We introduce magic measures to quantify the nonstabilizerness of multiqubit quantum gates and establish lower bounds on the $T$ count for fault-tolerant quantum computation. First, we introduce the stabilizer nullity of multi-qubit unitary, which is based on the subgroup of the quotient Pauli group associated with the unitary. This unitary stabilizer nullity extends the state-stabilizer nullity by Beverland et al. to a dynamic version. In particular, we show this nonstabilizerness measure has desirable properties such as subadditivity under composition and additivity under tensor product. Second, we prove that a given unitary's stabilizer nullity is a lower bound for the $T$ count, utilizing the above properties in gate synthesis. Third, we compare the state- and the unitary-stabilizer nullity, proving that the lower bounds for the $T$ count obtained by the unitary-stabilizer nullity are never less than the state-stabilizer nullity. Moreover, we show an explicit $n$-qubit unitary family of unitary-stabilizer nullity $2n$, which implies that its $T$ count is at least $2n$. This gives an example where the bounds derived by the unitary-stabilizer nullity strictly outperform the state-stabilizer nullity by a factor of $2$. We finally showcase the advantages of unitary-stabilizer nullity in estimating the $T$ count of quantum gates with interests.

翻译：我们引入魔法测度以量化多量子比特量子门的非稳定子性，并建立容错量子计算中$T$计数的下界。首先，我们提出多量子比特幺正的稳定子零化度，该量基于与幺正相关的商泡利群子群。此幺正稳定子零化度将Beverland等人的态稳定子零化度推广至动态版本。特别地，我们证明该非稳定子性测度具有理想性质，如复合下的次可加性和张量积下的可加性。其次，利用上述性质在门综合中，我们证明给定幺正的稳定子零化度是$T$计数的下界。第三，我们比较态稳定子零化度与幺正稳定子零化度，证明由幺正稳定子零化度得到的$T$计数下界始终不小于态稳定子零化度。此外，我们展示了一个显式的$n$量子比特幺正族，其幺正稳定子零化度为$2n$，这意味着其$T$计数至少为$2n$。该示例表明，由幺正稳定子零化度导出的界在严格程度上比态稳定子零化度高出两倍。最后，我们展示了幺正稳定子零化度在估计具有应用价值的量子门$T$计数方面的优势。