Controlled gates for multi-level quantum computation

Multi-level (ML) quantum logic can potentially reduce the number of inputs/outputs or quantum cells in a quantum circuit which is a limitation in current quantum technology. In this paper we propose theorems about ML-quantum and reversible logic circuits. New efficient implementations for some basic controlled ML-quantum logic gates, such as three-qudit controlled NOT, Cycle, and Self Shift gates are proposed. We also propose lemmas about r-level quantum arrays and the number of required gates for an arbitrary n-qudit ML gate. An equivalent definition of quantum cost (QC) of binary quantum gates for ML-quantum gates is introduced and QC of controlled quantum gates is calculated.