Many quantum algorithms require coherent access to classical data, often modeled by quantum read-only memory (QROM). We initiate the study of the $\mathrm{T}$ count of sparse QROM, in which only $s$ of the $2^n$ addresses store nonzero data. We prove that the optimal $\mathrm{T}$ count is $Θ\left(n+\min\left\{s,\sqrt{s\left(m+\log(2^{n+1}/s)\right)}\right\}\right)$. 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+$\mathrm{T}$ circuits. The lower bounds thus hold even when mid-circuit measurements and classically controlled operations are allowed. As applications, we obtain matching $\mathrm{T}$-count bounds $Θ\left(\min\left\{s,\sqrt{s\log(2^{n+1}/s)}\right\} +\sqrt{s\log(1/\varepsilon)}+\log(1/\varepsilon)\right)$ for $s$-sparse state preparation and $Θ\left(\sqrt{2^n s\left(n+\log(1/\varepsilon_{\rm BE})\right)} +\log(1/\varepsilon_{\rm BE})\right)$ for block encoding of $s$-sparse matrices, where $\varepsilon$ and $\varepsilon_{\rm BE}$ are the precision of state preparation and block encoding, respectively.

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