Optimal T Counts under Sparsity: from QROM to State Preparation and Block Encoding

Abstract:Many quantum algorithms require coherent access to classical data, often modeled by quantum read-only memory (QROM). We initiate the study of the $T$ count of sparse QROM, in which only $s$ of the $2^n$ addresses store nonzero data. We prove asymptotically optimal $T$-count bounds $\Theta(\sqrt{sm} + \sqrt{sn})$ with square-root dependence on the support size $s$ and message length $m$. Our upper bounds use a multilevel hashing scheme, while our lower bounds reduce sparse QROM to state preparation and use counting arguments for adaptive Clifford+$T$ circuits. The lower bounds thus hold even when mid-circuit measurements and classically controlled operations are allowed. As applications, we obtain matching $T$-count bounds $\Theta(\sqrt{sn} + \sqrt{s\log(1/\varepsilon)} + \log(1/\varepsilon))$ for $s$-sparse state preparation and $\Theta( \sqrt{2^n sn} + \sqrt{2^n s\log(s/\varepsilon_{\mathrm{BE}})} + \log(s/\varepsilon_{\mathrm{BE}}))$ for block encoding of $s$-sparse matrices, where $\varepsilon$ and $\varepsilon_{\mathrm{BE}}$ are the precision of state preparation and block encoding, respectively.

Quantum Physics (quant-ph); Computational Complexity (cs.CC); Data Structures and Algorithms (cs.DS)

arXiv:2607.28260 [quant-ph]

(or arXiv:2607.28260v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2607.28260 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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