Aiming at the effective mapping problem of reversible circuits to quantum circuits, a big mutation adaptive genetic algorithm with tabu list is proposed for the synthesis of quantum reversible circuits. The quantum gate library is formed by choosing the quantum NOT gate, controlled-NOT gate (CNOT), controlled-V gate and controlled-V+ gate. The quantum circuit calculation model is established. The binary serial coding scheme is adopted, and the fitness function, evolutionary operators and optimization rules are designed. The synthesis of quantum reversible circuits that use big mutation adaptive genetic algorithm with tabu list is realized, and tested by the circuit library named Revlib. Results show that the comprehensive method can get multiple functional solutions at the same time, and the cost of the generated circuits is better than that of the circuits in Revlib. The correctness and effectiveness of the proposed algorithm for the synthesis of quantum reversible circuits are verified.