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Figure 3: The 6 possible C’ gates for 3-bit reversible circuits. There are 6 possible C gates for the 3-in/out reversible circuits. They act as edge mapping, for example, C?, acts as a mapping between the upper edge and lower edge of RF. A gate library with C3 gates can realize a total of 168 reversible circuits as shown in Table 1. The C gates for 3-in/out reversible circuits are as follows,

Figure 3 The 6 possible C’ gates for 3-bit reversible circuits. There are 6 possible C gates for the 3-in/out reversible circuits. They act as edge mapping, for example, C?, acts as a mapping between the upper edge and lower edge of RF. A gate library with C3 gates can realize a total of 168 reversible circuits as shown in Table 1. The C gates for 3-in/out reversible circuits are as follows,

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Figure 1: A hypercube network with (a) 2 vertices, (b) 4 vertices, (c) 8 vertices and (d) 16 vertices. The one-to-one correspondence between a nxn reversible circuit and a permutation on A = {1,2,..., N} is established as done in [26]. In the permutation group references, A begins from one, instead of zero. Therefore, we have the following relation [26]: < Xn, Xn-1,...,X1 >2= index(Xy,...,X1) —1. Using the integer coding, a permutation is considered as a bijective function f : {1,2,...,N}— {1,2,...,N}. Cascading two generators is equivalent to multiplying two permutations. In what follows, an xn reversible gate is not distinguished from a permutation in Sy.
Figure 1: A hypercube network with (a) 2 vertices, (b) 4 vertices, (c) 8 vertices and (d) 16 vertices. The one-to-one correspondence between a nxn reversible circuit and a permutation on A = {1,2,..., N} is established as done in [26]. In the permutation group references, A begins from one, instead of zero. Therefore, we have the following relation [26]: < Xn, Xn-1,...,X1 >2= index(Xy,...,X1) —1. Using the integer coding, a permutation is considered as a bijective function f : {1,2,...,N}— {1,2,...,N}. Cascading two generators is equivalent to multiplying two permutations. In what follows, an xn reversible gate is not distinguished from a permutation in Sy.
Figure 2: The 3 possible N gates for 3-bit reversible circuits. There are 40320 possible 3-in/out reversible circuits. The N gate and the C gate cannot be used to synthesize all the 3-in/out reversible circuits. There are 3 possible N gates as follows,
Figure 2: The 3 possible N gates for 3-bit reversible circuits. There are 40320 possible 3-in/out reversible circuits. The N gate and the C gate cannot be used to synthesize all the 3-in/out reversible circuits. There are 3 possible N gates as follows,
Figure 4: The 3 possible T gates for 3-bit reversible circuit.
Figure 4: The 3 possible T gates for 3-bit reversible circuit.
Figure 5: The 3 possible F' gates for 3-bit reversible circuits. Fredkin (F’) gate is another 3-bit gate. F' gate performs a conditional swap on two of its inputs if the third input is set to 1. For 3-in/out reversible circuits, there are 3 possible F' gates as follows,
Figure 5: The 3 possible F' gates for 3-bit reversible circuits. Fredkin (F’) gate is another 3-bit gate. F' gate performs a conditional swap on two of its inputs if the third input is set to 1. For 3-in/out reversible circuits, there are 3 possible F' gates as follows,
Figure 6: The 6 possible P gates for 3-bit reversible circuits.
Figure 6: The 6 possible P gates for 3-bit reversible circuits.
Figure 7: The reciprocal of Leibniz Harmonic Triangle. are set to 1, the 15 is a 5-bit with 4 control bits and single target bit. The 75 gate flips the target bit if all control bits are set to 1, and so on. The GT4 library is the GT library for 4 bits with 32 basic gates. It contains 4 N4 gates, 12 C4 gates, 12 T3* gates and 4 T4* gates. The GT5 library is the GT library for 5 bits with 80 basic gates. It contains 5 N° gates, 20 C® gates, 30 T3° gates, 20 T4° gates and 5 T'5° gates. The distribution of gates for the GT'n library is according to the reciprocal of Leibniz Harmonic triangle as shown in Fig.7. The total number of gates for the the G7'n library can be calculated as follows,
Figure 7: The reciprocal of Leibniz Harmonic Triangle. are set to 1, the 15 is a 5-bit with 4 control bits and single target bit. The 75 gate flips the target bit if all control bits are set to 1, and so on. The GT4 library is the GT library for 4 bits with 32 basic gates. It contains 4 N4 gates, 12 C4 gates, 12 T3* gates and 4 T4* gates. The GT5 library is the GT library for 5 bits with 80 basic gates. It contains 5 N° gates, 20 C® gates, 30 T3° gates, 20 T4° gates and 5 T'5° gates. The distribution of gates for the GT'n library is according to the reciprocal of Leibniz Harmonic triangle as shown in Fig.7. The total number of gates for the the G7'n library can be calculated as follows,
Figure 9: The 6 possible G3 gates for a 3-bit reversible circuit. The G3 gate is a 3-bit gate. It combines the action of NV, C and T3 in a single gate, i.e. one bit is flipped if the other two bits are set to 1 then second bit is flipped if one of the remaining bits is set to 1. The last bit is flipped unconditionally. For 3-in/out reversible circuits, there are 6 possible G3 gates as follows,
Figure 9: The 6 possible G3 gates for a 3-bit reversible circuit. The G3 gate is a 3-bit gate. It combines the action of NV, C and T3 in a single gate, i.e. one bit is flipped if the other two bits are set to 1 then second bit is flipped if one of the remaining bits is set to 1. The last bit is flipped unconditionally. For 3-in/out reversible circuits, there are 6 possible G3 gates as follows,
Figure 10: Realization of G3 gate using different gates. It can be shown using GAP that a permutation group with the 6 generators of G3 is of size 40320, i.e. a cascade of these 6 gates are sufficient to implement any 3-in/out reversible circuits. The main G3 gates library consists of 6 gates. There are 64 possible sub libraries of gates from the main G3 library, not every sub library is universal. There are 51 sub libraries that are universal for 3-in/out reversible circuits. The smallest sub library contains 2 gates. There are 9 universal sub libraries with 2 gates, these sub libraries contain any two G3 gates not starting with the same mapping, for example, a library with G3 03 and G3j32 is not universal since they start by the same mapping 1 +> 5, while a library with G3123 and G3213 is universal. It can be verfied using GAP that a group with generators {G3123, G3213} is of size 40320. The average size of circuits synthesized with the G3 library is 6.402 as shown in Table 4. This average is better than a comparable library with 6 gates which is NT’. The maximum number of gates to realize any 3-in/out reversible circuits is 8 similar to NCT and NCF and the size of the gate library is smaller than other gate libraries such as NCP, NCTF, NCPT and NC PF. The G3 gate library is the only pure gate library that contains only one type of gates. Realization of G3 gates using different gates is shown in Fig.10.
Figure 10: Realization of G3 gate using different gates. It can be shown using GAP that a permutation group with the 6 generators of G3 is of size 40320, i.e. a cascade of these 6 gates are sufficient to implement any 3-in/out reversible circuits. The main G3 gates library consists of 6 gates. There are 64 possible sub libraries of gates from the main G3 library, not every sub library is universal. There are 51 sub libraries that are universal for 3-in/out reversible circuits. The smallest sub library contains 2 gates. There are 9 universal sub libraries with 2 gates, these sub libraries contain any two G3 gates not starting with the same mapping, for example, a library with G3 03 and G3j32 is not universal since they start by the same mapping 1 +> 5, while a library with G3123 and G3213 is universal. It can be verfied using GAP that a group with generators {G3123, G3213} is of size 40320. The average size of circuits synthesized with the G3 library is 6.402 as shown in Table 4. This average is better than a comparable library with 6 gates which is NT’. The maximum number of gates to realize any 3-in/out reversible circuits is 8 similar to NCT and NCF and the size of the gate library is smaller than other gate libraries such as NCP, NCTF, NCPT and NC PF. The G3 gate library is the only pure gate library that contains only one type of gates. Realization of G3 gates using different gates is shown in Fig.10.
Table 2: Utilization of gates in universal sub libraries. Table 1: The universality of different libraries for 3-in/out reversible circuits.
Table 2: Utilization of gates in universal sub libraries. Table 1: The universality of different libraries for 3-in/out reversible circuits.
Table 3: Utilization of gates in smallest universal sub libraries for 3-in/out reversible circuits
Table 3: Utilization of gates in smallest universal sub libraries for 3-in/out reversible circuits
Table 4: Minimum size of 3-bit reversible circuits using different libraries.
Table 4: Minimum size of 3-bit reversible circuits using different libraries.
Extending the G gate for n-bits is trivial as shown in Fig. 11. It can be shown using GAP that a permutation group with the n! generators of G'n is of size 2”! , ie. a cascade of these n! gates are sufficient to implement any n-in/out reversible circuits. The main Gn gates library consists of n! gates. There are 2” possible sub libraries of gates from the main Gn library. The smallest universal sub library always contains only 2 gates, for example, the sub library {G'42341, G41234} is universal for 4-in/out reversible circuits. The sub library {G512345, G@423451} is universal for 5-in/out reversible circuits and the sub library {G6234561, G6514326 } is universal for 6-in/out reversible circuits. It has been verified for n < 10 using a random gate generator on GAP that a sub library with 2 gates of size n is universal for n-in/out reversible circuits.
Extending the G gate for n-bits is trivial as shown in Fig. 11. It can be shown using GAP that a permutation group with the n! generators of G'n is of size 2”! , ie. a cascade of these n! gates are sufficient to implement any n-in/out reversible circuits. The main Gn gates library consists of n! gates. There are 2” possible sub libraries of gates from the main Gn library. The smallest universal sub library always contains only 2 gates, for example, the sub library {G'42341, G41234} is universal for 4-in/out reversible circuits. The sub library {G512345, G@423451} is universal for 5-in/out reversible circuits and the sub library {G6234561, G6514326 } is universal for 6-in/out reversible circuits. It has been verified for n < 10 using a random gate generator on GAP that a sub library with 2 gates of size n is universal for n-in/out reversible circuits.
Related topics:
Group TheoryCircuit Optimization
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