Variational Quantum Self-Organizing Map

We propose a novel quantum neural network architecture for unsupervised learning of classical and quantum data based on the kernelized version of Kohonen's self-organizing map. The central idea behind our algorithm is to replace the Euclidean distance metric with the fidelity between quantum states to identify the best matching unit from the low-dimensional grid of output neurons in the self-organizing map. The fidelities between the unknown quantum state and the quantum states containing the variational parameters are estimated by computing the transition probability on a quantum computer. The estimated fidelities are in turn used to adjust the variational parameters of the output neurons. Unlike $\mathcal{O}(N^{2})$ circuit evaluations needed in quantum kernel estimation, our algorithm requires $\mathcal{O}(N)$ circuit evaluations for $N$ data samples. Analogous to the classical version of the self-organizing map, our algorithm learns a mapping from a high-dimensional Hilbert space to a low-dimensional grid of lattice points while preserving the underlying topology of the Hilbert space. We showcase the effectiveness of our algorithm by constructing a two-dimensional visualization that accurately differentiates between the three distinct species of flowers in Fisher's Iris dataset. In addition, we demonstrate the efficacy of our approach on quantum data by creating a two-dimensional map that preserves the topology of the state space in the Schwinger model and distinguishes between the two separate phases of the model at $\theta = \pi$.