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

• URL: http://arxiv.org/abs/2607.28260v1
• Date: Thu, 30 Jul 2026 14:18:29 GMT
• Title: Optimal T Counts under Sparsity: from QROM to State Preparation and Block Encoding
• Abstract summary: Many quantum algorithms require coherent access to classical data, often modeled by quantum read-only memory (QROM)<n>We study the $T$ count of sparse QROM, in which only $s$ of the $2n$ addresses store nonzero data.<n>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.
• Score: 26.296743455346604
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $Θ(\sqrt{sm} + \sqrt{sn})$ 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 $Θ(\sqrt{sn} + \sqrt{s\log(1/\varepsilon)} + \log(1/\varepsilon))$ for $s$-sparse state preparation and $Θ( \sqrt{2^n sn} + \sqrt{2^n s\log(s/\varepsilon_{\mathrm{BE}})} + \log(s/\varepsilon_{\mathrm{BE}}))$ for block encoding of $s$-sparse matrices, where $\varepsilon$ and $\varepsilon_{\mathrm{BE}}$ are the precision of state preparation and block encoding, respectively.

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