Synthesis of GF(3) Based Reversible/QuantumLogic Circuits Without Garbage Output

Md. Mahmud Muntakim Khan, Ayan Kumar Biswas, Shuvro Chowdhury, Masud Hasan and Asif Islam Khanδ

Abstract—We present a method of synthesizing Ternary GaloisField (GF(3)) based reversible/quantum logic circuits without anyancillary trits/qutrits and hence without any garbage outputs. Werealize multi input ternary Toffoli gate and square functions ofGF(3) variables using linear ion trap realizable Muthukrishnan-Stroud (M-S) gates and shift gates in the absence of ancillaryqutrits. Then based on the Galois Field Sum of Products (GFSOP)expression of a multi-variable GF(3) function, we synthesize thecorresponding circuit.

Index Terms—Reversible/quantum computing, Ternary Galoisfield (GF3), trit, ternary Toffoli gate, Galois field sum of product(GFSOP).

I. INTRODUCTION

Multiple valued quantum logic (MVQL) provides a route toscale up quantum computer by simplifying the logical network[1]. Muthukrishnan et al. [1] showed that an arbitrary multiplevalued logic operations on any number of multiple valuedqudits (quantum digits) can be decomposed into elementarylogic gates that operate on only two qudits at a time. We referto these elementary two qudit gates as Muthukrishnan-Stroudor M-S gates. Ternary quantum systems offers several benefitover its binary and other multiple valued counterparts, suchas optimization of Hilbert space dimensionality for quantumcomputers [2], better security for some quantum cryptographicprotocols such as quantum bit commitment and coin-flippingprotocols [3], [4]. Proposals to built qutrit quantum computersappeared in literature [5], [6].

To synthesize quantum logic circuits, Al-Rabadi et al.[7], [8] proposed Galois field based approach. The particularadvantage of Galois field for MVQL is that arbitrary multiplevalued Galois field operations can be used in the Toffoli gatesof the ESOP-based (Exclusive Sums-of-Products) realizationof binary reversible circuits, where Galois addition and mul-tiplication replace the XOR and And gates respectively. Khanet al. [9], [10] presented a method of synthesizing ternaryGalois field sum of products (GFSOPs) using cascades ofmultiple input ternary Toffoli and swap gates. Realization of 3input ternary Toffoli gate on top of M-S gate without ancillaryqutrit was presented by Khan et al. [11]. Realization of k-input(k > 3) binary Toffoli gate was presented in [12]. A k-input

M. M. M. Khan, A. K. Biswas, S. Chowdhury are with the Department ofElectrical and Electronic Engineering, Bangladesh University of Engineeringand Technology, Dhaka 1000, Bangladesh.

M. Hasan is with the Department of Computer Science and Engineer-ing, Bangladesh University of Engineering and Technology, Dhaka 1000,Bangladesh.(e-mail: masudhasan@cse.buet.ac.bd.)

A. I. Khan is with the Department of Electrical Engineering and ComputerSciences, University of California, Berkeley, CA 94720, USA. (δe-mail:asif@eecs.berkeley.edu.)

TABLE IGF3 ADDITION AND MULTIPLICATION

+ 0 1 2 · 0 1 20 0 1 2 0 0 0 01 1 2 0 1 0 1 22 2 0 1 2 0 2 1

TABLE IIREVERSIBLE OPERATIONS IN GF(3). THE CORRESPONDING GATE

SYMBOLS ARE ALSO SHOWN.

y= y= y= y= y= y=x x x+1 x+2 2x 2x+1 2x+20 0 1 2 0 1 21 1 2 0 2 0 12 2 0 1 1 2 0

Gate Symbol = +1 +2 12 01 02

(k > 3) ternary Toffoli gate can be realized in a similar wayusing cascades of 3 input ternary Toffoli gate and (k − 2)ancillary bits. To design efficient ternary quantum circuits, it isnecessary to optimize both the number of gates and the numberof ancillary bits. To date, no work has reported a universalmethod to realize quantum ternary GFSOP expressions usingternary M-S gates without ancillary qutrits. In this paper, wepresent a method of synthesizing a quantum circuit realzinga multi-variable GF(3) function rom its truth table on top ofM-S gates in the absence of any ancillary qutrit.

II. TERNARY GALOIS FIELD ARITHMETICS

Galois field is an algebraic structure or field that containsonly finitely many elements. Two basic operations in Galoisfield are addition (denoted by +) and multiplication (denotedby · or absence of any operator). Ternary Galois Field (GF(3))consists of the set of elements T = 0, 1, 2 and addition andmultiplication in GF3 are defined in table I.

III. TERNARY GALOIS FIELD SUM OF PRODUCTSEXPRESSIONS

There are six reversible operations in GF(3) that operates ona single variable. They are tabulated in table II. Correspondingto these six reversible operations, there are six reversibleternary gates that operate on single qutrit. We call these gatesshift gates. The generic symbol of the shift gates are shownin figure 1.

39th International Symposium on Multiple-Valued Logic

0195-623X/09 $25.00 © 2009 IEEE

DOI 10.1109/ISMVL.2009.73

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Fig. 1. Ternary shift gates. P and Q are the input and output qutritsrespectively. z ∈ {=, +1, +2, +3, 12, 01, 02}.

TABLE IIIEXAMPLES OF SINGLE VARIABLE COMPOSITE FUNCTIONS.

x x2 = x.x x3 = x2.x x(x + 1) 2x2 + 10 0 0 0 21 1 1 2 12 1 2 0 1

With these reversible function, functions of a variable canbe formed which might not be reversible. Few examples ofsuch functions are shown in Table III.

We define a special irreversible single variable function, 1-Reduced Post Literals (RPL) as:

ix = 2(x + 2i)2 + 1 (1)

The truth table of the RPLs are shown in table III.For a ternary function f(x1, x2, ..., xi, ...xn), we define the

co-factors with respect to the variable xi as:

f0 = f(x1, x2, ..., xi = 0, ..., xn) (2)f1 = f(x1, x2, ..., xi = 1, ..., xn) (3)f2 = f(x1, x2, ..., xi = 2, ..., xn) (4)

Any ternary function f(x1, x2, , xi, , xn) can be expandedin terms of all the variables using RPLs and co-factors usingthe following relation.

f(x1, x2, ..., xi, ...xn) =0 xi · f0 +1 xi · f1 +2 xi · f2

=2∑

p=0

(pxi · fp)

=2∑

p1=0

2∑p2=0

. . .2∑

pn=0

n∏i=1

pixi · f(p1, p2, . . . pn)

(5)

where f(p1, p2, . . . pn) refers to the value off(x1, x2, . . . , xn) when x1 = p1, x2 = p2, . . . xn = pn. Wecall this expression the Galois field sum of products (GFSOP)form.

Table III shows the truth table of ternary functionf(A,B,C). The GFSOP expression of f(A,B,C) is shown

TABLE IVTERNARY 1-REDUCED POST LITERALS

0x = 1x = 2(x + 2)2 + 1 2x = 2(x + 1)2 + 1x 2x2 + 1 = 2x2 + 2x = 2x2 + x0 1 0 01 0 1 02 0 0 1

TABLE VTRUTH TABLE FOR f(A, B, C).

AB→ 00 01 02 10 11 12 20 21 22C↓

1 1 1 2 0 1 2 0 11 0 0 0 1 0 0 1 01 2 2 1 2 2 1 2 2

in Eq. 6. In the following sections, we will realize a M-S gaterealizable logic circuit without any ancillary qutrit.

f(A,B,C) =2∑

p=0

2∑q=0

2∑r=0

pA · qB · rC · f(p, q, r)

= A2BC + A2B + A2C + A2 + 2B2C + 1(6)

IV. QUANTUM GATES

An important gate for designing ternary quantum circuitsis the ternary M-S gates. M-S gate is realizable using linearion trap scheme for quantum computing [13]. The diagram ofa ternary M-S gate is shown in Figure 2. Here, input A isthe controlling input and input B is the controlled input. Theoutput P is equal to the input A. If A = 2, the other outputQ is the Z-transform of the input B, otherwise Q = B.

Figure 3 shows a modified ternary M-S gate. Here, if A = x,where, x = 0, 1, 2, the other output Q is the Z-transform ofthe input B, otherwise Q = B. Figure 4 shows the realizationof this circuit using M-S gate and shift gates.

Another important gate is the ternary Feynman gate (Fig.5(a)). It has two inputs A and B and two outputs P and Q.The outputs are equal to A and the GF3 sum of A and Brespectively. Its realization by M-S gates in Figure 5(b).

V. THREE INPUT TERNARY TOFFOLI GATE

Diagram of a ternary Toffoli gate and its truth table areshown in Figure 6. The inputs A and B are the controlling

Fig. 2. Ternary Muthukrishnan-Stroud gate.

Fig. 3. Modified ternary Muthukrishnan-Stroud gate. Here, Z ∈ {+1, +2,01, 02, 12}

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Fig. 4. Realization of modified ternary Muthukrishnan-Stroud gate. Here, uand v follows the GF3 relations, u + x = 2 and 2 + v = x. For example, ifx = 1, then u = 1 and v = 2.

Fig. 5. Ternary Feynman gate. (a) Realization using M-S gates. (b)

input and C is the controlled input. The output P and Q areequal to A and B, respectively, and R is equal to A ·B + C.An efficient realization of this gate was presented in [11]. Therealization is shown in Fig. 7 using modified M-S gates andFeynman gates. Note that this circuit does not require anyancillary qutrit.

VI. k-INPUT TERNARY TOFFOLI GATE, k > 3Figure 8 shows the diagram of a ternary Toffoli gate with

(N +1) inputs. The first N inputs, A1 to AN pass unchangedfrom input to output, while A1 · A2 · A3 . . . · AN + AN+1

is obtained at the (N + 1)-th wire. This gate can easily berealized in terms of three input Toffoli gates. Realization of thebinary version of this gate has been presented in [12], whichis also applicable for the ternary case. However, realization inthis method will require (N −1) ancillary qutrits and produce(N − 1) garbage outputs.

Figure 9 shows the realization of a four input Toffoli gate.The correctness of this circuit can be verified from the truthtable at the intermediate points of the circuit. This design

Fig. 6. Ternary Toffoli gate. (a) Its truth table. (b)

Fig. 7. Realization of ternary Toffoli gate using modified M-S gates andFeynman gates. The truth tables at the intermediate points are also shown.The circuit can be verified immediately from the truth tables.

Fig. 8. Ternary Toffoli gate with (N + 1) inputs.

methodology can be extended to implement Toffoli gates withhigher number of inputs. Figure 10 shows the realization ofa five input ternary Toffoli gate using two four input Toffoligates and modified M-S gates. Figure 11 shows the realizationof an (N +1) input ternary Toffoli gate using N input Toffoligates. Since the realization of a three input ternary Toffoli gatedoes not require any ancillary qutrit, this circuit also does not

Fig. 9. Realization of four input ternary Toffoli gate.

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Fig. 10. Realization of five input ternary Toffoli gate.

require any ancillary qutrit and hence does not produce anygarbage output.

VII. SQUARE OPERATION IN QUANTUM TERNARY LOGIC

Maximum power of a variable in the GFSOP expansion of aternary function is 2, since for any ternary variable A, A3 = A(see Table III). Figures 12 and 13 shows the realization A2+Dand A2B+D in the wire of the variable, D. The correctness ofthe circuits can be verified by examining the truth tables at theintermediate points and the truth table of square operation (seeTable III). Figures 14 and15 show the realization of A2B2+Dand A2B2C2 + D respectively.

VIII. REALIZATION OF THE CIRCUIT FROM THE TRUTHTABLE OF A MULTI-VARIABLE GF(3) FUNCTION WITHOUT

ANY ANCILLARY QUTRIT

Now we can realize the ternary circuit corresponding to amulti-variable GF(3) function in truth table form. Given thetruth table of a function, we can expand the function in termsof all the variables in GFSOP form using the method described

Fig. 11. Realization of (N + 1) input ternary Toffoli gate.

Fig. 12. Realization of A2 + D.

Fig. 13. Realization of A2B + D.

Fig. 14. Realization of A2B2 + D.

Fig. 15. Realization of A2B2C2 + D.

in Section III. Using the method described in Sections IV andV, the circuit can be generated using modified M-S gates andmulti-input Toffoli gates. Since both of these types of gatescan be synthesized using only shift gates and M-S gates andwithout ancillary qutrits, the circuit realization of the circuitcontains only shift gates and M-S gates and no ancillary qutritsand hence does not produce any garbage output. Figure 16shows the realization of the function, f(A,B,C) presented inSection III.

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Fig. 16. Realization of f(A, B, C).

IX. CONCLUSION

We presented a framework for realizing the circuit of amulti-variable GF(3) function from its truth table using onlyM-S gates and shift gates without ancillary qutrits and hencewithout garbage outputs. Since M-S and shift gates are linearion trap realizable, the circuits will also be linear ion traprealizable. However, the number of gates to realize a N + 1input Toffoli gate increases exponentially with the numberof inputs. Given the relative cost of implementation of M-S gates, shift gates and ancillary qutrits in a circuit, it wouldbe worthwhile to see whether it is possible to optimize thecost function a circuit allowing a few ancillary qutrits.

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