Permutation Algebra for Constructing Reversible Circuits

doi:10.4236/jqis.2012.23011

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Journal of Quantum Information Science, 2012, 2, 61-65

http://dx.doi.org/10.4236/jqis.2012.23011 Published Online September 2012 (http://www.SciRP.org/journal/jqis)

Permutation Algebra for Constructing Reversible Circuits

Shah Mohammad Bahauddin, Abu Ashik Md. Irfan

Department of Applied Physics, Electronics & Communication Engineering, Faculty of Engineering & Technology,

University of Dhaka, Dhaka, Bangladesh

Email: bahauddin_omar@ieee.org

Received April 17, 2012; revised May 9, 2012; accepted June 4, 2012

ABSTRACT

In this paper, we show that the algebra of permutation group is one of the inherent structures of reversible logic for

quantum computation. In this venture, we discuss necessary properties of cycle and transposition to reveal the potential

of permutation algebra for reversible logic. Then we present an efficient method which naturally interconnects the struc-

ture of reversible logic with the expression of cycle and corresponding transpositions. Finally we discuss several exam-

ples which show that the algebra can be effectively used to construct complex gates as well.

Keywords: Control NOT; Taffoli-Fredkin Gate; Full Adder

1. Introduction

In recent years, reversible logic has become more and

more prominent technology having its applications in

quantum computing and other aspects of nanotechnology.

Moreover, reversibility can also be a solution to the en-

ergy dissipation issue in chip level integrated systems.

Logically reversible gates are those where the inputs can

be inferred from knowledge of the outputs, meaning the

logic function implemented by the gate has a unique in-

verse. In this paper, we present a method which inter-

connects the structure of reversible logic with permuta-

tion algebra. By applying the properties of cycle and

transposition, we also show that any complex arithmetic

circuit such as full adder can be easily achieved and effi-

ciently constructed.

2. Reversibility & Permutation Group

An n-bit reversible logic gate is a block which performs a

permutation on the 2n possible states of bits that pass

through it. Hence it is a one-to-one correspondence from

a 2n state onto itself. Thus specifying a reversible logic

gate is nothing but specifying the mapping that it per-

forms. In quantum computing, these computational states

are represented by quantum mechanical state. For exam-

ple, let us define an arbitrary finite set S having n-ele-

ments (which are in this case well defined quantum me-

chanical states). Then any reversible logic of such set can

be expressed by introducing





where



is a one-

to-one mapping of S onto itself. Now, one can represent

the set of bits



with the set of kets



,ab





0,1 i

(such that i = 0, 1, 2, 3) which are the four computational

states expressed by two input bits of the measurement

gate. This allows us to represent any logic (in this exam-

ple XOR or more specifically Control NOT) by means of

specific basis state as

acontrol btarget |iinput > acontrol b

target |ioutput >

0 0 0 0 0 0

0 1 1 0 1 1

1 0 2 1 1 3

1 1 3 1 0 2

A closer examination immediately reveals that this

representation can be shown as an element under the

permutation S4. Hence we can say C-NOT can be repre-

sented as a one-to-one mapping of S4, and in terms of

energy state, the above table can now be represented by

the permutation.

0123

0132









Since the S4 contains 4! permutations it means that

there exists 24 different logic gates for a 4-state system.

Generalizing the above statements we can say that, any

N-level quantum system can be permutated in N! com-

putationally distinct logic sets. In other words, these per-

mutations of degree N form a group of order N! under

composite operation of permutation. This permutation

group, SN is the collection of all bijective maps

:SS



 of the interval , with com-

position maps () as the group operation. We now intend

to recall some basic definitions for describing permuta-

tions [1].



1,2, 3,S



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

Cycle of a Permutation: For k > 1, a k-cycle is a

permutation



12 1

,, ,,

iii iii i

ii i















that acts on S in the following way



maps



121 12

:,,,

iii iiiii

jjjii i

 









Here k is called the length of the cycle. Cycles of

length 1 are redundant since they reduce to identity. The

order of the entries in a cycle matters, but cycle notation

is not unique as the following







12 1

23 1

34 12

,, ,,

,, ,,,

,, ,,

kkk

iii i

ii ii

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.



,.,, ,



kjkjkjk kj

ii iiIiiiiii





A product of disjoint cycle or transpositions is com-

mutative.







,, ,,

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,





,,,,

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

010



NOT

01 0,1











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



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,

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



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

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)),



.

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



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



(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



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

[5]. Moreover, th

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