Synthesizing hybrid quantum circuits without ancilla qudits

This paper investigates the synthesis of quantum networks built to realize hybrid switching circuits in the absence of ancilla qudits. We prove that all hybrid reversible circuits can be constructed by hybrid Not and Multiple-Controlled-Not gates. We also prove that any hybrid reversible circuit with only 1 or 2 binary qudits and arbitrary number of other qudits, can be constructed by hybrid Not and Controlled-Not gates. We present two construction-based algorithms to synthesize hybrid reversible circuits without ancilla qudits. The algorithms use hybrid Not and Multiple-Controlled-Not gates or hybrid Not and `1´-Controlled-Not gates, which are exponentially lower than breadth-first search based synthesis algorithms with respect to the input number.