Systematic construction of multi-controlled Pauli gate decompositions with optimal $T$-count

Abstract: Multi-controlled Pauli gates are typical high-level qubit operations that appear in the quantum circuits of various quantum algorithms. We find multi-controlled Pauli gate decompositions with smaller CNOT-count or $T$-depth while keeping the currently known minimum $T$-count. For example, for the CCCZ gate, we find decompositions with CNOT-count 7 or $T$-depth 2 while keeping the $T$-count at the currently known minimum of 6. The discovery of these efficient decompositions improves the computational efficiency of many quantum algorithms. What led to this discovery is the systematic procedure for constructing multi-controlled Pauli gate decompositions. This procedure not only deepens our theoretical understanding of quantum gate decomposition but also leads to more efficient decompositions that have yet to be discovered.