Quantum Resources Required to Block-Encode a Matrix of Classical Data

Abstract

We provide a modular circuit-level implementation and resource estimates for several methods of block-encoding a dense <inline-formula><tex-math notation="LaTeX">$N\times N$</tex-math></inline-formula> matrix of classical data to precision <inline-formula><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula>; the minimal-depth method achieves a <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula>-depth of <inline-formula><tex-math notation="LaTeX">$\mathcal {O}(\log (N/\epsilon)),$</tex-math></inline-formula> while the minimal-count method achieves a <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula>-count of <inline-formula><tex-math notation="LaTeX">$\mathcal{O} (N \log(\log(N)/\epsilon))$</tex-math></inline-formula>. We examine resource tradeoffs between the different approaches, and we explore implementations of two separate models of quantum random access memory. As a part of this analysis, we provide a novel state preparation routine with <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula>-depth <inline-formula><tex-math notation="LaTeX">$\mathcal {O}(\log (N/\epsilon))$</tex-math></inline-formula>, improving on previous constructions with scaling <inline-formula><tex-math notation="LaTeX">$\mathcal {O}(\log ^{2} (N/\epsilon))$</tex-math></inline-formula>. Our results go beyond simple query complexity and provide a clear picture into the resource costs when large amounts of classical data are assumed to be accessible to quantum algorithms.