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

Abstract

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