Optimization of $S$-boxes GOST R 34.12-2015 «Magma» quantum circuits without ancilla qubits

We study the ways to implement $S$-boxes by quantum circuits with a minimal number of logical qubits and logical quantum gates without using ancilla qubits. New quantum circuits that implement the $S$-boxes of the GOST R 34.12-2015 “Magma” on 4 logical qubits are constructed. It means that for substitutions $s \in S(V_n)$ with a large number of cycles there exist quantum circuits on $n$ logical qubits that implement the substitution $s$ with fewer logical quantum gates compared with substitutions $g \in S(V_n)$ with a small number of cycles.