Efficient Algorithms for Optimal 4-Bit Reversible Logic System Synthesis

Algorithm 3

Quantum_Minimum _Representation (QMR).

Input: Quantum Gate Library 𝐿, permutation 𝑝

Output: Circuit of permutation 𝑝 with mini-length

1: if  ∃𝑙∈{0,1,2,…,max𝑙},GetMin(𝑝,𝜎,𝑏)∈𝑆[𝑙]  then

2: 𝐺=MQR(GetMin(𝑝,𝜎,𝑏),𝑙)

3: return  𝑅_𝑏(𝑃_𝜎−1(𝐺))

4: else if   ∃𝑖∈𝑆𝐹,∃𝑏∈𝑆𝐵,∃𝑙∈{1,2,…,max𝑙},

 ∃Node𝑥∈𝑆[𝑙],

   GetMin(𝑅_𝑏𝑃_𝜎𝑖𝑝∘Node𝑥.cpm⁻¹,𝜏,𝑐)

    ∈𝑆[max𝑙]  then

5: 𝐺₂=MQR(Node𝑥.cpm,𝑙)

6: 𝜌=GetMin𝑅_𝑏𝑃_𝜎𝑖(𝑃)∘Node𝑥.cpm⁻¹,𝜏,𝑐

7: 𝐺₁=MQR𝜌,max𝑙

8: return  𝑅_𝑏𝑃_𝜎𝑖⁻¹𝑅_𝑐𝑃_𝜏−1𝐺₁𝐺₂

9: else

10: return  NULL

11: end if