A Meet-in-the-Middle Algorithm for Fast Synthesis of Depth-Optimal Quantum Circuits

We present an algorithm for computing depth-optimal decompositions of logical operations, leveraging a meet-in-the-middle technique to provide a significant speedup over simple brute force algorithms. As an illustration of our method, we implemented this algorithm and found factorizations of commonly used quantum logical operations into elementary gates in the Clifford+
T
set. In particular, we report a decomposition of the Toffoli gate over the set of Clifford and
T
gates. Our decomposition achieves a total
T
-depth of 3, thereby providing a 40% reduction over the previously best known decomposition for the Toffoli gate. Due to the size of the search space, the algorithm is only practical for small parameters, such as the number of qubits, and the number of gates in an optimal implementation.