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
Abstract
We provide a modular circuit-level implementation and resource estimates for several methods of block-encoding a dense <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N\times N$</tex-math></inline-formula> matrix of classical data to precision <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> ; the minimal-depth method achieves a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> -depth of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal{O}(\log (N/\epsilon)),$</tex-math></inline-formula> while the minimal-count method achieves a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> -count of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal{O}(N\log (1/\epsilon))$</tex-math></inline-formula> . 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> -depth <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal{O}(\log (N/\epsilon)),$</tex-math></inline-formula> improving on previous constructions with scaling <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal{O}(\log ^{2} (N/\epsilon))$</tex-math></inline-formula> . 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.