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×N matrix of classical data to precision ϵ; the minimal-depth method achieves a T-depth of O(log(N/ϵ)), while the minimal-count method achieves a T-count of O(Nlog(1/ϵ)). 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 O(log(N/ϵ)), improving on previous constructions with scaling O(log²(N/ϵ)). 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.