Lower Bound for the T Count Via Unitary Stabilizer Nullity

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 multiqubit 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. [Quantum Sci. Technol. 5, 035009 (2020)] 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.