Variational Quantum Algorithm for Partial Singular Value Decomposition*

Quantum entanglement is a key description of multi-particle correlations in various quantum systems and constitutes the fundamental basis for emerging technologies such as quantum computation and communication. Specifically, the entanglement spectrum, quantifying quantum entanglement, serves as a crucial criterion for identifying topological quantum phases and assessing quantum criticality around phase transitions. Utilizing quantum computing for Singular Value Decomposition (SVD) plays a pivotal role in understanding quantum entanglement in large-scale quantum systems beyond the capability of classical computers.

The Variational Quantum Algorithm (VQA), executable with reduced gate operations, emerges as a promising solution for current Noisy Intermediate-Scale Quantum (NISQ) devices, where the error accumulation due to gate operations poses challenges. Numerous researchers have proposed VQAs for SVD, centered on approximating the unitary matrix in SVD using quantum circuits. However, applying such quantum algorithms for highly entangled states is challenging, demanding an exponentially large number of gate operations.

Our study revisited this challenge, redefining VQA as an approximation problem for a specific quantum circuit. We discovered that our partial SVD algorithm, approximating quantum circuits corresponding to individual left-right singular vectors, accurately determines singular values with fewer gate operations than conventional quantum SVD algorithms.