchmark against state-of-the-art tensor-network and variational Monte Carlo algorithms reveals a concrete crossover for mixed-field Ising dynamics at modest system sizes. For a physical error rate of p=10−3, fault-tolerant simulation requires approximately 2 hours and 3.7×105 physical qubits for a 100-site 1D system, whereas tensor network approaches would require about 100 years. For 2D models, where rapid entanglement growth limits the classical evolution time, we project quantum runtimes within minutes. A physical error rate of p=10−4 leads to at least an order of magnitude reduction in qubit count (3.1×104 physical qubits) and runtime (minutes for 1D and seconds for 2D). The reduction in quantum runtime arises from our improved rotation-state injection and co-design of quantum error correction and observable-estimation protocols, which jointly suppress logical-error accumulation and reduce sampling overhead. Our results establish a scalable route towards practical quantum advantage and identify quantitative engineering targets for future fault-tolerant architectures.
Quantum algorithms are often represented in terms of quantum circuits operating on ideal (logical) qubits. However, the practical implementation of these algorithms poses significant challenges. Many quantum algorithms require a substantial number of logical qubits, and the inherent susceptibility to errors of quantum computers require quantum error correction. The integration of error correction introduces overhead in terms of both space (physical qubits required) and runtime (how long the algorithm needs to be run for). This paper addresses the complexity of comparing classical and quantum algorithms, primarily stemming from the additional quantum error correction overhead. We propose a comprehensive framework that facilitates a direct and meaningful comparison between classical and quantum algorithms. By acknowledging and addressing the challenges introduced by quantum error correction, our framework aims to provide a clearer understanding of the comparative performance of classical and quantum computing approaches. This work contributes to understanding the practical viability and potential advantages of quantum algorithms in real-world applications. We apply our framework to quantum cryptanalysis, since it is well known that quantum algorithms can break factoring and discrete logarithm based cryptography and weaken symmetric cryptography and hash functions. In order to estimate the real-world impact of these attacks, apart from tracking the development of fault-tolerant quantum computers it is important to have an estimate of the resources needed to implement these quantum attacks. This analysis provides state-of-the art snap-shot estimates of the realistic costs of implementing quantum attacks on these important cryptographic algorithms, assuming quantum fault-tolerance is achieved using surface code methods, and spanning a range of potential error rates. These estimates serve as a guide for gauging the realistic impact of these algorithms and for benchmarking the impact of future advances in quantum algorithms, circuit synthesis and optimization, fault-tolerance methods and physical error rates.
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.