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

2026-07-30

AI Summary

This work addresses the optimization of T-gate complexity for sparse quantum read-only memory (QROM) and its applications to sparse state preparation and block encoding. By introducing a multi-level hashing scheme, the authors establish the first asymptotically tight bound on the T-count for sparse QROM, namely Θ(√(sm) + √(sn)), where s denotes sparsity and m, n are problem dimensions. Furthermore, through a reduction to adaptive Clifford+T circuits and state preparation, they derive a matching lower bound that remains valid even in the presence of intermediate measurements and classical control. These results extend to s-sparse state preparation and s-sparse matrix block encoding, revealing a square-root dependence of the T-count on the support size s and achieving tight asymptotic bounds for both upper and lower limits.

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.

Problem

• sparse QROM
• T-count
• state preparation
• block encoding
• quantum algorithms

Innovation

• sparse QROM
• T-count
• state preparation
• block encoding
• multilevel hashing