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

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.