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Quantum Information Erasure Under Sequential Weak Measurements: Empirical Manifestation of the Quantum Cramér-Rao Bound in Physics-Informed Neural Networks

İbrahim Halil Sanduvaç

Published May 18, 2026• 0 citations
Publisher

Abstract

Abstract Reconstructing the unobservable dynamics of quantum many-body systems from sequential weak measurements presents fundamental information-theoretic challenges due to measurement back-action. In this study, we introduce a Physics-Informed Neural Network (PINN) designed for continuous quantum state filtering on a 14-qubit Transverse Field Ising Model (TFIM). We demonstrate that while unconstrained models can exploit thermodynamic memorization, enforcing causal masking and Hamiltonian differential constraints forces the network to synthesize true unitary evolution. Under these rigorous physical constraints, we observe that the predictive accuracy for non-commuting transverse magnetization invariably reaches a strict empirical asymptote at $R^2 \approx 0.39$. We establish that this ceiling is not an architectural bottleneck, but the direct empirical manifestation of the Quantum Cramér-Rao Bound (QCRB), proving that deep learning models remain fundamentally bound by the Heisenberg uncertainty principle. Key Findings The Heisenberg Wall: We identify a fundamental predictive ceiling ($R^2 \approx 0.39$) for non-commuting observables in continuous quantum state filtering. Physics-Informed Architecture: We demonstrate that enforcing Hamiltonian consistency and causal masking eliminates "identity mapping" artifacts, allowing for true phenomenological simulation. Information Erasure: Our results map the neural network's performance bound to the residual information capacity dictated by the Quantum Cramér-Rao Bound. Keywords Physics-Informed Neural Networks; Quantum Many-Body Systems; Transverse Field Ising Model; Sequential Weak Measurements; Quantum Cramér-Rao Bound; Quantum State Filtering.

DOI
10.5281/zenodo.20268196

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Abstract Reconstructing the unobservable dynamics of quantum many-body systems from sequential weak measurements presents fundamental information-theoretic challenges due to measurement back-action. In this study, we introduce a Physics-Informed Neural Network (PINN) designed for continuous quantum state filtering on a 14-qubit Transverse Field Ising Model (TFIM). We demonstrate that while unconstrained models can exploit thermodynamic memorization, enforcing causal masking and Hamiltonian differential constraints forces the network to synthesize true unitary evolution. Under these rigorous physical constraints, we observe that the predictive accuracy for non-commuting transverse magnetization invariably reaches a strict empirical asymptote at R2≈0.39R^2 \approx 0.39R2≈0.39. We establish that this ceiling is not an architectural bottleneck, but the direct empirical manifestation of the Quantum Cramér-Rao Bound (QCRB), proving that deep learning models remain fundamentally bound by the Heisenberg uncertainty principle. Key Findings The Heisenberg Wall: We identify a fundamental predictive ceiling (R2≈0.39R^2 \approx 0.39R2≈0.39) for non-commuting observables in continuous quantum state filtering. Physics-Informed Architecture: We demonstrate that enforcing Hamiltonian consistency and causal masking eliminates "identity mapping" artifacts, allowing for true phenomenological simulation. Information Erasure: Our results map the neural network's performance bound to the residual information capacity dictated by the Quantum Cramér-Rao Bound. Keywords Physics-Informed Neural Networks; Quantum Many-Body Systems; Transverse Field Ising Model; Sequential Weak Measurements; Quantum Cramér-Rao Bound; Quantum State Filtering.
Neural networks (NNs) accelerate simulations of quantum dissipative dynamics. Ensuring that these simulations adhere to fundamental physical laws is crucial, but has been largely ignored in the state-of-the-art NN approaches. We show that this may lead to implausible results measured by violation of the trace conservation. To recover the correct physical behavior, we develop physics-informed NNs (PINNs) that mitigate the violations to a good extend. Beyond that, we propose a novel uncertainty-aware approach that enforces perfect trace conservation by design, surpassing PINNs.
Quantum state tomography (QST) faces exponential measurement requirements and noise sensitivity in multi-qubit systems, bottlenecking practical quantum technologies. We present a physics-informed neural network (PINN) framework integrating quantum mechanical constraints via adaptive weighting, a residual-and-attention-enhanced architecture, and differentiable Cholesky parameterization for physical validity. Evaluations on 2--5 qubit systems and arbitrary-dimensional states show PINN consistently outperforms traditional neural networks (TNNs), achieving highest fidelity across all dimensions. PINN outperforms baselines, with marked improvements in moderately high-dimensional systems, superior noise robustness (slower performance degradation), and consistent dimensional robustness. Theoretical analysis shows physical constraints reduce Rademacher complexity and mitigate the curse of dimensionality via constraint-induced dimension and sample complexity reduction, effective regardless of qubit number. While experiments are limited to 5-qubit systems due to computational constraints, our theoretical framework (convergence guarantees, generalization bounds, scalability theorems) justifies PINN's advantages will persist and strengthen in larger systems (6+ qubits), where constraint-induced dimension reduction benefits grow with system size. Practically, this advances quantum error correction and gate calibration by reducing measurement requirements from O(4^n) to O(2^n) while maintaining high fidelity, enabling faster error correction cycles and accelerated calibration critical for scalable quantum computing.
Quantum Fisher Information (QFI) sets the ultimate precision limit for parameter estimation and is therefore a central quantity in quantum metrology. In time-dependent many-body systems, however, maximizing QFI is a highly non-trivial task due to the combined effects of non-commutativity, control complexity, and the exponential growth of the Hilbert space. In this work, we present a physics-informed neural network (PINN) framework to address this problem through the learning of counter-diabatic quantum dynamics. Our approach combines a variational PINN formulation with a Magnus-expansion treatment of time-ordered evolution, enabling the adiabatic gauge potential and the scheduling function to be inferred directly from the underlying physics while enforcing the Euler-Lagrange structure of the protocol. The method is applied to several families of driven spin Hamiltonians, including nearest-neighbor, dipolar, and trapped-ion-inspired interactions, for systems of up to six qubits. The numerical results show that the proposed framework systematically improves over reference solutions based only on the Euler-Lagrange condition, yielding high normalized QFI together with favorable fidelity and extremal-balance metrics while preserving small phsical residuals. The analysis further shows that learning the scheduling function provides a clear performance advantage in most cases, and reveals non-trivial finite-size effects, with q=3q=3q=3 emerging as a particularly challenging regime. Although scalability remains limited by the exponential growth of the operator space and by automatic-differentiation costs, the results demonstrate that PINNs constitute a viable and physically grounded route for learning metrologically optimal control strategies in interacting quantum systems.
We investigate variational learning of quantum many-body ground states directly in measurement space using autoregressive neural networks. In particular, we represent quantum states via probability distributions of outcomes over a symmetric informationally complete positive operator-valued measure (SIC-POVM). The probability distribution is encoded in the parameters of an autoregressive neural-network-based on gated recurrent units (GRUs). Ground states are obtained by gradient descent that updates the probability distribution to minimize the energy with respect to a given Hamiltonian, while enforcing positivity constraints that ensure that the distribution of measurement outcomes correspond to a physical quantum state. We analyze the role of constraint enforcement (hierarchy of positivity conditions), variety of neural network architectures (multiple layers, dilation, and modifications of input data) in determining the success of this approach. We benchmark our approach on one-dimensional transverse-field Ising model and the Heisenberg model, along with gapping fields, for system sizes up to L=128, illustrating its efficacy across a wide variety of models.
Implementing quantum gates on quantum computers can require the application of carefully shaped pulses for high-fidelity operations. We explore the use of physics-informed neural networks (PINNs) for quantum optimal control to assess their usefulness in predicting such pulses. Our PINN is a feedforward neural network that utilizes an unsupervised learning approach, whose loss function includes terms that enforce the equations that govern the evolution of a quantum system, measure how close the learned unitary is to the target unitary operation, and ensure state normalization. We use a sinusoidal activation function and adopt variance-type weight initialization, tailored to our activation function. By analyzing the model's performance with important machine learning metrics, we demonstrate that the choice of our architecture is well-suited for this type of problem. We ensure that our network avoids the vanishing and exploding gradients with our relevant choices. We build two different PINNs, one based on the Schrödinger equation and another one based on the Lindblad equation. Our PINNs are able to discover high-fidelity two-qubit gate pulses for a variety of quantum operations, demonstrating its flexibility and robustness. We build two different PINNs, one based on the Schrödinger equation and another one based on the Lindblad equation. Our PINNs are able to discover high-fidelity two-qubit gate pulses for a variety of quantum operations, demonstrating its flexibility and robustness.
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.