T-count and T-depth of any multi-qubit unitary

Abstract We design an algorithm to determine the (minimum) T-count of any n -qubit ( ≥ 1) unitary W size 2 × , over Clifford+T gate set. The space and time complexity our are $$O\left({2}^{2n}\right)$$ O 2 n $$O\left({2}^{2n{{{{\mathcal{T}}}}}_{\epsilon }(W)+4n}\right)$$ T ϵ ( W ) + 4 respectively. $${{{{\mathcal{T}}}}}_{\epsilon }(W)$$ ϵ -T-count) is exactly implementable U $${{{\mathcal{T}}}}(U)$$ U ), such that d ) ≤ $${{{\mathcal{T}}}}(U)\le {{{\mathcal{T}}}}({U}^{{\prime} })$$ ≤ ′ where $${U}^{{\prime} }$$ with $$d({U}^{{\prime} },W)\le \epsilon$$ d , . (. .) global phase invariant distance. Our can also be used T-depth as well minimum non-Clifford-gate count or depth required implement multi-qubit a finite universal set like Clifford+CS, Clifford+V, etc. For small enough we synthesize optimal circuits.