The standard theory of quantum decoherence treats the environment as an external reservoir imposing a fixed decoherence rate. We propose a fundamental reinterpretation: within the Lie-Observable-Dependent Renormalization (LA-ODR) framework, the decoherence rate Γτ = Γ₀(χτ)^β is governed by the temporal susceptibility χτ — a state-dependent order parameter of phase-ensemble desynchronization. We derive the Kyiv–Ohnishi hybrid flow dχτ/dt = -αΓ₀(χτ)^(β+1) - λ_ctrlχτ + λ_nat(1-χτ), establish four theorems of global asymptotic stability via Lyapunov analysis, and prove a triple algebraic–geometric–thermodynamic correspondence at the fixed point χτ* = λ_nat/(λ_ctrl + λ_nat). The effective state-space metric dsτ² = (1-χτ)c²dt² - (1+χτ)dx² is a Riemannian analogue of the Fubini–Study metric on susceptibility space, not a physical spacetime metric. The framework provides the mathematical foundation for active coherence control in NISQ quantum devices.
The standard theory of quantum decoherence treats the environment as an external reservoir imposing a fixed decoherence rate. We propose a fundamental reinterpretation: within the Lie-Observable-Dependent Renormalization (LA-ODR) framework, the decoherence rate Γτ = Γ₀(χτ)^β is governed by the temporal susceptibility χτ — a state-dependent order parameter of phase-ensemble desynchronization. We derive the Kyiv–Ohnishi hybrid flow dχτ/dt = -αΓ₀(χτ)^(β+1) - λ_ctrlχτ + λ_nat(1-χτ), establish four theorems of global asymptotic stability via Lyapunov analysis, and prove a triple algebraic–geometric–thermodynamic correspondence at the fixed point χτ* = λ_nat/(λ_ctrl + λ_nat). The effective state-space metric dsτ² = (1-χτ)c²dt² - (1+χτ)dx² is a Riemannian analogue of the Fubini–Study metric on susceptibility space, not a physical spacetime metric. The framework provides the mathematical foundation for active coherence control in NISQ quantum devices.
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 [21] remains fundamentally limited by decoherence, which conventional dynamical decoupling [25] and quantum error correction [26, 28] 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 renormalization (LA-ODR) framework [5, 6, 7], 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 (illustrative) 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). The model predicts two experimentally distinguishable signatures: (i) Phase Inertia — non-exponential initial coherence suppression; and (ii) Avalanche Collapse — a second-order phase transition at χ_τ > χ_crit. These signatures distinguish the TPLL regime from standard Lindblad decay and passive discrete time crystals. Keywords: nonlinear phase stabilization; temporal susceptibility; LA-ODR; Kyiv–Ohnishi flow; TPLL; coherence plateau; Avalanche Collapse; Phase Inertia; NISQ devices; discrete time crystals; Krylov tower invariance
We present a rigorous Lie‑algebraic realization of the temporal interpretation of quantum decoherence recently proposed by Lemeshko et al. (2026). By promoting the temporal drift δτ(t) to an observable‑dependent generator Xtime inside the dynamical Lie algebra g = Lie{ iH, Lk, L†k, Oi } we perform systematic Lie closure to extract the maximal Lie‑observable‑dependent symmetry (Lie‑ODS) subalgebra whose adjoint action leaves the entire Krylov tower of phase‑synchronization observables { Re(ac), Im(ac), a†cac } invariant. Embedding this construction into the hierarchical Lie‑observable‑dependent renormalization (LA‑ODR) flow yields a universal fixed‑point Lindbladian L˜* of remarkably small and constant dimension d* = 4–9, independent of the ultraviolet cutoff. The resulting model exactly reproduces the exponential phase‑damping law ρij(t) = ρij(0) · exp(−Γτ · t) of the original temporal picture, with the decoherence rate γ ≡ Γτ emerging as a universal fixed‑point parameter. Non‑Markovian memory effects and 1/f noise appear naturally as residual temporal correlations between successive RG scales, while complete positivity and trace preservation are rigorously maintained at every coarse‑graining step. When augmented with variational quantum minimal models (VQMM), the framework enables NISQ photonic devices to self‑optimize their own temporal synchronization in real time, achieving Bloch‑vector errors below 3.2 × 10⁻⁴. The surviving Lie‑ODS subalgebra supplies a precise open‑system analogue of Noether’s theorem for temporal symmetries and opens a direct route to topological diagnostics via cyclic multi‑information measures. This work closes the long‑standing interpretive gap between phenomenological temporal pictures and exact effective theories in the thermodynamic limit. It unifies the temporal desynchronization ansatz with the same LA‑ODR machinery previously applied to quantum frequency combs and exciton‑polariton condensation, revealing the remarkable universality of observable‑dependent renormalization. The resulting fixed‑point Lindbladian is computationally trivial yet physically complete, offering powerful new tools for noise analysis, phase stability, and quantum information processing in complex open systems. Future extensions include spatially inhomogeneous drifts, dynamical fermions, and holographic analogues of quantum gravity in open quantum systems.
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
We develop a Lie-Jordan geometric formulation of finite-dimensional open quantum dynamics, building on the algebraic framework introduced in our previous work (arXiv:2606.26477). The Hilbert-Schmidt operator space is endowed with an orthonormal Hermitian basis, in which the commutator and anticommutator are encoded by the structure tensors Cμνλ and Bμνλ. Within this formulation, the von Neumann and Gorini-Kossakowski-Lindblad-Sudarshan equations admit a direct component representation. Our central result is the identification of a basis-independent universal trilinear dissipative map, D(X,Y)Z=XZY−21{YX,Z}, whose components define a universal operator-space tensor depending only on the Lie-Jordan structure tensors. The physical dissipator is obtained by contracting this tensor with the expansion coefficients of the Lindblad operators and the Kossakowski matrix, thereby separating the universal algebraic structure from the model-dependent physical information. We further show that the combinations (B+C) and (B−C) generate internal left and right transports, allowing the elementary dissipative map to be expressed as a left-right bimodule action corrected by an ordered Jordan contribution. The universal map satisfies basis-independent trace and Hermitian-conjugation identities, from which trace preservation, Hermiticity preservation of the complete dissipator, and the reality of its component representation in a Hermitian Hilbert-Schmidt basis follow. We also derive its Hilbert-Schmidt adjoint and illustrate the formalism for a qubit with pure dephasing and amplitude-damping channels. This construction provides a tensorial and affine-geometric interpretation of the universal superoperator structure derived in arXiv:2606.26477 while keeping the algebraic and model-dependent sectors explicitly separated.
Abstract We present a mathematically controlled, model‑independent proof of the Norm–Expectation Correspondence in compressed Lie‑algebraic observable‑dependent renormalization (LA‑ODR) algebras. The correspondence relates the Hilbert‑Schmidt (HS) norm of the projected temporal‑drift generator ∥ΠA* (kg_time)∥²HS to the synchronization deviation 1 − Tr(ρ₁ P_sync) from the synchronized stationary state. Combining (i) the Wedderburn decomposition of the output algebra AODS, (ii) the spectral theory of primitive quantum dynamical semigroups (HS‑contractivity and variational spectral gap λ > 0), and (iii) the Riemannian geometry of the quantum statistical manifold with the monotone HS metric and Amari–Nagaoka α‑connections, we prove the Universal Bridge Theorem: ∥ΠA* (kg_time)∥²HS = λ (1 − Tr(ρ₁ P_sync)) + R, where |R| ≤ C κ(A*) · d⁴HS(ρ₁, ρ∞) is controlled by the quantum Fisher‑information curvature κ. The remainder R vanishes exactly under quantum detailed balance or flatness of the α = 1 connection. This extended preprint adds: (a) a new HS self‑adjointness lemma closing the integration identity in the proof; (b) an independent spectral verification of the two‑qubit finite‑dimensional example; (c) a self‑contained review of Amari–Nagaoka information geometry; (d) a summary of the TTU framework; and (e) an application to molecular spin qubits using the temporal susceptibility χτ = 1 − Tr(ρ₁ P_sync) as a universal decoherence marker. Keywords: LA‑ODR algebras, Universal Bridge Theorem, primitive quantum dynamical semigroups, Hilbert–Schmidt geometry, information geometry, quantum synchronization, temporal susceptibility, emergent spacetime.