GF(4) Based Synthesis of Quaternary Reversible/Quantum Logic Circuits

Mozammel H. A. KhanEast West University, Dhaka, Bangladesh

mhakhan@ewubd.edu

Marek A. PerkowskiPortland State University, Portland, OR, USA

mperkows@ee.pdx.edu

Introduction

• D-level (multiple-valued) quantum circuits have many advantages

• There is not much published about the practical circuit realization for such circuits

• MV logic functions having many inputs can be expressed as GFSOP

• GFSOP can be realized as cascade of Feynman and Toffoli gates

• No work has yet been done on expressing quaternary logic function as QGFSOP

• No work has yet been done on realizing QGFSOP as cascade of quaternary Feynman and Toffoli gates

Contribution of the Paper

• We have developed nine QGFEs (QGFE1 – QGFE9)

• We show way of constructing QGFDDsQGFDDs using QGFEs

• We show method of generating QGFSOPQGFSOP by flattening QGFDD

• We show technique of realizing QGFSOP as a cascade cascade of quaternary 1-qudit, Feynman, and Toffoli gates

Contribution of the Paper (contd)

• We show way of 2-bit encoded quaternary realization of binary functions

– We have developed circuit for binary-to-quaternary encoding– We have developed circuit for quaternary-to-binary decoding

Quaternary Galois field arithmetic Quaternary Galois field arithmetic Q = {0, 1, 2, 3}

Table 1. GF(4) operations

+ 0 1 2 3 0 1 2 3

0 0 1 2 3 0 0 0 0 0

1 1 0 3 2 1 0 1 2 3

2 2 3 0 1 2 0 2 3 1

3 3 2 1 0 3 0 3 1 2

Example: (2 x+1) 2= (2 2) x + (1 2) = 3 x + 2

Quaternary Galois field sum of products expressionQuaternary Galois field sum of products expressionTable 2. Basic quaternary reversible-literals

Input x x+1 x+2 x+3

0123

0123

1032

2301

3210

Input 2x 2x+1 2x+2 2x+3

0123

0231

1320

2013

3102

Input 3x 3x+1 3x+2 3x+3

0123

0312

1203

2130

3021

Input x2 x2+1 x2+2 x2+3

0123

0132

1023

2310

3201

Input 2x2 2x2+1 2x2+2 2x2+3

0123

0213

1302

2031

3120

Input 3x2 3x2+1 3x2+2 3x2+3

0123

0321

1230

2103

3012

Example of one-qutrit gate3x2+1Example of one-qutrit gate

Quaternary Galois field sum of products expression (contd)

Table 3. Products of basic quaternary reversible-literals and the constant 2Table 3. Products of basic quaternary reversible-literals and the constant 2

literal x x+1 x+2 x+3

2(literal) 2x 2x+2 2x+3 2x+1

literal 2x 2x+1 2x+2 2x+3

2(literal) 3x 3x+2 3x+3 3x+1

literal 3x 3x+1 3x+2 3x+3

2(literal) x x+2 x+3 x+1

literal x2 x2+1 x2+2 x2+3

2(literal) 2x2 2x2+2 2x2+3 2x2+1

literal 2x2 2x2+1 2x2+2 2x2+3

2(literal) 3x2 3x2+2 3x2+3 3x2+1

literal 3x2 3x2+1 3x2+2 3x2+3

2(literal) x2 x2+2 x2+3 x2+1

Example: (2 x+1) 2= (2 2) x + (1 2) = 3 x + 2

Quaternary Galois field sum of products expression (contd)

Table 4. Product of basic quaternary reversible-literal and the constant 3

literal x x+1 x+2 x+3

3(literal) 3x 3x+3 3x+1 3x+2

literal 2x 2x+1 2x+2 2x+3

3(literal) x x+3 x+1 x+2

literal 3x 3x+1 3x+2 3x+3

3(literal) 2x 2x+3 2x+1 2x+2

literal x2 x2+1 x2+2 x2+3

3(literal) 3x2 3x2+3 3x2+1 3x2+2

literal 2x2 2x2+1 2x2+2 2x2+3

3(literal) x2 x2+3 x2+1 x2+2

literal 3x2 3x2+1 3x2+2 3x2+3

3(literal) 2x2 2x2+3 2x2+1 2x2+2

Quaternary Galois field sum of products expression (contd)

• Product of two or more basic quaternary reversible-literals is called a QGFP.

(2x+2)(3x2+2)(2x2)

• Sum of two or more QGFP is called a QGFSOP

(2x+2)(3x2+2) + (3x+1)(2x) + x

These may be functions of one or more variablesThese may be functions of one or more variables

Quaternary Galois field expansionsQuaternary Galois field expansions

• Cofactors

),,0,,,( 210 nxxxff

),,1,,,( 211 nxxxff

),,2,,,( 212 nxxxff

),,3,,,( 213 nxxxff

Quaternary Galois field expansion Quaternary Galois field expansion (contd)(contd)

• Composite Cofactors

1001 fff

2002 fff

3003 fff

2112 fff

3113 fff

3223 fff

210012 ffff

310013 ffff 320023 ffff

321123 ffff

32100123 fffff

321)33)(22(1 32 ffff

321)32)(23(1 23 ffff

See notation for some See notation for some composite cofactorscomposite cofactors

Quaternary Galois field expansions (contd)Quaternary Galois field expansions (contd)

•QGFE 1:

•QGFE 2:

•QGFE 3:

•QGFE 4:

0302

010

)2)(1()3)(1(

)3)(2(

fxxxfxxx

fxxxff

1312

011

)2)(1()3)(1(

)3)(2)(1(

fxxxfxxx

fxxxff

2312

022

)2)(1()3)(2(

)3)(2)(1(

fxxxfxxx

fxxxff

2313

033

)3)(1()3)(2(

)3)(2)(1(

fxxxfxxx

fxxxff

First four Quaternary Expansions – they are generalizations of the First four Quaternary Expansions – they are generalizations of the familiar Shannon and Davio expansionsfamiliar Shannon and Davio expansions

Can be derived from inverted from quaternary Shannon Expansion. Can be derived from inverted from quaternary Shannon Expansion.

Quaternary Galois field expansions (contd)

• QGFE 5:

232

132

032

012

)2)(3(

)1)(1(

fxx

fxxxfxff

232

132

032

012

)13)(12(

)33)(22(32

fxx

fxxxfxf

232

132

032

012

)32)(23(

)22)(33(23

fxx

fxxxfxf

Quaternary Galois field expansions (contd)

• QGFE 6:

232

122

022

013

)3)(2(

)1)(1(

fxx

fxxxfxff

232

122

022

013

)23)(32(

)33)(22(32

fxx

fxxxfxf

232

122

022

013

)12)(13(

)22)(33(23

fxx

fxxxfxf

Quaternary Galois field expansions (contd)

• QGFE 7:

132

122

012

023

)3)(2(

)2)(3(

fxx

fxxxfxff

132

122

012

023

)23)(32(

)13)(12(32

fxx

fxxxfxf

132

122

012

023

)12)(13(

)32)(23(23

fxx

fxxxfxf

Quaternary Galois field expansions (contd)

• QGFE 8:

• QGFE 9:

032

022

012

123

)3)(2()2)(3(

)1)(1(

fxxfxx

fxxff

032

022

012

123

)23)(32()13)(12(

)33)(22(

fxxfxx

fxxf

032

022

012

123

)12)(13()32)(23(

)22)(33(

fxxfxx

fxxf

)32)(23(1)33)(22(12

01232

0 xffxxfxff

Quaternary Galois field decision diagramsQuaternary Galois field decision diagrams

Table 5.Table 5. Truth Table of an example quaternary function

F = x + y (GF4)

x y 00 01 02 03 10 11 12 13

f(x,y) 0 1 2 3 1 0 3 2

xy 20 21 22 23 30 31 32 33

f(x,y) 2 3 0 1 3 2 1 0

x y

Quaternary Galois field decision diagrams (contd)

Figure 1. QGFDD for the function of Table 5 using QGFE1 and QGFE2

f

x

y

0 1 2 3

001 02

03

01/112

131

13/12/011

13/12/01 13/12/01

1

1 QGFE

QGFE2 QGFE2 QGFE2 QGFE2

)2)(1(3)3)(1(2

)3)(2()2)(1(2

)3)(1(3)3)(2)(1(1

xxxxxx

xxxyyy

yyyyyyf

x

y

Two expansion variables, x and y

Quaternary Galois field decision diagrams (contd)

Figure 2. QGFDD for the function of Table 5 using QGFE9

f

x

y

0 1 2 3

0

0123)33)(22(1

)32)(23(1

)33)(22(1

/0123/0

)32)(23(1

0)32)(23(1

/)33)(22(1

/0123

QGFE9

QGFE9 QGFE9

yxf x

y

Quaternary Galois Field Decision Diagrams

• Similarly to KFDDs, the order of variables and the choice of expansion type for every level affects the number of nodes (size) of the decision diagram.

Quaternary 1-qudit Quaternary 1-qudit reversible/quantum gatesreversible/quantum gates

• Each of the 24 quaternary reversible-literals can be implemented as 1-qudit gates using quantum technology

Figure 3. Representation of quaternary reversible 1-qudit gates

x exp QGF y

Quaternary 2-qudit Muthukrishnan-Stroud gate family

Figure 4. Quaternary Muthukrishnan-Stroud gate family

z

A

B

AP

otherwise

3 if of transform-

B

ABzQ

2 Table of sformqudit tran-1any is z

Quaternary Feynman gate

Figure 5. Quaternary Feynman gate

Figure 6. Realization of quaternary Feynman gate

A

B

AP

(GF4) BAQ

1 2 3

A

B

1a2 3 12a 3a

AP

[GF(4)] BAQ

Quaternary Toffoli gate

Figure 7. Quaternary Toffoli gate

Figure 8. Realization of quaternary Toffoli gate

A

B

C

AP BQ

[GF(4)] CABR

A

B

0

C

2 3 1 BQ

2 3 1 2 2 3 1 2 2 3 1 2 AP

1 2 3 1 2 3 1 2 3 2 3 1

1 2 3

AB

[GF(4)] CABR

1a

1b

1t

2a

2b

2t

3a

3b

ABt 3

1Segment 2Segment 3Segment 4Segment

One One ancilla bitancilla bit

Quaternary Toffoli gate (contd)

Figure 9. Four-input quaternary Toffoli gate

A

B

0

C

D

AB

A

B

0

CDABC

A

B

CDABC

A

B

C

D

Symbol (a)nRealizatio (b) Two Two

ancilla ancilla bitsbits

Synthesis of QGFSOP expressionsSynthesis of QGFSOP expressions

Figure 10. Realization of QGFSOP expression

)2)(1(3)3)(1(2

)3)(2()2)(1(2

)3)(1(3)3)(2)(1(1

xxxxxx

xxxyyy

yyyyyyf

x

0

0

0

x

x

x

1x

x

x

2x

3x

x2

x3

y

0

0

0

y1y

2y

3y

y2

y3

0

0

0

0

1f

y

y

y

y

Binary-to-quaternary encoder and quaternary-to-binary decoder circuits

Figure 11. Binary-to-quaternary encoder circuit

Figure 12. Quaternary-to-binary decoder circuit

0A

1A

0 B

2

2

21

*0A*1A

1

1 1

1 1

2

B0

0

0

0A

1A

0B

1BB

output

input

inputs

outputs

garbage

Are d-level quantum circuit an advantage?

• Benchmarking is necessary.

• In some cases quaternary circuit is much simpler than binary.

• These applications include especially circuits with many arithmetic blocks and comparators.

• Control should be binary, data path should be multiple-valued.

• We need hybrid circuits that convert from binary to d-level and vice versa. This is relatively easy in quantum.

Oracle for Quantum Map of Europe ColoringOracle for Quantum Map of Europe Coloring

Spain

France

Germany

Switzerland

Spain

France

Germany

Switzerland

Good Good coloringcoloring

quaternary

Oracle for Quantum Map of Europe ColoringOracle for Quantum Map of Europe Coloring

0 1 2 3

1 0 3 2

2 3 0 1

3 2 1 0

0+1=1 1+1=0 2+1=3 3=1=2

0+0=0 1+0=1 2+0=2 3+0=3

0+3=3 1+3=2 2+3=1 3+3=0

0+2=2 1+2=3 2+2=0 3+2=1

0

1

2

3

0 1 2 3

0 1 2 3

0 1 2 3

0

1

2

3

+2

+3

0

1

2

3

+1

+3

+2

Quaternary Feynman Quaternary input/binary output comparator of equality

1 when A = B

A

B+1

Oracle for Quantum Map of Europe ColoringOracle for Quantum Map of Europe Coloring

0

1

2

3

+1

+3

+2

A

B

+11

1

0

1

2

3

+1

+3

+2

C

D

+11

1

1 1 -- when control 1

1 -- for controls 0,2 and 3

Binary qudit =1 for frontier AB when countries A and B have different colors

0

Binary signal 1 when all frontiers well colored

Comparator for Comparator for each frontiereach frontier

Quaternary controlled binary target gate

Binary Toffoli

Conclusion Conclusion • We have developed nine QGFEs

• These QGFEs can be used for constructing QGFDDs

• By flattening the QGFDD we can generate QGFSOP

• We have shown example of implementation of QGFSOP as cascade of quaternary 1-qudit gate, Feynman gate, and Toffoli gate

Conclusion (contd)

• For QGFSOP based quantum realization of functions with many input variables, we need to use quantum gates with many inputs. • Quantum gates with more than two inputs are very difficult to realize as a primitive gate

• We have shown the quantum realization of macro-level quaternary 2-qudit Feynman and 3-qudit Toffoli gates on the top of theoretically liquid ion-trap realizable 1-qudit gates and 2-qudit Muthukrishnan-Stroud primitive gates •We also show the realization of m-qudit (m > 3) Toffoli gates using 3-qudit Toffoli gates

Conclusion (contd)

• The quaternary base is very useful for 2-bit encoded realization of binary function • We show quantum circuit for binary-to-quaternary encoder and quaternary-to-binary decoder for this purpose

Conclusion (contd)

• The presented method is especially applicable to quantum oracles • The developed method performs a conversion of a non-reversible function to a reversible one as a byproduct of the synthesis process

• Our method can be used for large functions

• As it is using Galois logic, the circuits are highly testable

Conclusion (contd)• Our future research includes

(1) developing more QGFEs, if such expansions exist

(2) developing algorithms for(i) selecting expansion for each variable(ii) variable ordering(iii) constructing QGFDD (Kronecker and

pseudo-Kronecker types) for both single-output and multi-output functions

(3) Building gates on the level of Pauli Rotations, similarly as it was done in our published paper – Soonchil Lee et al.

Conclusion (contd)• Building gates on the level of Pauli

Rotations will require deciding in which points on the Bloch Sphere are the basic states

– Paper in RM 2007Paper in RM 2007

• This will affect rotations between these states, which means complexity of single-qudit and two-qudit gates.

• For each of created gates we will calculate quantum cost – numbers of Pauli rotations and Interaction gates (Controlled Z).

Conclusion (contd)

• Building blocks on the levels of first these gates and next Pauli Rotations to analyse what is the real gain of using mv concepts – example is comparator in Graph Coloring Oracle for Grover.

– Adders, multipliers, comparators of order, counting circuits, converters between various representations, etc.

• Practical realization of Pauli rotations and Interaction gates (Controlled Z) in NMR, ion trap, one-way and other technologies.

Research Questions1. What is the best location of basic states?

2. Can Galois Field mathematical theory be used for more efficient factorization or expansion, in general for synthesis?

3. Can we generalize Galois Field to circuits with 6 basic states (for which good placement on Bloch Sphere exists)?

4. Can we create some kind of algebra to allow expansion, factorization and other algebraic rules directly applied to easily realizable MS gates, rather than complex gates such as based on Galois Fields?

Thanks

Questions?