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 $N\times N$ 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 $\mathcal{O} (N \log(\log(N)/\epsilon))$. 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 $T$-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.

Keywords

• Block encoding
• quantum circuit synthesis
• quantum circuits
• quantum computing
• quantum random access memory (QRAM)
• quantum resources