We introduce magic measures for multi-qubit quantum gates and establish lower bounds on the non-Clifford resources for fault-tolerant quantum computation. First, we introduce the stabilizer nullity of an arbitrary 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. We in particular show this magic measure has desirable properties such as sub-additivity 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 further connect the unitary stabilizer nullity and the state stabilizer nullity with auxiliary systems, showing that adding auxiliary systems and choosing proper stabilizer states can strictly improving the lower bound obtained by the state stabilizer nullity.

翻译：我们引入了多种量子门的魔力措施, 并在非克利福德资源上设定了较低的限制, 用于错误容忍量的计算。 首先, 我们引入了任意的多度比特单一单元的稳定性无效性, 以与单元一致的保利商集团的分组为基础。 这一单一稳定性无效性将Beverland et al. 的州稳定性无效性扩展为动态版本。 我们特别展示了这一神奇措施的特性, 如成份下的次增加性, 以及高产品下的相加性。 第二, 我们证明给定的统一稳定剂的无效性是T计的较低约束性, 使用上述组合的属性。 第三, 我们比较了国家和单一稳定剂的单一稳定性统一性, 证明由统一稳定性获得的T的较低约束性。 此外, 我们展示了一个明确的美元- 平面统一性统一性统一性稳定性组合, 意味着其固定性为至少2美元, 严格地将固定性稳定性与固定性稳定性国家体系连接起来。