Papers › T-count and T-depth of any multi-qubit unitary
T-count and T-depth of any multi-qubit unitary
Vlad Gheorghiu, Michele Mosca, Priyanka Mukhopadhyay
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While implementing a quantum algorithm it is crucial to reduce the quantum resources, in order to obtain the desired computational advantage. For most fault-tolerant quantum error-correcting codes the cost of implementing the non-Clifford gate is the highest among all the gates in a universal fault-tolerant gate set. In this paper we design provable algorithm to determine 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(2²ⁿ) and O(2^(2n𝒯_ϵ(W)+4n)) respectively. 𝒯_ϵ(W) (ϵ-T-count) is the (minimum possible) T-count of an exactly implementable unitary U i.e. 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 possible) T-depth of any multi-qubit unitary and the complexity has exponential dependence on n and ϵ-T-depth. This is the first algorithm that gives T-count or T-depth of any multi-qubit (n≥1) unitary. For small enough ϵ, we can synthesize the T-count and T-depth-optimal circuits. Our results can be used to determine the minimum count (or depth) of non-Clifford gates required to implement any multi-qubit unitary with a universal gate set consisting of Clifford and non-Clifford gates like Clifford+CS, Clifford+V, etc. To the best of our knowledge, there were no such optimal-synthesis algorithm for arbitrary multi-qubit unitaries in any universal gate set.
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