QubitLens: An Interactive Learning Tool for Quantum State Tomography

Quantum state tomography is a fundamental task in quantum computing, involving the reconstruction of an unknown quantum state from measurement outcomes. Although essential, it is typically introduced at the graduate level due to its reliance on advanced concepts such as the density matrix formalism, tensor product structures, and partial trace operations. This complexity often creates a barrier for students and early learners. In this work, we introduce QubitLens, an interactive visualization tool designed to make quantum state tomography more accessible and intuitive. QubitLens leverages maximum likelihood estimation (MLE), a classical statistical method, to estimate pure quantum states from projective measurement outcomes in the X, Y, and Z bases. The tool emphasizes conceptual clarity through visual representations, including Bloch sphere plots of true and reconstructed qubit states, bar charts comparing parameter estimates, and fidelity gauges that quantify reconstruction accuracy. QubitLens offers a hands-on approach to learning quantum tomography without requiring deep prior knowledge of density matrices or optimization theory. The tool supports both single- and multi-qubit systems and is intended to bridge the gap between theory and practice in quantum computing education.

Quantum Natural Stochastic Pairwise Coordinate Descent

Quantum machine learning through variational quantum algorithms (VQAs) has gained substantial attention in recent years. VQAs employ parameterized quantum circuits, which are typically optimized using gradient-based methods. However, these methods often exhibit sub-optimal convergence performance due to their dependence on Euclidean geometry. The quantum natural gradient descent (QNGD) optimization method, which considers the geometry of the quantum state space via a quantum information (Riemannian) metric tensor, provides a more effective optimization strategy. Despite its advantages, QNGD encounters notable challenges for learning from quantum data, including the no-cloning principle, which prohibits the replication of quantum data, state collapse, and the measurement postulate, which leads to the stochastic loss function. This paper introduces the quantum natural stochastic pairwise coordinate descent (2-QNSCD) optimization method. This method leverages the curved geometry of the quantum state space through a novel ensemble-based quantum information metric tensor, offering a more physically realizable optimization strategy for learning from quantum data. To improve computational efficiency and reduce sample complexity, we develop a highly sparse unbiased estimator of the novel metric tensor using a quantum circuit with gate complexity $\Theta(1)$ times that of the parameterized quantum circuit and single-shot quantum measurements. Our approach avoids the need for multiple copies of quantum data, thus adhering to the no-cloning principle. We provide a detailed theoretical foundation for our optimization method, along with an exponential convergence analysis. Additionally, we validate the utility of our method through a series of numerical experiments.