FROM SINGULAR VALUES TO QUBITS: A PEDAGOGICAL INTRODUCTION TO VARIATIONAL QUANTUM SVD AND TUCKER DECOMPOSITION, WITH A CLASSICAL SIMULATION STUDY

Abstract

The singular value decomposition (SVD) and its multi-way generalisation, the Tucker decomposition, compress data and reveal which directions carry information. For quantum states, however, the matrix dimension doubles with every added qubit, so classical SVD routines become exponentially expensive in the number of qubits. We give a pedagogical account of two hybrid quantum-classical algorithms designed to avoid this wall: the Variational Quantum Singular Value Decomposer (QSVD) and its tensor extension, Quantum Tucker Decomposition (QTucker). We build the necessary background from first principles and present the variational circuit and cost function. In noiseless classical simulations on bipartite states of up to ten qubits and on a small third-order tensor, the post-processed spectra and entanglement entropies agree with the classical benchmark to numerical precision. This agreement is guaranteed by the invariance of singular values under local unitaries, so it is the training cost, not the spectrum, that measures the accuracy of the circuit. All results come from classical simulation; a computational advantage on quantum hardware is an asymptotic projection that is not demonstrated here.