Nonparametric Quantum State Estimation Using the Nearest Neighbor Approach

Quantum state estimation is a central task in quantum information theory, aiming to reconstruct an unknown density matrix from a finite number of measurement outcomes. Conventional approaches such as kernel smoothing or maximum likelihood estimation (MLE) often rely on restrictive parametric assumptions and become computationally prohibitive for high-dimensional quantum systems. In this work, we introduce a nonparametric quantum state estimation framework based on the k-Nearest Neighbor (k-NN) approach. The method leverages the local geometric structure of measurement data within the quantum feature space to infer the underlying quantum state without any predefined model. Using intrinsic quantum distance metrics—such as the trace distance and quantum fidelity—the proposed estimator identifies neighboring states and performs a weighted aggregation, followed by a positive-semidefinite projection to ensure physical admissibility. We establish the consistency and asymptotic convergence of the NN-based estimator under mild regularity conditions. Numerical simulations on single- and two-qubit systems confirm that the proposed method achieves high reconstruction accuracy, robustness to measurement noise, and superior computational efficiency compared to kernel- and likelihood-based estimators. Overall, this study highlights the potential of data-driven, geometry-aware, nonparametric learning for scalable quantum state reconstruction, providing a promising alternative for modern quantum information processing.