We study a deterministic modulation protocol, Q-Loop, that replaces multi-axis, measurementdependent control with a single scalar waveform and is implementable as an open-loop schedule on current NISQ hardware. Building on earlier work where Q-Loop was shown to reduce both the dephasing functional χ(τ ) and the minimum control amplitude required to stabilize few-qubit GHZ states under classical 1/f noise, we revisit the protocol with a more conservative analytic treatment and a quantitative comparison to alternative empirical scaling forms for modest system sizes (n = 2–8 qubits). In a pure-dephasing (or effectively commuting) regime we derive the filter-function structure of an idealized Q-Loop waveform and show that it produces a discrete comb of sidebands with tunable low-frequency suppression relative to free evolution and standard Carr–Purcell (CP) sequences. For more general noise channels the scalar filter-function picture is used as a qualitative guide, and quantitative statements are obtained from stochastic-Hamiltonian simulations with classical 1/f γ noise. Using QuTiP-based simulations of Schrödinger dynamics with time-dependent Hamiltonians and classical 1/f γ local and correlated noise, we compute the minimum control amplitude β ∗ (n) needed to maintain a fixed target fidelity as a function of system size n for n = 2–8 qubits. Over this range the ensemble-averaged β ∗ (n) data are empirically well described by β ∗ (n) ≈ Aeαn with α ≃ 0.23–0.25, while quadratic and power-law forms yield systematically higher residuals and poorer goodness-of-fit metrics (including R2 , RMSE, and Bayesian information criterion), with BIC values for the exponential model typically 10–15 units lower than for polynomial alternatives. We emphasize that this exponential behaviour is an empirical description over a finite range, not an asymptotic law. We further map the Q-Loop-induced reduction in decoherence onto a toy physical error rate p and propagate this into a repetition-code capacity model. For a fixed phenomenological mapping, Q-Loop preprocessing yields an apparent 5–6× elevation of toy thresholds relative to an amplitudematched CP baseline, although these numbers are not intended as accurate thresholds for any specific fault-tolerant architecture. Finally, we discuss deployment on superconducting, trapped-ion, and photonic platforms using a Python-based waveform generator with PyVISA AWG integration, and outline how platform-specific noise spectra could be used to tune Q-Loop parameters in situ
We study a deterministic modulation protocol, Q-Loop, that replaces multi-axis, measurementdependent control with a single scalar waveform and is implementable as an open-loop schedule on current NISQ hardware. Building on earlier work where Q-Loop was shown to reduce both the dephasing functional χ(τ ) and the minimum control amplitude required to stabilize few-qubit GHZ states under classical 1/f noise, we revisit the protocol with a more conservative analytic treatment and a quantitative comparison to alternative empirical scaling forms for modest system sizes (n = 2–8 qubits). In a pure-dephasing (or effectively commuting) regime we derive the filter-function structure of an idealized Q-Loop waveform and show that it produces a discrete comb of sidebands with tunable low-frequency suppression relative to free evolution and standard Carr–Purcell (CP) sequences. For more general noise channels the scalar filter-function picture is used as a qualitative guide, and quantitative statements are obtained from stochastic-Hamiltonian simulations with classical 1/f γ noise. Using QuTiP-based simulations of Schrödinger dynamics with time-dependent Hamiltonians and classical 1/f γ local and correlated noise, we compute the minimum control amplitude β ∗ (n) needed to maintain a fixed target fidelity as a function of system size n for n = 2–8 qubits. Over this range the ensemble-averaged β ∗ (n) data are empirically well described by β ∗ (n) ≈ Aeαn with α ≃ 0.23–0.25, while quadratic and power-law forms yield systematically higher residuals and poorer goodness-of-fit metrics (including R2 , RMSE, and Bayesian information criterion), with BIC values for the exponential model typically 10–15 units lower than for polynomial alternatives. We emphasize that this exponential behaviour is an empirical description over a finite range, not an asymptotic law. We further map the Q-Loop-induced reduction in decoherence onto a toy physical error rate p and propagate this into a repetition-code capacity model. For a fixed phenomenological mapping, Q-Loop preprocessing yields an apparent 5–6× elevation of toy thresholds relative to an amplitudematched CP baseline, although these numbers are not intended as accurate thresholds for any specific fault-tolerant architecture. Finally, we discuss deployment on superconducting, trapped-ion, and photonic platforms using a Python-based waveform generator with PyVISA AWG integration, and outline how platform-specific noise spectra could be used to tune Q-Loop parameters in situ
Dynamical decoupling and shaped control are standard tools for mitigating dephasing noise in solid-state qubits. Most analyses either assume commuting pure dephasing with longitudinal control, where filter-function theory applies exactly, or move to fully numerical non-commuting dynamics with limited interpretability. Here I introduce Q-Loop, a scalar modulation protocol designed to suppress low-frequency 1/f γ dephasing more efficiently than standard Carr–Purcell (CP) sequences, and I connect its performance across both regimes. First, in the commuting pure-dephasing limit I compute the filter functions F(ω, τ ) and decoherence functionals χ(τ ) for free evolution, CP, and Q-Loop under 1/f γ noise with γ ∈ {0.5, 0.9, 1.2}. For γ = 0.9 I find that Q-Loop reduces the integrated noise susceptibility to χQ-Loop/χCP ≈ 0.36, with even stronger suppression for steeper spectra (χQ-Loop/χCP ≈ 0.14 at γ = 1.2). Monte Carlo simulations of classical noise trajectories verify that the resulting coherence W(τ ) = ⟨e −iϕ⟩ matches the analytic prediction e −χ(τ) . Second, I move beyond the commuting approximation and study an n-qubit GHZ state evolving under a time-dependent Hamiltonian with transverse control and longitudinal dephasing noise, H(t) = β 2 y(t) Xn i=1 Xi + 1 2 Xn i=1 Bi(t)Zi, where Bi(t) are mixed local+collective classical 1/f 0.9 fields and y(t) is either Q-Loop or CP. Using QuTiP to simulate unitary dynamics conditioned on classical noise, I determine the minimum control amplitude β ∗ (n) required to maintain fidelity F ≥ 0.9 with the GHZ state at a fixed interrogation time. For n = 2, 3, 4 qubits I obtain β ∗ Q-Loop ≃ (2.03, 2.28, 2.53) and β ∗ CP ≃ (15.53, 15.53, 16.00), corresponding to an average efficiency gain ⟨β ∗ CP/β∗ Q-Loop⟩ ≈ 6.9. Thus the spectral advantage of Q-Loop in reducing χ(τ ) is reflected in a substantial reduction of transverse control power required to stabilize entanglement under realistic non-commuting dynamics. I also discuss the limitations of the present analysis—classical noise, finite system size, and the restricted range n = 2–4—and outline how the framework can be extended to larger systems and more microscopic noise models.
Q-Loop: Deterministic Coherence Scaling Without Feedback in NISQ Systems We present the Q-Loop protocol, a deterministic modulation scheme that achieves coherence scaling without measurement-based feedback in noisy intermediate-scale quantum (NISQ) systems. Q-Loop applies a folded, periodic control waveform to collective parity operators, producing destructive interference with dominant decoherence channels. Unlike adaptive protocols such as QEC-C, which require rapid measurement and real-time gain updates, Q-Loop is fully open-loop and relies only on deterministic waveforms implementable with existing arbitrary waveform generators or global control fields. Using filter-function analysis and numerical simulations across superconducting, ion-trap, and photonic models, we show that Q-Loop realizes an exponential scaling of an effective control amplitude beta(n) ~ A * exp(alpha n), with alpha comparable to that obtained in adaptive-feedback schemes, while requiring no ancilla qubits and no real-time classical processing. The protocol reshapes 1/f-type noise into a structured spectrum with suppressed mid-band power and yields substantial fidelity improvements and effective fault-tolerance threshold elevation when used as a preprocessing layer for standard error-correcting codes. This establishes Q-Loop as a simple, hardware-compatible route to deterministic coherence enhancement in current NISQ hardware.
This repository contains the simulation and analysis code accompanying the manuscript “Q-Loop-Enabled Deterministic Coherence Management for Portable Quantum Sensors”. It provides a fully reproducible Python/NumPy/QuTiP workflow to evaluate Q-Loop, a continuous phase-modulation protocol for coherence protection in NV-centre magnetometers and related qubit sensors operating under “robotic” composite noise.The code computes filter functions (|F(\omega,\tau)|^2) and decoherence functionals (\chi(\tau)) for free evolution, Q-Loop, and standard pulsed dynamical decoupling sequences (CPMG, XY8, UDD), under both Lorentzian and composite (1/f +) line-rich noise spectra. It generates the main figures in the paper, including: (i) filter-function and noise-weighting plots, and (ii) jitter–flip-angle robustness heatmaps comparing Q-Loop to XY8/CPMG. A Colab-friendly script is included so all figures can be regenerated on a laptop without specialized hardware. The repository also provides example waveform generators (I/Q baseband) for implementing Q-Loop and pulsed sequences on standard AWGs, together with reference parameter sets ((\eta \simeq 3.6, T \simeq 0.97\tau_c)) that reproduce the reported (\sim 5)–(6\times) coherence extension. This archive is intended as a starting point for both numerical studies of coherence management under realistic platform noise and for experimental implementation on ensemble NV magnetometers or other driven qubit sensors.
QEC-C 2: Adaptive-Scaling Quantum Error Correction for NISQ Systems QEC-C 2 is a low-power quantum error correction framework tailored to noisy intermediate-scale quantum (NISQ) devices, in which the strength of error protection adapts dynamically to the observed noise environment. Instead of operating at a fixed code distance and fixed-cycle syndrome extraction rate, QEC-C 2 maintains an online estimate of effective noise strength and modulates the frequency and intensity of checks according to an exponential gain law beta_star(n) = A * exp(alpha n). This adaptive scaling allows near-exponential improvements in logical coherence time while keeping ancilla counts, measurement load, and classical processing within NISQ-compatible limits. Using filter-function analysis and Monte Carlo error-tracking with mixed local, correlated, and collective noise, we show that QEC-C 2 can raise effective fault-tolerance thresholds by factors of order five to six relative to unoptimized repetition-code baselines. The scheme interpolates between dynamical decoupling and full fault-tolerant quantum error correction, providing a practical pathway for stepwise protection on current hardware and a natural benchmark for comparing deterministic open-loop protocols such as Q-Loop.
Quantum error correction (QEC) faces a significant challenge in realistic noisy intermediate-scale quantum (NISQ) devices: correlated decoherence. Building upon the foundational QEC-C protocol, we introduce QEC-C II, a scalable framework that delivers robust fault tolerance against spatially and temporally correlated noise. By integrating decoherence-free subspace (DFS) principles and correlation-aware adaptive feedback, QEC-C II achieves exponential error suppression in simulation under correlated noise models, with fewer physical qubits than typical topological baselines for the same target error in our benchmarks. The protocol leverages an enhanced scaling law, β(n) = 0.788 e 0.360n , and efficiently suppresses noise with minimal computational overhead. We validate QEC-C II on 28-qubit GHZ states, extend it to 56-qubit subspaces, and conduct 84-qubit stress-test simulations under correlated noise models. The results demonstrate sustained fidelities exceeding 99.99 % in simulations under realistic dephasing, amplitude-damping, burst-error, and correlated decoherence scenarios, with rapid convergence (typically ≤ 2 iterations in our simulations). QEC-C II significantly reduces both resource and control complexity, simplifying hardware integration and improving performance with increasing system size. This work provides a practical pathway toward resource-efficient fault tolerance in quantum processors operating in real-world correlated environments.
Portable solid-state quantum sensors, such as nitrogen-vacancy (NV) center magnetometers operated on moving platforms, must contend with strongly nonstationary “robotic” noise: a composite of low-frequency drift, narrow spectral lines, and mechanically induced sidebands that evolves on timescales of seconds to minutes. Recently, the Q-Loop protocol—a continuous, open-loop phasemodulation scheme in which a smooth folded waveform y(t) replaces discrete π pulses—was introduced and shown to provide substantial coherence enhancement under composite robotic noise in static conditions [1]. However, in realistic field operation the underlying noise correlation time and line amplitudes drift, and a fixed choice of Q-Loop parameters (η, T) can lose most of its advantage. Here we propose and analyze adaptive Q-Loop, a self-tuning continuous dynamical decoupling protocol that compensates slow spectral drift using low-overhead calibration and a precomputed control policy. We model the dephasing noise as a parametric power spectral density with three dominant components (Ornstein–Uhlenbeck band, line spectrum, and regularized 1/f tail), and treat the corresponding parameters θ(t) as slowly varying stochastic processes. Within the standard filterfunction formalism (Gaussian, weak noise, quasi-stationary over a single interrogation), we first map the static coherence-extension factor R(η, T; θ) and show that Q-Loop exhibits broad performance plateaus in (η, T) for realistic composite spectra. We then construct an offline-optimized policy P that assigns near-optimal (η, T) to each point in a low-dimensional parameter space (τc, Alines, A1/f ) and combine it with a lightweight online estimator based on a small set of short Ramsey and bandselective probe sequences. Numerical simulations using time-dependent Hamiltonian evolution (QuTiP) with drifting correlation time and line amplitudes demonstrate that, for factor-of-five drifts in τc and order-unity variations in line strength, a static Q-Loop tuned at t = 0 can degrade from an initial ∼ 50× coherence extension down to ∼ 2×. In contrast, adaptive Q-Loop maintains ∼ 40× extension and recovers roughly 80–90% of the coherence advantage of an oracle policy that knows θ(t) exactly, using a calibration overhead fcal ≲ 10% of the total shots. This performance far exceeds that of fixed CPMG/XY8 sequences with realistic timing and amplitude errors under the same drifting noise. Adaptive Q-Loop therefore realizes a practical, low-complexity “firmware layer” for plug-andforget coherence protection in field-deployable quantum sensors.