We investigate the problem of synthesizing T-depth optimal quantum circuits for exactly implementable unitaries over the Clifford+T gate set. We construct a subset, \({{\mathbb{V}}}_{n}\), of T-depth 1 unitaries. T-depth-optimal decomposition of unitary U is \({e}^{i\phi }\left({\prod }_{i}{V}_{i}\right)C\), \({V}_{i}\in {{\mathbb{V}}}_{n}\), C is Clifford and \(| {{\mathbb{V}}}_{n}| \,\le \,n\cdot {2}^{5.6n}\). We use nested meet-in-the-middle technique to synthesize provably depth-optimal and T-depth-optimal circuits. For the latter, we achieve space and time complexity \(O({({4}^{{n}^{2}})}^{\lceil d/c\rceil })\) and \(O({({4}^{{n}^{2}})}^{(c-1)\lceil d/c\rceil })\) respectively (d is the minimum T-depth, c ≥ 2 a constant). The previous best algorithm had complexity \(O({({3}^{n}\cdot {2}^{k{n}^{2}})}^{\lceil \frac{d}{2}\rceil }\cdot {2}^{k{n}^{2}})\)(k > 2.5 a constant). We design a more efficient algorithm with space and time complexity poly(n, 2^5.6n, d) (or \({{{\rm{poly}}}}({n}^{\log n},{2}^{5.6n},d)\) with weaker assumptions). The claimed efficiency, optimality depends on conjectures.

我们研究了在 Clifford+T 门集上为完全可实现的酉合成 T 深度最优量子电路的问题。我们构造了一个 T-depth 1 酉子集\({{\mathbb{V}}}_{n}\)。酉U的 T 深度最优分解是\({e}^{i\phi }\left({\prod }_{i}{V}_{i}\right)C\) , \({V }_{i}\in {{\mathbb{V}}}_{n}\) , C是 Clifford 和\(| {{\mathbb{V}}}_{n}| \,\le \, n\cdot {2}^{5.6n}\)。我们使用嵌套的中间相遇技术来合成可证明的深度最优和 T 深度最优电路。对于后者，我们实现了空间和时间复杂度\(O({({4}^{{n}^{2}})}^{\lceil d/c\rceil })\)和\(O({({4}^{{n}^{2}})}^{(c-1)\lceil d/c\rceil })\)分别（d是最小T深度，c  ≥ 2 个常数）。之前最好的算法复杂度\(O({({3}^{n}\cdot {2}^{k{n}^{2}})}^{\lceil \frac{d}{2}\ rceil }\cdot {2}^{k{n}^{2}})\)（k  > 2.5 一个常数）。我们设计了一种更有效的算法，其空间和时间复杂度为 poly( n , 2 ^{5.6 n} ,  d ) (或\({{{\rm{poly}}}}({n}^{\log n},{2} ^{5.6n},d)\)假设较弱）。声称的效率、最优性取决于猜想。