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Quantifying Accessible Advantage in Quantum Spectral Computation: A Task-Matched, Device-Calibrated Metric With a 6G Channel-Estimation Case Study

Sanjay Kumar Sharma, Ashutosh Anand, S. Natya

Published Jan 1, 2026• 0 citations
Publisher

Abstract

The Quantum Fourier Transform (QFT) uses $\mathcal {O}(n^{2})$ gates against the $\mathcal {O}(N\log N)$ operations of the Fast Fourier Transform, but this advantage is contingent on state preparation, readout, connectivity, decoherence, and on the classical algorithm chosen for comparison. We introduce the Quantum Fourier Advantage Index (QFAI), a task-, data- and reference-aware resource metric combining state preparation, task-dependent readout, native transpiled gate counts on real coupling maps, and a two-term calibrated noise model with relaxation. The QFAI is a conditional estimate of accessible advantage under stated input-access, task, reference-algorithm and stochastic-noise assumptions, not an intrinsic property of the QFT, and we separate asymptotic, resource-accounting and wall-clock advantage throughout. Across 9 datasets and 7 processors, the conventional dense-FFT reference yields broad windows on all-to-all hardware (peak QFAI 37.68, Quantinuum H2). Two findings qualify this. First, under a common synthetic noise convention and once SWAP-routing permutations are excluded from the measured infidelity, the gate-error amplification constant is modality-independent ( $\kappa _{g}=0.786$ , 0.778, 0.770); against per-qubit calibration snapshots of 4 real IBM processors it spans 0.349 to 0.769 and must be refitted per device. Second, replacing the dense FFT by the task-matched classical algorithm, a sublinear sparse FFT for dominant-mode extraction, reduces the peak to a near-break-even 0.17, whose sign reverses within a factor-of-two change of the unmeasured sparse-FFT constant; we prove this ratio is bounded by a constant, so no asymptotic separation exists, and we propagate fit, calibration and reference uncertainties into explicit bands. A 6G OFDM channel-estimation case study, an end-to-end model of the quantum processing pipeline rather than a measured receiver, shows the framework as a design tool: the quantum route recovers delay taps with precision up to 1.00 yet misses the $\mathrm {125.00~\mu \text {s} }$ slot deadline by five to eleven orders of magnitude, with the per-shot cost of state loading, reset and readout, not gate fidelity, the binding constraint, while a measured single-core dense FFT meets it at $N\le 4096$ . All results are simulation-based, with stated evidence tiers and a released hardware-validation campaign.

DOI
10.1109/access.2026.3743557

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Learning problems involving quantum data are natural candidates for demonstrating an advantage in quantum machine learning. Recent results indicate that, for certain tasks and under noiseless conditions, coherent processing of quantum data outperforms fixed-measurement schemes followed by classical processing. It remained uncertain whether this performance gap persists at a finite scale, and in the presence of noise that is unavoidable with current quantum devices. In this work, we present simulations and analysis of the performance of existing hardware on a learning problem known to exhibit asymptotic advantage, now subjected to noisy quantum data. Comparing coherent quantum processing directly against fixed-measurement schemes, our results demonstrate a clear performance separation at a scale of just 30 to 40 noisy qubits. Already at this scale, the fundamental bottleneck is no longer classical computation but data acquisition; matching the noisy coherent protocol with measure-first strategies would still require months or even years of measurements. By systematically evaluating hardware constraints such as state preparation, gate errors, readout errors, connectivity, and coherence times, we provide evidence that a demonstration of such a strong learning advantage is accessible on near-term devices.

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Cloud-accessible quantum processors enable direct execution of quantum algorithms on heterogeneous hardware platforms. Unlike classical systems, however, identical quantum circuits may exhibit substantially different behavior across devices due to architectural variations in qubit connectivity, gate fidelity, and coherence times. In this work, we systematically benchmark five representative quantum algorithms - Bell state preparation, GHZ state generation, Quantum Fourier Transform (QFT), Grover's Search, and the Quantum Approximate Optimization Algorithm (QAOA) - across trapped-ion, superconducting, and simulator backends using Amazon Braket. Performance metrics including fidelity, CHSH violation, success probability, circuit depth, and gate counts are evaluated. Our results demonstrate a strong dependence of algorithmic performance on hardware topology and noise characteristics. For example, 10-qubit GHZ states achieved fidelities above 0.8 on trapped-ion hardware, while superconducting platforms dropped below 0.15 due to routing overhead and accumulated two-qubit gate errors. These findings highlight the importance of hardware-aware algorithm selection and provide practical guidance for benchmarking in the NISQ era.

quant-pharXiv:2601.05286•Jan 8, 2026•0 citations
The Quantum Fourier Transform (QFT) uses O(n2)\mathcal {O}(n^{2})O(n2) gates against the O(Nlog⁡N)\mathcal {O}(N\log N)O(NlogN) operations of the Fast Fourier Transform, but this advantage is contingent on state preparation, readout, connectivity, decoherence, and on the classical algorithm chosen for comparison. We introduce the Quantum Fourier Advantage Index (QFAI), a task-, data- and reference-aware resource metric combining state preparation, task-dependent readout, native transpiled gate counts on real coupling maps, and a two-term calibrated noise model with relaxation. The QFAI is a conditional estimate of accessible advantage under stated input-access, task, reference-algorithm and stochastic-noise assumptions, not an intrinsic property of the QFT, and we separate asymptotic, resource-accounting and wall-clock advantage throughout. Across 9 datasets and 7 processors, the conventional dense-FFT reference yields broad windows on all-to-all hardware (peak QFAI 37.68, Quantinuum H2). Two findings qualify this. First, under a common synthetic noise convention and once SWAP-routing permutations are excluded from the measured infidelity, the gate-error amplification constant is modality-independent ( κg=0.786\kappa _{g}=0.786κg​=0.786 , 0.778, 0.770); against per-qubit calibration snapshots of 4 real IBM processors it spans 0.349 to 0.769 and must be refitted per device. Second, replacing the dense FFT by the task-matched classical algorithm, a sublinear sparse FFT for dominant-mode extraction, reduces the peak to a near-break-even 0.17, whose sign reverses within a factor-of-two change of the unmeasured sparse-FFT constant; we prove this ratio is bounded by a constant, so no asymptotic separation exists, and we propagate fit, calibration and reference uncertainties into explicit bands. A 6G OFDM channel-estimation case study, an end-to-end model of the quantum processing pipeline rather than a measured receiver, shows the framework as a design tool: the quantum route recovers delay taps with precision up to 1.00 yet misses the 125.00 μs\mathrm {125.00~\mu \text {s} }125.00 μs slot deadline by five to eleven orders of magnitude, with the per-shot cost of state loading, reset and readout, not gate fidelity, the binding constraint, while a measured single-core dense FFT meets it at N≤4096N\le 4096N≤4096 . All results are simulation-based, with stated evidence tiers and a released hardware-validation campaign.
Across public tabular benchmarks, quantum machine-learning (QML) models usually lose to carefully tuned classical baselines. We argue that this is a structural property of the datasets rather than merely a limitation of current models. Because an angle-encoded quantum neural network is a partial Fourier series, a genuine advantage can arise only when the target spectrum is simultaneously off-grid, of interaction order at least three, high-frequency, supported by near-independent features, and dense beyond practical enumeration. We call the failure to satisfy these conditions the Fourier wall. We operationalize the conditions as SPECTRA, a two-tier certificate: a simulator-free structural screen followed by a decisive comparison between a matched quantum model and five tuned classical twins using paired-bootstrap confidence bounds. On industrial smart-meter data, SPECTRA correctly refuses the real peak-load target, for which gradient-boosted trees reach a held-out ROC-AUC of 0.999. On the same real energy phases with labels generated by an interacting quantum process, the dynamics-matched quantum model reaches 0.994 versus 0.699 for the best generic classical baseline, and collapses to chance when its interaction couplings are ablated. An exact classical simulator ties the quantum model at small width but incurs measured exponential evaluation cost, with an estimated hardware crossover near 13-19 sites. These results provide a practical recipe for identifying, engineering, and deploying quantum-advantage candidates in tabular data.
Quantum Fourier sampling may help audit the spectral learnability of delay-based physical unclonable functions (PUFs). We ask whether that promise survives access matching, a strong classical comparator, and oracle synthesis. Three gates structure the evaluation. Structure: low degree is not small support at reachable challenge lengths; for 4-XOR at n=14n=14n=14, degree ≤df(0.1)\le d_f(0.1)≤df​(0.1) admits 91%91\%91% of all 2n2^n2n characters and the median 90%90\%90%-mass set spans a third of the spectrum. Algorithmics: constructing the phase oracle logically implies classical membership access, making Kushilevitz--Mansour the correct baseline; across 45 tasks it exhausts each finite domain, and no 4-XOR ideal-sampling case reaches 90%90\%90% mass within 2n2^n2n calls. A quantum-kernel diagnostic appears more favorable, with geometric difference rising to 2.1512.1512.151 at N=512N=512N=512 challenges, but it correlates 0.9910.9910.991 with 1/λmin⁡(KC)1/\sqrt{λ_{\min}(K_C)}1/λmin​(KC​)​ for the classical Gram matrix KCK_CKC​, and the 4-XOR label-complexity ratio does not exceed a balance-preserving permutation null (p=0.930p=0.930p=0.930). Trace-normalized geometric difference can therefore grow through classical ill-conditioning alone, without task-label alignment. Implementation: a simulator-validated fixed-point phase oracle based on the quantum Fourier transform admits an 18.9%18.9\%18.9% routed-depth reduction, yet the least certified precisions have estimated durations of 1.181.181.18--1.55×1.55\times1.55× the median dephasing time T2T_2T2​ of the mapped qubits on a static backend snapshot, without hardware execution. We find no end-to-end advantage in the evaluated regime, although ideal sampling does use fewer coherent calls on the thresholded task. The contribution is the Three-Gate Quantum Audit Protocol: a reproducible procedure separating an ideal query advantage from a realizable security benefit. This is not a claim about deployed silicon and not an impossibility result.
Learning problems involving quantum data are natural candidates for demonstrating an advantage in quantum machine learning. Recent results indicate that, for certain tasks and under noiseless conditions, coherent processing of quantum data outperforms fixed-measurement schemes followed by classical processing. It remained uncertain whether this performance gap persists at a finite scale, and in the presence of noise that is unavoidable with current quantum devices. In this work, we present simulations and analysis of the performance of existing hardware on a learning problem known to exhibit asymptotic advantage, now subjected to noisy quantum data. Comparing coherent quantum processing directly against fixed-measurement schemes, our results demonstrate a clear performance separation at a scale of just 30 to 40 noisy qubits. Already at this scale, the fundamental bottleneck is no longer classical computation but data acquisition; matching the noisy coherent protocol with measure-first strategies would still require months or even years of measurements. By systematically evaluating hardware constraints such as state preparation, gate errors, readout errors, connectivity, and coherence times, we provide evidence that a demonstration of such a strong learning advantage is accessible on near-term devices.
The Quantum Pi Benchmark (QPB) is a family of verification, search, counting, frequency-estimation, and collision tasks over the digits of Pi. It is designed as an architecture-neutral specification of mathematically fixed instances, classical comparators, resource boundaries, and admissible claims. The specification fixes zero-based indexing, binary and hexadecimal digit conventions, dense and sparse candidate domains, and task-specific payloads for target-bearing and collision problems. It separates three independent properties that benchmark descriptions can conflate: digit access (lookup or direct), oracle-construction provenance (P0–P4), and controller blindness (B0 or B1). A non-total capability grid retains codes R0–R12 as stable claim labels while stating their named dependencies; a higher-numbered code does not automatically inherit every lower-numbered capability. The evidence protocol requires pre-execution validation, routed-resource accounting, preservation of raw outcomes, outcome-independent eligibility, visible conditioning, uncertainty estimates, protocol separation, and formal record freezing. Collision scaling is stated as a heuristic candidate-domain model: birthday behavior gives D_h = Θ(16^(h/2)), while element distinctness gives O(D_h^(2/3)) coherent queries only under declared access and memory assumptions. These query bounds do not establish the earliest repeat, a quantum frontier, or end-to-end advantage. Frontier status depends on a current classical post-hoc resource audit; advantage requires a predeclared matched cost comparison that includes preparation, oracle construction, error correction, sampling, readout, and verification. The resulting framework classifies future Pi-derived experiments without importing device-specific performance into the foundational definition.
We present an analysis of quantum circuit fidelity across the full compilation stack, from high-level gate optimization through pulse-level control. We connect a C++ circuit optimizer to a per-gate Lindblad master-equation fidelity model whose decoherence channels are cross-validated against qiskit-dynamics and whose absolute predictions are benchmarked against execution on real hardware. Across a campaign of 4,452 experiment runs over 371 benchmark circuits, gate cancellation provides the dominant improvement (d=1.66d = 1.66d=1.66, 72% of circuits improved), while circuit size and pulse duration are the strongest negative predictors of process fidelity (input gates r=−0.78r = -0.78r=−0.78; pulse duration r=−0.73r = -0.73r=−0.73, R2=0.53R^2 = 0.53R2=0.53). A formal ablation study shows that pass ordering has no significant effect on two-qubit gate reduction (Kruskal--Wallis p=0.302p = 0.302p=0.302). Comparing against Qiskit transpilation levels, we show that two-qubit gate count, not total gate count, is the hardware-relevant metric: our optimizer attains superior two-qubit reduction on structured circuits (87.8% on QFT, 100% on QAOA) whereas Qiskit's larger total-gate reduction is dominated by single-qubit (u3u_3u3​) consolidation. Finally, executing eight circuits on the IQM Resonance Garnet processor (8/8 jobs completed, job identifiers released) reveals that the model is a consistent upper bound: it preserves the relative difficulty ordering of circuits but overestimates absolute fidelity by a mean of 0.49, quantifying the error budget (crosstalk, leakage, readout) outside a T1T_1T1​/T2T_2T2​/depolarizing model. We release the framework, data, and scripts as open source.
Cloud-accessible quantum processors enable direct execution of quantum algorithms on heterogeneous hardware platforms. Unlike classical systems, however, identical quantum circuits may exhibit substantially different behavior across devices due to architectural variations in qubit connectivity, gate fidelity, and coherence times. In this work, we systematically benchmark five representative quantum algorithms - Bell state preparation, GHZ state generation, Quantum Fourier Transform (QFT), Grover's Search, and the Quantum Approximate Optimization Algorithm (QAOA) - across trapped-ion, superconducting, and simulator backends using Amazon Braket. Performance metrics including fidelity, CHSH violation, success probability, circuit depth, and gate counts are evaluated. Our results demonstrate a strong dependence of algorithmic performance on hardware topology and noise characteristics. For example, 10-qubit GHZ states achieved fidelities above 0.8 on trapped-ion hardware, while superconducting platforms dropped below 0.15 due to routing overhead and accumulated two-qubit gate errors. These findings highlight the importance of hardware-aware algorithm selection and provide practical guidance for benchmarking in the NISQ era.