Trading Memory for Depth: Tight Bounds for LCU Block-Encodings

arXiv:2611.15229v1 Announce Type: new Abstract: An LCU block-encoding of $A=\sum_j\alpha_j U_j$ runs the $U_j$ in series on one register, so its select depth is the total work $\sum_j d_j$. Depth was the one block-encoding constraint with no knob: grouping the $T$ terms into $g$ registers that run in parallel gives it one, and the binning is the knob. For a $g$-branched realisation (up to $g$ oracles in parallel) the select depth is tight at $\Theta(C+\log(g\,\max\{n,m\}))$ for $g\ge2$, with $C$ the binning makespan and $m$ the largest bin, in the $k$-local atomic-oracle model. The work floor is a two-way reduction to classical multiprocessor scheduling: every realisation \emph{induces} a $g$-machine schedule, and the construction realises any binning, so an optimal binning attains it.