Abstract 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.
Abstract 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.
Continuous-variable quantum systems are foundational to quantum computation, communication, and sensing. While traditional representations using wave functions or density matrices are often impractical, the tomographic picture of quantum mechanics provides an accessible alternative by associating quantum states with classical probability distribution functions called tomograms. Despite its advantages, including compatibility with classical statistical methods, the tomographic method remains underutilized due to a lack of robust estimation techniques. This Letter addresses this gap by introducing a nonparametric kernel quantum state estimation (KQSE) framework for reconstructing quantum states and their trace characteristics from noisy data, without prior knowledge of the state. In contrast to existing methods, KQSE yields estimates of the density matrix in various bases, as well as trace quantities such as purity, higher moments, overlap, and trace distance, with a near-optimal convergence rate of O[over ˜](T^{-1}), where T is the total number of measurements. KQSE is robust for multimodal, non-Gaussian states, making it particularly well suited for characterizing states essential for quantum science.
We further develop Kernel Quantum State Estimation (KQSE), a fully data-driven nonparametric method for continuous variable quantum state reconstruction and characterization, introduced in our recent work and rooted in nonparametric kernel density estimation (KDE). Unlike approaches relying on finite-dimensional basis truncations, parametric ansätze, or prior models, KQSE combines a new kernel estimator of the characteristic function of the tomographic quadrature distribution with suitable kernel integral transformations. The characteristic function is estimated directly from experimental homodyne or heterodyne data and subsequently transformed to estimate quantum state representations and characteristics, including the Wigner function and the density matrix kernel. Observing that several other physically relevant representations and characteristics admit transformations of a closely related form, we derive convergence rates for the corresponding broad class of KQSE-based estimators. The resulting framework covers all phase-space representations, the photon-number tomogram, trace products of quantum states, and purity. We derive mean squared error convergence rates for these kernel transformed estimators and show that the corresponding KQSE applications inherit the near optimal rate O(T^{-1}), where T is the total number of measurements. The proposed theory applies equally to Gaussian and non-Gaussian continuous variable quantum states without imposing a Fock space cutoff. Numerical experiments on simulated Gaussian and non-Gaussian states and real homodyne data demonstrate the advantages of KQSE over state-of-the-art methods, establishing it as a statistically consistent and computationally modest framework for continuous variable quantum state estimation and characterization, particularly in the non-Gaussian regime.
This paper presents a novel quantum K-nearest neighbors (QKNN) algorithm, which offers improved performance over the classical k-NN technique by incorporating quantum computing (QC) techniques to enhance classification accuracy, scalability, and robustness. The proposed modifications focus on optimizing quantum data encoding using Hadamard and rotation gates, ensuring more effective rendering of classical data in quantum states. In addition, the quantum feature extraction process is significantly enhanced by the use of entangled gates such as IsingXY and CNOT, which enables better feature interactions and class separability. A novel quantum distance metric, based on the swap test, is introduced to calculate similarity measures between various quantum states, offering superior accuracy and computational efficiency compared to traditional Euclidean distance metrics. We assess the achievement of the proposed QKNN algorithm on three benchmark datasets: Wisconsin Breast Cancer, Iris, and Bank Note Authentication, and have noted its superior performance relative to both classical k-NN (CKNN) and Quantum Neural Network (QNN). The proposed QKNN algorithm is found to achieve prediction accuracies of 98.25%, 100%, and 99.27% ,respectively, for the three datasets, while the customized QNN shows prediction accuracies of only 97.17%, 83.33%, and 86.18%, respectively. Furthermore, we address the challenges of quantum noise by incorporating a repetition encoding-based error mitigation strategy, which ensures the stability and resilience of the algorithm in noisy quantum environments. The results highlight the potential of the proposed QKNN as a scalable, efficient and robust quantum-enhanced machine learning algorithm, especially in high-dimensional and complex datasets, when traditional approaches frequently fail.
Abstract Quantum coherence, as one of the most fundamental non-classical features in quantum mechanics, plays a pivotal role in various quantum information processing tasks, including quantum computing and quantum metrology. The robustness of quantum coherence (RoC) offers an operational interpretation by quantifying the advantage provided by a quantum state in phase discrimination tasks. To achieve verification RoC with high precision via semidefinite programming (SDP), complete knowledge of quantum states is typically required. Relying solely on expectation values of observables may introduce significant errors in SDP-based estimations. To estimate RoC with high precision using limited data extracted from quantum states, firstly, we propose a semi-supervised K-nearest neighbor (KNN) algorithm (semi-KNN) and a semi-supervised method that combines the KNN and random forest (RF) models with a dynamical threshold (semi-KNN-RF). Then we implement the semi-KNN and semi-KNN-RF models to efficiently estimate quantum coherence by analyzing statistical data obtained from randomly generated local projective measurements performed on unknown quantum states. The semi-KNN-RF model performs better than the semi-KNN model. This innovative methodology allows for accurate coherence estimation, even for high-dimensional quantum systems.
ABSTRACT A quantum K‐nearest neighbors(QKNN) algorithm is proposed to offer superior performance compared to the classical KNN(CKNN) approach, improving classification accuracy, scalability, and robustness. Our approach optimizes Hadamard and rotation gates for quantum data encoding and efficiently embeds classical data into quantum states. Entangled gates, such as IsingXY and CNOT, enhance feature extraction and classification by enabling complex feature interactions. A new quantum distance metric based on swap test results is used to calculate similarity measures between quantum states. This algorithm offers superior accuracy and computational efficiency compared to traditional Euclidean distance metrics. We used three benchmark datasets to evaluate the suggested QKNN method. The results demonstrated that it outperformed the other two methods, classical KNN (CKNN) and quantum neural networks (QNN), as well as the more recent QKNN research. The proposed QKNN algorithm achieves prediction accuracies of 98.25%, 100%, and 99.27% for the three datasets, whereas the QNN achieves prediction accuracies of 97.17%, 83.33%, and 86.18%, respectively. Moreover, quantum noise challenges are addressed by integrating a Shor code‐based error mitigation strategy, which ensures stability of the algorithm and resilience to noisy quantum environments. The results demonstrate the scalability, efficiency, and robustness of the proposed QKNN algorithm.
Dealing with high-dimensional data is a well-known limitation for the classic K-Nearest Neighbors (KNN) algorithm, creating substantial computational and memory overhead. While Quantum Machine Learning (QML) presents a compelling alternative, many existing Quantum KNN (QKNN) frameworks are largely theoretical, remaining impractical for current hardware due to their significant qubit demands. This paper introduces the Fidelity-Based Quantum K-Nearest Neighbors (FQKNN) algorithm, a method specifically designed for feasibility on near-term devices. Our primary contribution is the use of quantum fidelity as a similarity function, which offers a more nuanced metric than conventional distance calculations. This is combined with an efficient amplitude encoding strategy that logarithmically scales the qubit requirement to just log2(N) for an N-dimensional feature vector. This combination makes FQKNN a resource-efficient and practical algorithm. We validated its performance on the benchmark Iris and Breast Cancer datasets, achieving classification accuracies of 96.7% and 93.3% respectively, surpassing the performance of the classical KNN baseline. These findings not only confirm the viability of our fidelity-based similarity measure but also underscore the demonstrable potential of QML to enhance classification tasks.