arXiv 2101.031422021-01quant-ph

综合T深度最优电路的（准）多项式时间启发式算法

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

Vlad Gheorghiu, Michele Mosca, Priyanka Mukhopadhyay

备注：: Published in Nature Partner Journal Quantum Information. Compared to v4: Minor changes. Not the exact journal version
DOI：: 10.1038/s41534-022-00624-1

摘要：

研究了Clifford+T门集上T深度最优量子电路的综合问题。首先，我们构造一个特殊的子集的T-深度1酉，这样就可以表示的T-深度最优分解的任何酉作为产品的酉从这个子集和克利福德（全球相位）。该子集的基数至多为$n\cdot 2^{5.6n}$。我们使用嵌套的中间相遇（MITM）技术开发的算法综合可证明\emph{depth-optimal}和\emph{T-depth-optimal}电路的精确可实现的酉。具体地说，对于T深度最优电路的综合，我们得到了一个空间复杂度为$O\left(\left(4^{n^2}\right)^{\lceil d/c\rceil}\right)$，时间复杂度为$O\left(\left(4^{n^2}\right)^{(c-1)\lceil d/c\rceil}\right)$的算法，其中$d$为最小T深度，$c\geq 2$为常数。这比Amy等人的算法复杂度好得多。（2013），具有复杂度$O\left(\left(3^n\cdot 2^{kn^2}\right)^{\lceil \frac{d}{2}\rceil}\cdot 2^{kn^2}\right)$的先前最佳，其中$k>2.5$是常数。我们设计了一个更有效的算法来合成T-深度优化电路。声称的效率和最优性取决于一些假设，这些假设是从Mosca和Mukhopadhyay（2020）的工作中得到启发的。据我们所知，这些照片与以前的工作无关。我们的算法具有空间和时间复杂度$poly(n,2^{5.6n},d)$（或$poly(n^{\log n},2^{5.6n},d)$在一些较弱的假设下）。

英文摘要：

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).

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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