A (quasi-)polynomial time heuristic algorithm for synthesizing T-depth optimal circuits

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