Abstract: Measurement-based feedback control is a foundational tool in quantum engineering, quantum thermodynamics, and quantum information processing. While feedback is known to stabilize small quantum systems, its scalability to larger systems remains poorly characterized. In this work, we identify a fundamental measurement–coherence scaling barrier for full-state feedback control in open quantum systems. We show that for generic n-qubit states the measurement resources required for real-time state reconstruction scale exponentially with system size, while the available coherence window remains finite. As a result, beyond a system-size threshold ncrit, the time required to acquire sufficient state information exceeds the decoherence time, preventing closure of the feedback loop. We derive an explicit closed-form expression for ncrit in terms of three hardware parameters—coherence time, single-shot measurement time, and reconstruction accuracy—and show that under realistic device conditions this threshold is typically in the range 2–7 qubits. This provides a quantitative information-theoretic explanation for why full- state tomographic feedback has only been demonstrated at very small scales. Importantly, the barrier is architectural rather than fundamental: syndrome-based control, stabilizer tracking, and measurement-efficient protocols such as classical shadows evade it by exploiting structure and extracting only operationally relevant information. Our results formalize a design principle for scalable quantum control and clarify the operational boundaries of feedback in noisy intermediate-scale quantum (NISQ) devices.
Abstract: Measurement-based feedback control is a foundational tool in quantum engineering, quantum thermodynamics, and quantum information processing. While feedback is known to stabilize small quantum systems, its scalability to larger systems remains poorly characterized. In this work, we identify a fundamental measurement–coherence scaling barrier for full-state feedback control in open quantum systems. We show that for generic n-qubit states the measurement resources required for real-time state reconstruction scale exponentially with system size, while the available coherence window remains finite. As a result, beyond a system-size threshold ncrit, the time required to acquire sufficient state information exceeds the decoherence time, preventing closure of the feedback loop. We derive an explicit closed-form expression for ncrit in terms of three hardware parameters—coherence time, single-shot measurement time, and reconstruction accuracy—and show that under realistic device conditions this threshold is typically in the range 2–7 qubits. This provides a quantitative information-theoretic explanation for why full- state tomographic feedback has only been demonstrated at very small scales. Importantly, the barrier is architectural rather than fundamental: syndrome-based control, stabilizer tracking, and measurement-efficient protocols such as classical shadows evade it by exploiting structure and extracting only operationally relevant information. Our results formalize a design principle for scalable quantum control and clarify the operational boundaries of feedback in noisy intermediate-scale quantum (NISQ) devices.
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.
Measurement-induced phase transitions (MIPT) offer a novel paradigm for exploring quantum criticality, yet their experimental observation remains stifled by the "post-selection problem"—the exponential cost of resolving quantum trajectories. Here, we propose a scalable protocol to observe feedback-stabilized criticality on current Noisy Intermediate-Scale Quantum (NISQ) processors, eliminating the need for post-selection. By coupling local measurements to real-time unitary feedback, we demonstrate that a many-body quantum system can be steered toward the universal 1/e information horizon, effectively turning the measurement back-action from a source of decoherence into a resource for complexity. Extensive numerical simulations incorporating realistic depolarizing noise reveal a robust stability plateau extending to gate error rates of p ≈ 10⁻², compatible with state-of-the-art superconducting architectures such as IBM Heron. Furthermore, we introduce a verification scheme based on classical shadows to efficiently estimate the Schmidt concentration ⟨K_c⟩ without full tomography. Our results establish that thermodynamic error correction is achievable in the NISQ era, providing a blueprint for the first experimental observation of a feedback-induced phase transition at the mesoscopic scale (N ~ 100).
Abstract: The scalability of Noisy Intermediate-Scale Quantum (NISQ) devices remains fundamentally limited by decoherence, which conventional dynamical decoupling and quantum error correction only partially mitigate. This paper presents a transition from passive noise suppression to active dynamical stabilization through controlled modulation of the effective state-space metric. Within the Lie-algebraic observable-dependent renormalization (LA-ODR) framework, we introduce the Temporal Phase-Locked Loop (TPLL) — an active feedback controller that drives the temporal susceptibility χ_τ toward the globally stable fixed point χ_τ* = λ_nat/(λ_ctrl + λ_nat) via the Kyiv–Ohnishi hybrid flow: dχ_τ/dt = −α·Γ₀·(χ_τ)^(β+1) − λ_ctrl·χ_τ + λ_nat·(1−χ_τ). Numerical simulations demonstrate the emergence of metastable coherence plateaus extending coherence times from ~100 µs to ~1.5 ms under realistic hardware benchmarks (FPGA latency Δt < 500 ns). Three operationally distinct zones are characterized by χ_τ thresholds, and two experimentally distinguishable signatures — Phase Inertia and Avalanche Collapse — are predicted. The TPLL is compared with dynamical decoupling, quantum error correction, and discrete time crystals, establishing complementarity rather than competition. The mathematical foundations rest on four global stability theorems via Lyapunov analysis and Chiba renormalization-group flow. Index Terms: Quantum coherence; decoherence control; temporal susceptibility; LA-ODR; Temporal Phase-Locked Loop (TPLL); Kyiv–Ohnishi flow; NISQ devices; open quantum systems; Lyapunov stability; Lie algebra; FPGA quantum control; nonlinear dynamics.
Abstract: The scalability of Noisy Intermediate-Scale Quantum (NISQ) devices remains fundamentally limited by stochastic decoherence, which traditional passive shielding and resource-heavy quantum error correction (QEC) struggle to overcome. Here, we propose a paradigm shift from passive noise mitigation to active internal phase stabilization. By integrating the Lie-algebraic renormalization (LA-ODR) framework with a dynamic model of temporal susceptibility (χ_τ), we derive a nonlinear control law for quantum coherence. We demonstrate that decoherence can be treated as a deterministic dynamical process governed by the feedback equation dχ_τ / dt = −α · Γ_τ · χ_τ + λ · (1 − χ_τ). Our numerical simulations confirm the formation of metastable coherence plateaus, where the effective qubit lifetime is extended significantly beyond the standard Lindbladian limit (e.g., from 100 µs to 1.5 ms in transmon-based models). We further specify a hardware-ready Temporal Phase-Locked Loop (TPLL) architecture, proving that with control latencies below 500 ns, active synchronization of the temporal metric becomes feasible on modern FPGA-based controllers. This framework provides a path toward fault-tolerant quantum computing by enabling fidelity-by-design at the physical layer, potentially providing a substantial reduction in QEC overhead by maintaining high physical fidelity prior to logical encoding. Keywords nonlinear phase stabilization; temporal susceptibility χτ; Lie-observable-dependent renormalization (LA-ODR); fixed-point Lindbladian; quantum decoherence; active feedback control; temporal phase-locked loop (TPLL); NISQ devices; superconducting qubits; cold atom arrays
Quantum computers hold immense promise for solving problems intractable for classical machines, yet their practical realization is hindered by the fragility of quantum states. Quantum Error Correction (QEC) is indispensable for protecting quantum information from environmental noise and operational imperfections. Traditional QEC schemes often rely on measurement-based feedback, which involves performing measurements, processing the classical outcomes, and then applying corrective operations. This process introduces latency, consumes significant resources, and can itself be a source of errors due to the invasive nature of quantum measurement and the classical-quantum interface. Coherent feedback control offers a compelling alternative by allowing the controller to directly interact with the quantum system using quantum fields, without intermediate measurement and classical processing. This maintains the coherence of the quantum information throughout the feedback loop, potentially leading to more robust and faster error correction. This paper explores the theoretical framework and potential advantages of employing coherent feedback control for autonomous QEC. We review existing QEC paradigms, delve into the principles of coherent feedback, and propose how this approach can be integrated into autonomous QEC architectures. Specifically, we investigate its application to bosonic codes and stabilizer codes, discussing the engineering challenges and the prospects for achieving truly fault-tolerant quantum computation by minimizing the classical overhead and enhancing real-time error suppression.
This review develops a control-theoretic perspective on measurement-based feedback for continuously monitored open quantum systems, with the main analysis focused on finite-dimensional systems governed by diffusive stochastic master equations. We introduce the relevant state-space, invariant-subspace, and quantum non-demolition structures, interpret quantum filtering as nonlinear observer dynamics, and review open-loop asymptotics, filter stability, state-feedback stabilization, robustness, and reduced-order observer-based control. Particular emphasis is placed on a recurrence--contraction framework, in which Hamiltonian feedback removes non-target invariant obstructions while measurement-induced dynamics provide local exponential contraction. Although the detailed analysis is developed for finite-dimensional diffusive models, the underlying measurement--estimation--feedback architecture is relevant across a broad range of quantum platforms. We further discuss implementation challenges and open problems involving scalable estimation, sampling and delay, adaptation, hybrid and non-Markovian dynamics, practical stability, optimal control, and learning-based design. By organizing these developments around invariance, estimation, recurrence, and contraction, the review provides a tutorial bridge between measurement-based quantum feedback and nonlinear stochastic control.