ii iii

iii i

1

Transposition: A cycle of length 2 is called a trans-

position. Every cycle of length k in SN (N > 1) is ex-

pressible as a product (k − 1) of transpositions in the fol-

lowing manner,

1211,111312

,, ,,,,,

kkk k

iiiiiiiiii ii

Since, the inverse of a permutation is the permutation

obtained by interchanging the rows between them. This

results to state that every transposition is the inverse of

its own.

1

,.,, ,

j

kjkjkjk kj

ii iiIiiiiii

A product of disjoint cycle or transpositions is com-

mutative.

,, ,,

j

kmnmn jk

iiiiii ii

An important property which will be particularly in-

teresting for this paper is that one can decompose a

transposition in the following product,

,,,,

j

ljkkljk

iiiiii ii

3. Algebra of Reversible Logic

Now we show that the algebra of reversible logic can be

introduced in terms of permutation group and the concept

of cycle works as the building block of reversible gate.

Hence, we adopt the following statements,

For every n-bit reversible logic there exist a unique

permutation of degree 2n. This means, the properties and

the operations of permutation groups are applicable for

reversible logic as well.

This statement is discussed in the beginning of the lit-

erature. Recall that as a subgroup of a symmetric group,

all the necessary group axioms are satisfied by permuta-

tion group; that is it contains the identity permutation, the

inverse permutation and the associative and closure

property. Since reversible logic mimics a permutation as

shown before, the composite operations between permu-

tations can also be adapted for reversible gates.

Since every reversible gate is expressible by corre-

sponding permutation, hence it is possible to express a

reversible gate as a product of disjoint cycles and nec-

essarily as a product of transpositions.

Let us examine the logic of an inverter which is

and 1 since it is a single bit logic, the gate

can be expressed as a permutation of degree 2 which is

010

NOT

01 0,1

10

It can be seen that the inverter is logically reversible,

but unfortunately is not a universal gate in the sense that

any arbitrary Boolean logic is not realizable in terms of

inverter alone. For quantum computation, we need uni-

versal reversible gates. One of such gates is the Taffoli-

Fredkin (T-F) gate [2,3] which is necessarily a 3-bit gate

with 3 inputs (a, b, c) and corresponding outputs (a', b',

c'). The relationship between the input and output is the

following

'

'

'

aa

bb

cc ab

Here

denotes the Exclusive OR and

denotes

the AND function. The variables a and b are control bits

where c is called the target bit. As seen from the logic,

the control bit suffers from no change while passing

through the gate. However the target bit flips if both con-

trol bits are logic 1. The permutation corresponding to

this gate is,

T-F

000001010011 100101 110111

000001010011100101 111 110

And the corresponding transposition is T-F

= (110,

111). We can also write the transposition in decimal

symbol which would be more convenient for calculation.

In case of T-F gate, the transposition is (6, 7).

The similar treatment can be accounted for more com-

plex gates. Such a gate is Peres gate [4], which is another

reversible and universal one. Peres gate is mainly inter-

esting due to its combination of T-F gate and C-NOT gate

which makes it an attractive candidate to realize complex

logic such as addition, subtraction etc. Peres gate is also a

3-bit gate which maintains relationship between the input

and output as following,

Copyright © 2012 SciRes. JQIS

S. M. BAHAUDDIN, A. A. MD. IRFAN

Copyright © 2012 SciRes. JQIS

63

'

'

'

aa

bab

cc ab

= (4, 6)(6, 7)(5, 6)

= (4, 6)(5, 7)(5, 7)(6, 7)(5, 7)(5, 7)(5, 6)

= (4, 6)(5, 7)(5, 6)(5, 7)(5, 6)

= (4, 6)(5, 7)(6, 7)

As seen from the logic, the outputs of bit b and c suffer

from CNOT and T-F operations respectively while bit a

remains unchanged. The permutation corresponding to

this gate is,

Now going from right to left, all we have to do is to sum

up the corresponding C-NOT gates as per transpositions.

Then the decomposition will denote Peres gate through

the joining of 3-bit C-NOT gates as shown below:

Peres

000001010011 100101 110111

000 001010 011110111101100

The corresponding cycle has a length of 4 and can be

decomposed in 3 transpositions,

In the above example, let the C-NOT of cycle (6, 7)

and (4, 6) (5, 7) are expressed as C1 and C2 respectively.

Then clearly C1, C2 are one-to-one mapping of

0,1,2,, 7S

C

onto itself and the Peres gate is merely

a composite (12

) of C1 and C2 which can mathe-

matically written by (12

)(x) = C1(C2(x)),

C

CC

x

S

.

Since this is the closure property of permutation group, it

is obvious that

Peres 100,110,101,111

100,111100,101100,110

In decimal symbol, the cycle of Peres gate is (4, 6, 5, 7)

= (4, 7)(4, 5)(4, 6).

4. Construction of General Gates 12

CC

S8.

Similarly, for n = 4, we have four options to choose

any one as a target bit and for each target bit, there exists

23 – 1 = 7 unique combinations of control bits. Hence

there are 7 × 4 = 28 numbers of C-NOT gate for n = 4

(Table 1). Again each of the 28 gates can be realized by

a unique product of transpositions as shown in Figure 2.

We showed earlier, with example, that the product of

cycles of a permutation is sufficient to realize any re-

versible logic. In an extension, let us introduce an n bit

C-NOT gate. In this case, we can realize the C-NOT op-

eration on any one bit of n while the control bits are the

other (n – 1) bits or (n – 2) and so on. If n = 3, then we can

choose any one as a target bit out of three and for each

target bit, there exist 22 – 1 = 3 unique combinations of

control bits. This means there are 9 possible C-NOT gate

for n = 3 (Figure 1), where each of them can be realized

by a unique product of transpositions.

At this juncture, let us imagine that we intend to real-

ize the logic of a full adder, which is evidently irreversi-

ble. By adding an extra bit, we can make it reversible,

which means it would be a 4-bit reversible logic. But

now the question lies where we want to put that extra bit.

Since there were 3 bits in an irreversible full adder, we

have different positions of significance (from MSB to

LSB) to put that extra bit. In our paper, we have intro-

duced 3 logics of such reversible Full Adder where the

extra bit is set at three different positions of significance

starting from LSB. The permutation corresponding to the

gates are given below. Here, in all three cases, the output

of a full adder logic is realized only when the extra bit is

zero. In addition, the least significant two bits of the

outputs are the sum and carry-out.

Now we will utilize the properties of cycles, to prove

that any reversible logic can be simulated by a combina-

tion of elementary gates. In other words, we will prove that

the decomposition of reversible logic is possible by exploit-

ing the properties of cycles under permutation algebra.

As an example, we can realize Peres gate in terms of

the 3-bit C-NOT gates

Peres

= (4, 6, 5, 7)

= (6, 5, 7, 4)

FA-1

0000000100100011 01000101011001111000 10011010 10111100 11011110 1111

00000011 0110010000100001010101111010 1011110111001001 100011111110

FA-2

00000001001000110100010101100111 10001001 10101011 11001101 11101111

00000010000100110110010101110100 1110 1101 11111100100110111000 1010

FA-3

0000000100100011010001010110011110001001 101010111100110111101111

0000001001100001 01000101001101111010 1001110110111100 1000 1110 1111

Now the corresponding cycles in decimal are, FA-3

= (1, 2, 6, 3)(8, 10, 13)

FA-1

= (1, 3, 4, 2, 6, 5)(8, 10, 13)(9, 11, 12)(14, 15) As we stated before, our intention is to decompose

these reversible adder logics by 4-bit C-NOT gates only.

= (1, 2)(4, 6, 7)(8, 14)(9, 13, 11, 12)(10, 15)

FA-2

S. M. BAHAUDDIN, A. A. MD. IRFAN

64

(6,7)(2,3)(6,7)(5,7)(4,6)(5,7)

(4,5)(6,7)

(2,6)(3,7)

(3,7)

(1,3 )(6, 7)

(1,5 )(3, 7)

Figure 1. Possible permutations and corresponding trans-

positions of 3-bit C-NOT gates.

Table 1. Transpositions of 4-bit C-NOT gates correspond-

ing to Figure 2.

Row 1 (target bit at 1st LSB) Row 3 (target bit at 3rd LSB)

No. Transpositions No. Transpositions

1 (14,15) 1 (11,15)

2 (6,7)(14,15) 2 (3,7)(11,15)

3 (10,11)(14,15) 3 (9,13)(11,15)

4 (12,13)(14,15) 4 (10,14)(11,15)

5 (8,9)(10,11)(12,13)(14,15) 5 (8,12)(9,13)(10,14)(11,15)

6 (4,5)(6,7)(12,13)(14,15) 6 (2,6)(3,7)(10,14)(11,15)

7 (2,3)(6,7)(10,11)(14,15) 7 (1,5)(3,7)(9,13)(11,15)

Row 2 (target bit at 2nd LSB) Row 4 (target bit at MSB)

No. Transpositions No. Transpositions

1 (13,15) 1 (7,15)

2 (5,7)(13,15) 2 (3,11)(7,15)

3 (9,11)(13,15) 3 (5,13)(7,15)

4 (12,14)(13,15) 4 (6,14)(7,15)

5 (8,10)(9,11)(12,14)(13,15) 5 (4,12)(5,13)(6,14)(7,15)

6 (4,6)(5,7)(12,14)(13,15) 6 (2,10)(3,11)(6,14)(7,15)

7 (1,3)(5,7)(9,11)(13,15) 7 (1,9)(3,11)(5,13)(7,15)

Figure 2. Possible permutations of 4-bit C-NOT gates.

Figure 3. Realization of full adderFA -3

expressed by 26

4-bit C-NOT gates.

Here, we will utilize the expression of different C-NOT

gates in terms of corresponding transpositions and we

will operate permutation algebra on these transpositions

according to the lemmas to obtain the logic of reversible

adder. So, let us first examine the reversible logic of φFA-3

and then decompose it into elementary transpositions

which are available for C-NOT operation in following

manner,

FA-3

= (1, 2, 6, 3)(8, 10, 13)

= (3, 1, 2, 6)(8, 10, 13)

= (3, 6)(3, 2)(3, 1)(8, 13)(8, 10)

= (3, 7)(6, 7)(3, 7)(2, 3)(1, 3)(12, 13)(8, 12)(12, 13)

(8, 10)

Now, if we give a closure look at the figures of C-

NOT gates previously sketched before with their corre-

sponding product of transpositions, we can immediately

reveal that the first transposition (3, 7) cannot be realized

alone by any C-NOT gate rather it can be construted by a

combination of (3, 7) (11, 15) and (11, 15). According to

the previous statement, we can say that both of the

transpositons (6, 7) and (12, 13) also require a combina-

tion of two C-NOT gates while each of the four transpo-

sitions (1, 3), (2, 3), (8, 10) and (8, 12) requires a combi-

nation of four distinct C-NOT gates. In summary, we have

to add a total of 26 C-NOT gates to realize the logic of

FA-3

(Figure 3).

It is now evident that we could also simulate the logic

of FA-2

(shown in Figure 4) and FA-1

(not shown) as

well. Our calculation shows that the FA-2

can be con-

structed by 40 C-NOT gates while FA-1

will be com-

prised of 53 C-NOT gates.

5. Conclusion

The decomposition of complex gate by means of the

product of transposition and corresponding n-bit C-NOT

gate gives us further opportunity to express the logic in

more elementary gates. This is because any n-bit C-NOT

gate and corresponding unitary operations can be ex-

pressed as a composition of 2 bit C-NOT and the T-F

Figure 4. Realization of full adder expressed by 40 4-bit C-NOT gates.

FA -2

Copyright © 2012 SciRes. JQIS

S. M. BAHAUDDIN, A. A. MD. IRFAN 65

[5]. Moreover, th

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