Speaker
Description
The emergence of Quantum Numerical Linear Algebra (qNLA) offers a paradigm shift in solving large-scale linear systems and matrix functions. However, the practical utility of these algorithms, such as the seminal HHL, is fundamentally bottlenecked by the "input problem", namely the efficient representation of classical matrices as quantum circuits.
In this talk, we explore two distinct methodologies for addressing this challenge. First, we present a systematic framework for constructing block encodings from state preparation circuits. This is achieved by introducing efficient basis-transformation protocols and low-overhead, constant-depth Pauli multiplexers. We then establish a bidirectional operational equivalence by demonstrating the converse: converting block encodings back into state preparation circuits using quantum linear algebra operations.
Furthermore, we introduce a structurally-driven approach based on the factorization of matrix state preparation. We prove that when the state preparation algorithm is applied to vectorized classical data, this algorithm naturally decomposes into a standard Norm Loader and a Direction Multiplexer. This observation reveals that a well-known approach for creating block encodings was effectively "hidden" within the classical logic of state preparation. By leveraging this hidden structure, we present a new method for block encoding that offers unique advantages in qubit utilization and circuit depth.