Simulating open quantum multi-body systems governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation is essential for quantum optimal control, superconducting qubit characterization, and quantum error mitigation. However, when deploying Physics-Informed Neural Networks (PINNs) or Neural Ordinary Differential Equations (Neural ODEs) as accelerated surrogates for open quantum dynamics, long-horizon autoregressive rollouts suffer from secular drift: (1) trace conservation breakdown (|Tr(ρ) − 1| > 0), violating total probability conservation; (2) loss of positive semi-definiteness (λmin(ρ) < 0), producing unphysical negative quantum measurement probabilities; and (3) distortion of purity and decoherence envelopes. While static metric projection of noisy density matrices onto the state spectrahedron is established in quantum tomography (Smolin et al., PRL 2012; Michelot 1986), its rigorous dynamical integration into autoregressive scientific machine learning (SciML) time-stepping loops remains unexplored. In this paper, we formulate an exact, non-dissipative dynamical framework: Idempotent Quantum State Manifold Projections (ΠS2 = ΠS) integrated via operator splitting into neural surrogate rollouts. We prove that ΠS is firmly non-expansive, preserving the high-order convergence rate of the underlying differential integrator while unconditionally eradicating secular trace runaway and negative probabilities without damping physical coherences. On a canonical benchmark of a driven-dissipative superconducting qubit under spontaneous relaxation and dephasing (4,000 integration steps, t = 20.0, replicated over 10 independent random seeds): Unconstrained neural rollouts suffer severe trace drift (|Tr(ρ) − 1| = (3.91 ± 1.25) × 10−3) and negative eigenvalues (λmin = (−1.21 ± 0.82) × 10−4); A standard Cholesky regularization baseline (ρ = L L† / Tr(L L†) with adaptive diagonal jitter εI) requires artificial diagonal entropy injection that dilutes low-entropy states; IdemSolver ΠS maintains exact unit trace ((8.44 ± 0.89) × 10−16), strictly non-negative eigenvalues (λmin ≥ 0), pristine Rabi coherence, exact steady-state purity (0.5244 ± 0.0005 matching the exact physical Lindblad baseline 0.5244), and machine-zero idempotency residual ((1.10 ± 0.08) × 10−15); and Achieves a step latency of 122.70 ± 4.70 μs (>8.1 kHz update rate), establishing a verified algebraic safeguard for real-time quantum control and SciML digital twins.
Simulating open quantum multi-body systems governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation is essential for quantum optimal control, superconducting qubit characterization, and quantum error mitigation. However, when deploying Physics-Informed Neural Networks (PINNs) or Neural Ordinary Differential Equations (Neural ODEs) as accelerated surrogates for open quantum dynamics, long-horizon autoregressive rollouts suffer from secular drift: (1) trace conservation breakdown (|Tr(ρ) − 1| > 0), violating total probability conservation; (2) loss of positive semi-definiteness (λmin(ρ) < 0), producing unphysical negative quantum measurement probabilities; and (3) distortion of purity and decoherence envelopes. While static metric projection of noisy density matrices onto the state spectrahedron is established in quantum tomography (Smolin et al., PRL 2012; Michelot 1986), its rigorous dynamical integration into autoregressive scientific machine learning (SciML) time-stepping loops remains unexplored. In this paper, we formulate an exact, non-dissipative dynamical framework: Idempotent Quantum State Manifold Projections (ΠS2 = ΠS) integrated via operator splitting into neural surrogate rollouts. We prove that ΠS is firmly non-expansive, preserving the high-order convergence rate of the underlying differential integrator while unconditionally eradicating secular trace runaway and negative probabilities without damping physical coherences. On a canonical benchmark of a driven-dissipative superconducting qubit under spontaneous relaxation and dephasing (4,000 integration steps, t = 20.0, replicated over 10 independent random seeds): Unconstrained neural rollouts suffer severe trace drift (|Tr(ρ) − 1| = (3.91 ± 1.25) × 10−3) and negative eigenvalues (λmin = (−1.21 ± 0.82) × 10−4); A standard Cholesky regularization baseline (ρ = L L† / Tr(L L†) with adaptive diagonal jitter εI) requires artificial diagonal entropy injection that dilutes low-entropy states; IdemSolver ΠS maintains exact unit trace ((8.44 ± 0.89) × 10−16), strictly non-negative eigenvalues (λmin ≥ 0), pristine Rabi coherence, exact steady-state purity (0.5244 ± 0.0005 matching the exact physical Lindblad baseline 0.5244), and machine-zero idempotency residual ((1.10 ± 0.08) × 10−15); and Achieves a step latency of 122.70 ± 4.70 μs (>8.1 kHz update rate), establishing a verified algebraic safeguard for real-time quantum control and SciML digital twins.
Reconstruction of the quantum density operator ρ from finite projective measurement statistics is a foundational requirement for quantum characterization, verification, and benchmarking (QCVB) in modern quantum processors. However, standard linear inversion systematically fails non-negativity (λmin < 0) in the presence of statistical shot noise, yielding unphysical indefinite operators that violate the Born rule and predict negative probabilities. Conversely, standard iterative Maximum Likelihood Estimation (MLE) via Hradil's RρR map or Semidefinite Programming (SDP) incurs high computational overhead and exhibits sensitivity to rank deficiencies, creating severe latency bottlenecks in automated quantum processing unit (QPU) calibration sweeps. Here, we present IdemQST, a deterministic quantum state tomography framework governed by idempotent projection operator theory (Π2 = Π). In an informationally complete orthogonal Pauli basis, sequential projection across measurement hyperplanes collapses in exactly one pass to the unconstrained linear inversion estimate ρlin, and subsequent Euclidean simplex spectral projection ΠΔ yields the exact global Frobenius-closest physical state on the quantum state cone Sd, establishing mathematical equivalence to Projected Linear Inversion (PLI; Smolin, Gambetta, and Smith, 2012). Rather than asserting novel algorithmic optimality for the simplex step, IdemQST re-articulates this foundation within metric idempotent operator algebra, integrates zero-heap static workspace caching (IdemKV), provides empirical dynamic drift cancellation via Dynamic Permutation Networks (DPN), and extends to incomplete or shadow tomography (M < 4N - 1) via cyclic Projections Onto Convex Sets (POCS). Across simulated projective Born measurements on one-, two-, and three-qubit systems under realistic Poisson shot noise (Nshots ∈ [102, 2 × 104]), IdemQST unconditionally eliminates negative eigenvalues (λmin ≥ 0), conserves unit trace to machine precision (10-15), and achieves state fidelities F > 0.994 under pure prior projection or large shot budgets. Crucially, across N=30 repeated timing benchmarks, IdemQST executes the core simplex spectral projection in 96.7 ± 44.3 μs for two-qubit Bell states (39.3× speedup over Hradil MLE and 41.1× over Diluted MLE) and 0.145 ± 0.035 ms (145.0 μs) for three-qubit GHZ states (d=8), achieving a 111.0× speedup over iterative Hradil MLE (16.08 ± 2.17 ms) and 113.6× over accelerated Diluted MLE (16.46 ± 1.58 ms; Šeháček et al., 2007). Furthermore, we incorporate State Preparation and Measurement (SPAM) readout mitigation via simplex-constrained assignment inversion, providing a deterministic, hardware-ready platform for real-time automated QPU calibration.
Real-time reconstruction of conditional quantum states from continuous measurement records is a fundamental requirement for quantum feedback control, yet standard stochastic master equation (SME) solvers require exact model specification, known system parameters, and are sensitive to parameter mismatch. While neural sequence models can fit these stochastic dynamics, the unconstrained predictors can violate physicality such as positivity or trace constraints, leading to unstable rollouts and unphysical estimates. We propose a Kraus-structured output layer that converts the hidden representation of a generic sequence backbone into a completely positive trace preserving (CPTP) quantum operation, yielding physically valid state updates by construction. We instantiate this layer across diverse backbones, RNN, GRU, LSTM, TCN, ESN and Mamba; including Neural ODE as a comparative baseline, on stochastic trajectories characterized by parameter drift. Our evaluation reveals distinct trade-offs between gating mechanisms, linear recurrence, and global attention. Across all models, Kraus-LSTM achieves the strongest results, improving state estimation quality by 7% over its unconstrained counterpart while guaranteeing physically valid predictions in non-stationary regimes.
Machine learning has emerged as a promising tool for simulating quantum dissipative dynamics. However, existing methods often struggle to enforce key physical constraints, such as trace conservation, when modeling reduced density matrices (RDMs). While physics-informed neural networks (PINN) aim to address these challenges, they frequently fail to achieve full physical consistency. In this work, we introduce a novel approach that leverages the su(n) Lie algebra to represent RDMs as a combination of an identity matrix and n2 - 1 Hermitian, traceless, and orthogonal basis operators, where n is the system's dimension. By learning only the coefficients associated with the operators, our framework inherently ensures exact trace conservation, as the traceless nature of the operators restricts the trace contribution solely to the identity matrix. This eliminates the need for explicit trace-preserving penalty terms in the loss function, simplifying optimization and improving learning efficiency. We validate our approach on two benchmark quantum systems: the spin-boson model and the Fenna-Matthews-Olson complex. By comparing the performance of four neural network architectures-purely data-driven physics-uninformed neural networks (PUNN), su(n) Lie algebra-based PUNN (su(n)-PUNN), traditional PINN, and su(n) Lie algebra-based PINN (su(n)-PINN)-we highlight the limitations of conventional methods and demonstrate the superior accuracy, robustness, and efficiency of our approach in learning quantum dissipative dynamics.
Modeling the dynamics of open quantum systems on noisy intermediate-scale quantum (NISQ) devices constitutes a major challenge, as high noise levels and environmental degradations lead to the decay of pure quantum states (decoherence) and energy losses. This situation represents one of the most important problems in the field of quantum information technologies. While existing data-driven methods struggle to generalize beyond the training data (extrapolation), physics-informed neural networks (PINNs) require predefined governing equations, which limit their discovery capability when the underlying physics is incomplete or unknown. In this work, we present the ParaQNN (ParaQuantum neural network) architecture, an equation-free framework for physical discovery. ParaQNN disentangles multi-scale dynamics without relying on a priori laws by employing a dialetheist logic layer that models coherent signal and decoherent noise as independent yet interacting channels. Through extensive benchmark tests performed on Rabi oscillations, Lindblad dynamics, and particularly complex "mixed regimes" where relaxation and dephasing processes compete, we show that ParaQNN exhibits a consistent performance advantage compared to Random Forest, XGBoost, and PINN models with incomplete physical information. Unlike its competitors, ParaQNN succeeds in maintaining oscillatory and damping dynamics with high accuracy even in extrapolation regions where training data are unavailable, by "discovering" the underlying structural invariants from noisy measurements. These results demonstrate that paraconsistent logic provides a structurally more stable epistemic foundation than classical methods for learning quantum behavior in situations where mathematical equations prove insufficient.
This monograph delivers a rigorous, non-perturbative closed-form analytical solution to the generalized non-Markovian Lindblad-Nakajima-Zwanzig master equation governing open quantum-classical systems coupled to arbitrary non-equilibrium thermal baths. By lifting the dissipative Liouville-von Neumann evolution onto an extended non-Hermitian supersymmetric (SUSY) fibre bundle equipped with a Cartan-Darboux connection, the environmental memory kernel is mapped onto the characteristic flow of an exceptional point foliation. We prove two foundational theorems: (1) Global Closed-Form Integrability, reducing non-local integro-differential dynamics to exact meromorphic Wilson holonomy loops with O(N^3) deterministic complexity, and (2) Strict Complete Positivity and Trace Preservation (CPTP) across all finite and asymptotic temporal regimes, definitively overcoming the unphysical negative-probability breakdown during informational backflow intervals. The formulation includes a production-certified, zero-heap embedded ISO C99 deterministic kernel executing in 1.85 microseconds on standard bare-metal controllers, establishing a new foundation for decoherence-resilient autonomous cybernetic computing.
This manuscript addresses a long-standing challenge in the theory of open quantum systems: the loss of complete positivity (CP) in the non-secular Redfield master equation when energy-level splittings become comparable to relaxation rates. While the standard Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) framework guarantees complete positivity through full secular truncation, it systematically suppresses physically relevant low-frequency quantum coherences. We present a rigorous coarse-graining framework that bridges the gap between exact non-Markovian Nakajima-Zwanzig dynamics and Markovian quantum master equations. By introducing a finite time-averaging window as a physically motivated coarse-graining scale, we demonstrate that partial secularization acts as a controlled frequency filter, preserving the completely positive and trace-preserving (CPTP) structure of the generator while retaining coherence effects that are lost under full secularization. Analytical bounds for the coarse-grained approximation are derived, and the resulting generator is shown to provide a physically consistent and highly accurate description of open-system dynamics in regimes where the standard GKSL approximation becomes overly restrictive. The formalism establishes a mathematically robust intermediate layer between Redfield and GKSL descriptions and offers a natural framework for the investigation of structured environments, near-degenerate quantum systems, and scale-dependent dissipative quantum flows.