Lower Bound for theTCount Via Unitary Stabilizer Nullity br

We introduce magic measures to quantify the nonstabilizerness of multiqubit quantum gates and estab-lish lower bounds on theTcount for fault-tolerant quantum computation. First, we introduce the stabilizernullity of multiqubit unitary, which is based on the subgroup of the quotient Pauli group associated withthe unitary. This unitary stabilizer nullity extends the state-stabilizer nullity by Beverlandet al.[QuantumSci. Technol. 5, 035009 (2020)] to a dynamic version. In particular, we show this nonstabilizerness mea-sure 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 theTcount, utilizing theabove properties in gate synthesis. Third, we compare the state and the unitary stabilizer nullity, provingthat the lower bounds for theTcount obtained by the unitary stabilizer nullity are never less than thestate-stabilizer nullity. Moreover, we show an explicitn-qubit unitary family of unitary stabilizer nullity2n, which implies that itsTcount is at least 2n. This gives an example where the bounds derived by theunitary stabilizer nullity strictly outperform the state-stabilizer nullity by a factor of 2. We finally show-case the advantages of unitary stabilizer nullity in estimating theTcount of quantum gates with interests