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 (n ≥ 1) unitary W of size 2n × 2n, over the Clifford+T gate set. The space and time complexity of our algorithm are O(2 2n) and O(22nTϵ(W)+4n), respectively. Tϵ(W) (ϵ-T-count) is the (minimum) T-count of an exactly implementable unitary U (T(U)), such that d(U,W) ≤ ϵ and T(U) ≤ T(U′) where U′ is any exactly implementable unitary with 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.