1. GF(4) Based Synthesis of Quaternary Reversible/Quantum Logic Circuits
Mozammel H. A. Khan
East West University, Dhaka, Bangladesh
Marek A. Perkowski
Portland State University, Portland, OR, USA

2. 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

3. Contribution of the Paper
We have developed nine QGFEs (QGFE1 – QGFE9)
We show way of constructing QGFDDs using QGFEs
We show method of generating QGFSOP by flattening QGFDD
We show technique of realizing QGFSOP as a cascade of quaternary 1-qudit, Feynman, and Toffoli gates

4. 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

5. Quaternary Galois field arithmetic
Table 1. GF(4) operations + 1 2 3 
Example: (2  x+1) 2= (2  2)  x + (1  2) = 3  x + 2

6. Quaternary Galois field sum of products expression
Table 2. Basic quaternary reversible-literals
Input x
x+1
x+2
x+3 1 2 3 2x
2x+1
2x+2
2x+3 3x
3x+1
3x+2
3x+3
Input x2
x2+1
x2+2
x2+3 1 2 3
2x2
2x2+1
2x2+2
2x2+3
3x2
3x2+1
3x2+2
3x2+3
Example of one-qutrit gate
3x2+1
Example of one-qutrit gate

7. Quaternary Galois field sum of products expression (contd)
Table 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 3x
3x+2
3x+3
3x+1 x2
x2+1
x2+2
x2+3
2x2
2x2+2
2x2+3
2x2+1
3x2
3x2+2
3x2+3
3x2+1
Example: (2  x+1) 2= (2  2)  x + (1  2) = 3  x + 2

8. 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 2x
2x+1
2x+2
2x+3 x2
x2+1
x2+2
x2+3
3x2
3x2+3
3x2+1
3x2+2
2x2
2x2+1
2x2+2
2x2+3

9. 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 variables

10. Quaternary Galois field expansions
Cofactors

11. Quaternary Galois field expansion (contd)
Composite Cofactors
See notation for some composite cofactors

12. Quaternary Galois field expansions (contd)
First four Quaternary Expansions – they are generalizations of the familiar Shannon and Davio expansions
Can be derived from inverted from quaternary Shannon Expansion.
QGFE 1:
QGFE 2:
QGFE 3:
QGFE 4:

13. Quaternary Galois field expansions (contd)
QGFE 5:

14. Quaternary Galois field expansions (contd)
QGFE 6:

15. Quaternary Galois field expansions (contd)
QGFE 7:

16. Quaternary Galois field expansions (contd)
QGFE 8:
QGFE 9:

17. Quaternary Galois field decision diagrams
Table 5. Truth Table of an example quaternary function
F = x + y (GF4) x y
x y 00 01 02 03 10 11 12 13
f(x,y) 1 2 3 xy 20 21 22 23 30 31 32 33

18. Quaternary Galois field decision diagrams (contd)x
Two expansion variables, x and y y
Figure 1. QGFDD for the function of Table 5 using QGFE1 and QGFE2

19. Quaternary Galois field decision diagrams (contd)x y
Figure 2. QGFDD for the function of Table 5 using QGFE9

20. 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.

21. Quaternary 1-qudit reversible/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

22. Quaternary 2-qudit Muthukrishnan-Stroud gate family
Figure 4. Quaternary Muthukrishnan-Stroud gate family

23. Quaternary Feynman gate
Figure 5. Quaternary Feynman gate
Figure 6. Realization of quaternary Feynman gate

24. Quaternary Toffoli gate
Figure 7. Quaternary Toffoli gate
Figure 8. Realization of quaternary Toffoli gate
One ancilla bit

25. Quaternary Toffoli gate (contd)
Figure 9. Four-input quaternary Toffoli gate
Two ancilla bits

26. Synthesis of QGFSOP expressions
Figure 10. Realization of QGFSOP expression

27. Binary-to-quaternary encoder and quaternary-to-binary decoder circuits
Figure 11. Binary-to-quaternary encoder circuit
Figure 12. Quaternary-to-binary decoder circuit
garbage
inputs
output
outputs
input

28. 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.

29. Oracle for Quantum Map of Europe Coloring
Germany
France
Switzerland
Spain
quaternary
Spain
France
Germany
Switzerland    
Good coloring

30. Oracle for Quantum Map of Europe Coloring 1 2 3 1 2 3
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 A +1
1 when A = B 1 2 3 1 2 3 +1 B +2 +3 +3 +2
Quaternary Feynman
Quaternary input/binary output comparator of equality

31. Oracle for Quantum Map of Europe Coloring
Comparator for each frontier 1 2 3 +1 +3 +2 A B
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 1 2 3 +1 +3 +2 C D
Binary signal 1 when all frontiers well colored
Quaternary controlled binary target gate
Binary Toffoli

32. 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

33. 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

34. 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

35. 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

36. Conclusion (contd) Our future research includes
developing more QGFEs, if such expansions exist
developing algorithms for
selecting expansion for each variable
variable ordering
constructing QGFDD (Kronecker and pseudo-Kronecker types) for both single-output and multi-output functions
Building gates on the level of Pauli Rotations, similarly as it was done in our published paper – Soonchil Lee et al.

37. 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 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).

38. 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.

39. Research Questions What is the best location of basic states?
Can Galois Field mathematical theory be used for more efficient factorization or expansion, in general for synthesis?
Can we generalize Galois Field to circuits with 6 basic states (for which good placement on Bloch Sphere exists)?
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?

40. Thanks
Questions?