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

Abstract

We provide modular circuit-level implementations and resource estimates for several methods of block-encoding a dense $N\times N$ matrix of classical data to precision $\hat{I}\mu$; the minimal-depth method achieves a $T$-depth of $\mathcal{O}(\log (N/\hat{I}\mu )),$ while the minimal-count method achieves a $T$-count of $\mathcal{O}(N\log (1/\hat{I}\mu ))$. We examine resource tradeoffs between the different approaches, and we explore implementations of two separate models of quantum random access memory (QRAM). As part of this analysis, we provide a novel state preparation routine with $T$-depth $\mathcal{O}(\log (N/\hat{I}\mu ))$, improving on previous constructions with scaling $\mathcal{O}({\log }^2(N/\hat{I}\mu ))$. 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.