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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 ($n\geq 1$) unitary $W$ of size $2^n\times 2^n$, over the Clifford+T gate set. The space and time complexity of our algorithm are $O\left(2^{2n}\right)$ and $O\left(2^{2n\mathcal{T}_{\epsilon}(W)+4n}\right)$ respectively. $\mathcal{T}_{\epsilon}(W)$ ($\epsilon$-T-count) is the (minimum possible) T-count of an exactly implementable unitary $U$ i.e. $\mathcal{T}(U)$, such that $d(U,W)\leq\epsilon$ and $\mathcal{T}(U)\leq\mathcal{T}(U')$ where $U'$ is any exactly implementable unitary with $d(U',W)\leq\epsilon$. $d(.,.)$ 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 $\epsilon$-T-depth. This is the first algorithm that gives T-count or T-depth of any multi-qubit ($n\geq 1$) unitary. For small enough $\epsilon$, 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., Comment: Accepted for publication in Nature Partner Journal Quantum Information. Not structured according to the journal policies. Compared to v3 : A note about implementation of 2-qubit QFT in Table 3

We investigate the problem of synthesizing T-depth optimal quantum circuits over the Clifford+T gate set. First we construct a special subset of T-depth 1 unitaries, such that it is possible to express the T-depth-optimal decomposition of any unitary as product of unitaries from this subset and a Clifford (up to global phase). The cardinality of this subset is at most $n\cdot 2^{5.6n}$. We use nested meet-in-the-middle (MITM) technique to develop algorithms for synthesizing provably \emph{depth-optimal} and \emph{T-depth-optimal} circuits for exactly implementable unitaries. Specifically, for synthesizing T-depth-optimal circuits, we get an algorithm with space and time complexity $O\left(\left(4^{n^2}\right)^{\lceil d/c\rceil}\right)$ and $O\left(\left(4^{n^2}\right)^{(c-1)\lceil d/c\rceil}\right)$ respectively, where $d$ is the minimum T-depth and $c\geq 2$ is a constant. This is much better than the complexity of the algorithm by Amy et al.(2013), the previous best with a complexity $O\left(\left(3^n\cdot 2^{kn^2}\right)^{\lceil \frac{d}{2}\rceil}\cdot 2^{kn^2}\right)$, where $k>2.5$ is a constant. We design an even more efficient algorithm for synthesizing T-depth-optimal circuits. The claimed efficiency and optimality depends on some conjectures, which have been inspired from the work of Mosca and Mukhopadhyay (2020). To the best of our knowledge, the conjectures are not related to the previous work. Our algorithm has space and time complexity $poly(n,2^{5.6n},d)$ (or $poly(n^{\log n},2^{5.6n},d)$ under some weaker assumptions)., Published in Nature Partner Journal Quantum Information. Compared to v4: Minor changes. Not the exact journal version