Quantum Resources Required to Block-Encode a Matrix of Classical Data
B. D. Clader, Alexander M. Dalzell, Nikitas Stamatopoulos, Grant Salton, Mario Berta, William J. Zeng · IEEE Transactions on Quantum Engineering · 2022
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$\mathcal{O}(N\log (1/\epsilon))$. 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.