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 $\epsilon$; the minimal-depth method achieves a $T$-depth of $\mathcal{O}{(\log (N/\epsilon))},$ while the minimal-count method achieves a $T$-count of $\mathcal{O}{(N\log(1/\epsilon))}$. 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 $\mathcal{O}{(\log (N/\epsilon))}$, improving on previous constructions with scaling $\mathcal{O}{(\log^2 (N/\epsilon))}$. 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.