Numerical Linear Algebra in the Quantum Era

Abstract

Quantum computing offers a unique opportunity to revolutionize numerical linear algebra and scientific computing. This is due to the capability of quantum computers to efficiently model complex structures, and their ability to represent and act on high-dimensional vectors and matrices using exponentially fewer qubits. However, the current landscape of quantum computing research emphasizes intricate, tailor-made circuit designs, created in an ad-hoc manner for specific mathematical challenges. In this talk, I will discuss our recent progress on developing a unified and systematic approach to utilizing quantum computing for numerical linear algebra. Our research centers around a novel Quantum Linear Algebra (qLA) framework offering fundamental matrix algebra building blocks, akin to BLAS but for Quantum Computers. This framework is designed to provide a high-level interface for writing and executing linear algebra subroutines, empowering developers by hiding low-level circuit complexities. I will present an initial version of the framework based on Matrix State Preparation (MSP) and discuss our recent work on expanding it with additional input models, specifically Block Encoding, and the foundational matrix algebra operations that connect them.