This project introduces a computational framework for quantum trajectory reconstruction using weak measurements and probabilistic inference. In standard quantum mechanics, physical systems are described by a wavefunction, and observable quantities are inherently probabilistic. As a result, particle trajectories are not directly accessible. This work proposes an operational approach in which trajectories are treated as latent variables inferred from noisy measurement data. The framework combines three core components: A numerical solver for the time-dependent Schrödinger equation (using FFT-based split-operator methods) A weak measurement model that simulates continuous, noisy observations of the system Inference engines, including Sequential Monte Carlo (particle filtering) and recurrent neural networks, to reconstruct the most probable trajectory The central idea is to reformulate quantum dynamics as an inference problem over hidden states, where the goal is to estimate the trajectory that best explains the observed measurements. This approach does not violate the principles of quantum mechanics, including the Principe d'incertitude de Heisenberg, but instead provides a new computational lens through which quantum systems can be analyzed. Potential applications include: Quantum sensing and metrology Quantum control and error correction Hybrid AI-quantum systems Advanced simulation of open quantum systems The project is fully reproducible and open-source, with modular architecture designed for research extensibility.
This project introduces a computational framework for quantum trajectory reconstruction using weak measurements and probabilistic inference. In standard quantum mechanics, physical systems are described by a wavefunction, and observable quantities are inherently probabilistic. As a result, particle trajectories are not directly accessible. This work proposes an operational approach in which trajectories are treated as latent variables inferred from noisy measurement data. The framework combines three core components: A numerical solver for the time-dependent Schrödinger equation (using FFT-based split-operator methods) A weak measurement model that simulates continuous, noisy observations of the system Inference engines, including Sequential Monte Carlo (particle filtering) and recurrent neural networks, to reconstruct the most probable trajectory The central idea is to reformulate quantum dynamics as an inference problem over hidden states, where the goal is to estimate the trajectory that best explains the observed measurements. This approach does not violate the principles of quantum mechanics, including the Principe d'incertitude de Heisenberg, but instead provides a new computational lens through which quantum systems can be analyzed. Potential applications include: Quantum sensing and metrology Quantum control and error correction Hybrid AI-quantum systems Advanced simulation of open quantum systems The project is fully reproducible and open-source, with modular architecture designed for research extensibility.
Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigen state will be observed in which experimental run. The probabilistic Born rule gives it an ensemble interpretation, predicting proportions of various outcomes over many experimental runs. Understanding gradual weak measurements requires replacing this scenario with a dynamical evolution equation for the collapse of the quantum state in individual experimental runs. In this work, I revisit the quantum trajectory framework that models quantum measurement as a continuous nonlinear stochastic process. I describe the ensemble of quantum trajectories as noise fluctuations on top of geodesics that attract the quantum state towards the measured operator eigen states. In this effective theory framework for the ensemble of quantum trajectories, the measurement interaction is specific to each system-apparatus pair—a context necessary for understanding weak measurements. Also in this framework, the constraint to reproduce projective measurement as per the Born rule in the appropriate limit, requires that the magnitudes of the noise and the attraction are precisely related, in a manner reminiscent of the fluctuation dissipation relation. This relation implies that both the noise and the attraction have a common origin in the underlying measurement interaction between the system and the apparatus. I analyse the quantum trajectory ensemble for the scenarios of quantum diffusion and binary quantum jump, and show that the ensemble distribution is completely determined in terms of a single evolution parameter.
I test the trajectory ensemble distribution predicted by the quantum diffusion model against the experimental data for weak measurement of superconducting transmon qubits. There is a good fit between theory and experiment for different initial states and several weak measurement couplings. This test vindicates the continuous stochastic measurement framework for quantum state collapse, where the rate of collapse is a characteristic parameter for each system-apparatus pair and is not a universal constant. Furthermore, it implies that the environment can influence the measurement outcomes only via the apparatus and not directly. These are important clues in construction of a complete theory of quantum measurement.
The framework of weak measurements can also be used to construct quantum error correction protocols that protect a quantum state from external disturbances. Unlike projective measurements, one can extract only partial information about the error syndrome
from the encoded state using weak measurements. I construct a feedback protocol that probabilistically corrects the error based on the extracted information. Using numerical simulations of one-qubit error correction codes, I show that the error correction succeeds for a range of the weak measurement strength, where (a) the error rate is below the threshold beyond which multiple errors dominate, and (b) the…
This paper explores the application of probabilistic logic to simulate wavefunction collapse, a fundamental process in quantum mechanics. We propose a novel logical system designed to explicitly model probabilities associated with quantum measurements, aiming to provide a deeper theoretical foundation for understanding the measurement problem. The core mechanism involves a shift from standard probabilistic models to a system that directly incorporates the dynamics of wavefunction evolution, allowing for a more complete and nuanced simulation of quantum phenomena. The paper details the design of this probabilistic logic system, its mathematical foundations, and preliminary results demonstrating its ability to reproduce key aspects of wave function collapse. Ultimately, this work seeks to bridge the gap between classical probability and the quantum realm, offering a new framework for analyzing measurement and its implications.
Quantum state vector reconstruction is a fundamental problem in quantum information theory, crucial for tasks like quantum state tomography and quantum error correction. Traditional methods often struggle with noisy measurements and lack a principled framework for uncertainty quantification. This paper introduces a novel algorithm, Quantum State Vector Reconstruction via Bayesian Inference, that leverages Bayesian inference to reconstruct a quantum state vector from a set of measurements, enhancing accuracy and offering a more robust approach compared to existing methods. We detail the core mechanism, emphasizing the integration of Bayesian estimation for uncertainty quantification and refinement of the reconstruction process. The algorithm incorporates a probabilistic approach to estimate the state vector distribution, improving reconstruction by incorporating uncertainty estimates directly into the reconstruction process. This results in a more reliable and accurate state vector reconstruction, particularly when dealing with noisy measurements. This work addresses the limitations of conventional methods by providing a principled Bayesian framework for state vector reconstruction, offering a significant advancement in this field.
The application of Model Predictive Control (MPC) to quantum systems faces fundamental challenges arising from measurement-induced disturbances and the probabilistic nature of quantum dynamics. Unlike classical nonlinear systems, quantum measurements inherently disturb the system state, making feedback control intrinsically model-dependent and stochastic. Within the framework of Positive Operator-Valued Measures (POVMs), this thesis focuses on strong quantum measurements, where each observation induces a complete state collapse on the system. When uncertainty is introduced, the measurement process itself becomes a dominant source of disturbance. To address this, a lower bound on the measurement uncertainty is imposed, quantified as the predicted deviation between the measured quantum state and the nominal MPC trajectory, ensuring that the quantum state can be steered along the designed MPC trajectory while maintaining physical realizability and stability. The thesis further develops a stochastic MPC (SMPC) formulation based on quantum filtering equations, describing the conditional evolution of quantum states under continuous observation. A theoretical equivalence between the proposed SMPC scheme and an optimal control problem is established, providing a tractable computational approach for feedback design in quantum stochastic systems. Finally, to address the limitations imposed by explicit identification of quantum systems in complex quantum systems, a data-driven MPC framework is introduced. Using locally observable dynamics and trajectory data, a data-driven MPC formulation is introduced to realize predictive quantum control without explicit system identification. This framework lays the foundation for measurement-aware and physically consistent quantum MPC.
Quantum OS v5 is a hybrid quantum-classical computational operating environment designed for real-time quantum state inference, probabilistic dynamical reconstruction, and distributed quantum orchestration. The platform combines: - weak-measurement reconstruction, - Bayesian inference, - Sequential Monte Carlo (Particle Filtering), - GRU-based recurrent neural networks, - reinforcement-learning scheduling, - distributed DAG execution, - GPU/CUDA acceleration, - photonic validation, - and quantum-control primitives. Unlike traditional operating systems, Quantum OS v5 is not intended to replace hardware-level kernels such as Linux or Windows. Instead, it acts as a high-level orchestration infrastructure specialized for: - quantum computation workflows, - probabilistic inference, - hybrid AI-assisted simulation, - and scalable distributed execution. The architecture integrates: - asynchronous execution kernels, - resource-aware scheduling, - telemetry monitoring, - distributed cluster consensus, - sandbox isolation, - secure orchestration layers, - and modular plugin extensibility. Quantum OS v5 enables operational reconstruction of latent quantum dynamical structures from weak observational streams without violating standard quantum mechanics or uncertainty principles. The system establishes a computational bridge between: - quantum mechanics, - information theory, - machine learning, - distributed systems engineering, - and hybrid computational infrastructures. Potential applications include: - quantum state estimation, - adaptive quantum control, - quantum error correction, - photonic quantum computing, - AI-assisted quantum diagnostics, - distributed quantum telemetry, - and next-generation hybrid quantum-classical computing systems. Official Website: https://quantum-os-v5.netlify.app/ Repository: https://github.com/BarackNdenga/Quantum-Trajectory-Reconstruction
Quantum measurements are described as instantaneous projections in textbooks. They can be stretched out in time using weak measurements, whereby one can observe the evolution of a quantum state as it heads towards one of the eigenstates of the measured operator. This evolution can be understood as a continuous nonlinear stochastic process, generating an ensemble of quantum trajectories, consisting of noisy fluctuations on top of geodesics that attract the quantum state towards the measured operator eigenstates. The rate of evolution is specific to each system-apparatus pair, and the Born rule constraint requires the magnitudes of the noise and the attraction to be precisely related. We experimentally observe the entire quantum trajectory distribution for weak measurements of a superconducting qubit in circuit QED architecture, quantify it, and demonstrate that it agrees very well with the predictions of a single-parameter white-noise stochastic process. This characterisation of quantum trajectories is a powerful clue to unraveling the dynamics of quantum measurement, beyond the conventional axiomatic quantum theory.