1. Ternary Deutsch’s, Deutsch-Jozsa and Affine functions Problems All those problems are not published yet

2. Ternary Deutsch’s Problem Determine whether ternary function f(x) of single variable is constant or balanced using as few queries to the oracle as possible. Ternary Deutsch-Jozsa Problem Determine whether ternary function f(x 1,..x n ) of n variables is constant or balanced using as few queries to the oracle as possible. Ternary Affine function separation Problem Determine for ternary affine function f(x 1,..x n ) of n variables what is the affine function with accuracy to adding a ternary constant For instance, functions X+Y, X+Y+1 and X+Y+2 are in the same category. Addition is modulo 3. M-valued Affine function separation Problem Determine for M-valued affine function f(x 1,..x n ) of n variables what is the affine function with accuracy to adding a ternary constant New problems to solve

3. Ternary Deutsch Classically we need to query the oracle two times to solve ternary Deutsch’s Problem  f f f(0)  f(1) 1 for balanced, 0 for constants 0 1 Three constant functions: F(x)=0, F(x)=1, F(x)=2 Six balanced functions: F(x)=x, F(x)=x+1, F(x)=x+2, F(x) = (01)(x) F(x)=(02)(x). F(x)=(12)(x) y +mod3 F(x) {0,1,2} equivalence

4. Balanced Functions of single variable 2 0 1 0 1 2 0 2 1 1 0 2 1 2 0 2 1 0 Constant Functions of single variable 0 0 0 1 1 1 2 2 2

5. Generalization So far, nothing has been published on generalizations of these ideas to ternary and in general multiple-valued quantum computing. We need the following: –A gate that would generalize Hadamard –Gates to build arbitrary ternary oracle –Gates for transform after oracle.

6. Butterfly for ternary Chrestenson 1 1 1 1 a a 2 1 a 2 a  1 1 1 1 1 a a a2a2 a2a2 a = e i 2  /3 Chrestenson generalizes Hadamard

7. Classical ternary Chrestenson 1 1 1 1 a a 2 1 a 2 a a 1 a 2 1 1 1 a 2 1 a a a 2 1 a 2 a 1 1 1 1 First new ternary Chrestenson Second new ternary Chrestenson

8. Butterfly for ternary Chrestenson 1 1 1 1 a a 2 1 a 2 a a a a a a 2 1 a 1 a 2 a 2 a 2 a 2 a 2 1 a a 2 a 1 1 1 1 1 a a 2 1 a 2 a a a a a a 2 1 a 1 a 2 a 2 a 2 a 2 a 2 1 a a 2 a 1 From Kronecker product we obtain this unitary matrix for a parallel connection of two Chrestensons:

9. Affine Ternary functions 0 a 2a b 2b a+b a+2b 2a+b 2a+2b 1+0 1+a 1+2a 1+b 1+2b 1+a+b 1+a+2b 1+2a+b 1+2a+2b Binary function of 2 variables has 2 2 = 4 spectral coefficients Binary function has 3 2 = 9 coefficients 2+0 2+a 2+2a 2+b 2+2b 2+a+b 2+a+2b 2+2a+b 2+2a+2b

10. Constant functions 1 1 1 1 1 1 1 1 1 a a a a a a a a a a a a a 2 a 2 a 2 1 1 1 a 2 a 2 a 2 1 1 1 a a a 1 1 1 a a a a 2 a 2 a 2 a a 2 1 a a 2 1 a a 2 1 1 a a 2 a a 2 1 a 2 1 a 1 a a 2 1 a a 2 1 a a 2 a 2 a 2 a 2 a 2 a 2 a 2 a 2 a 2 a 2 Examples of maps of functions of two ternary variables. 1=0 a=1 a 2 =2 0 1 2 X Y 0 a 2a b 2b a+b a+2b 2a+b 2a+2b a 2a 0 12

11. + + a a2a2 + a a2a2 1 1 1 1 a a 2 1 a 2 a Butterfly for ternary Chrestenson 1 1 1 1 a a 2 1 a 2 a

12. 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a Butterfly for ternary Chrestenson + + a a2a2 + a a2a2

13. 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a Butterfly for ternary Chrestenson 111111 111111 111111 3 1+a+a 2 =0 3 0 3 0 9 3(1+a+a 2 )=0 0 Ternary constant 1

14. 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a Butterfly for ternary Chrestenson aaaaaa aaaaaa aaaaaa 3a a(1+a+a 2 )=0 3a 0 3a 0 9a 0 0 0 Ternary constant a

15. 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a Butterfly for ternary Chrestenson a2a2a2a2a2a2 3a 2 a 2 (1+a+a 2 )=0 3a 2 0 3a 2 0 9a 2 0 0 0 Ternary constant a 2 a2a2a2a2a2a2 a2a2a2a2a2a2

16. 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a Butterfly for ternary Chrestenson aaaaaa 111111 a2a2a2a2a2a2 3a a(1+a+a 2 )=0 3 3a 2 0 3(a+1+a 2 )=0 3a+3a+3a 2 a 2 =9a 0 Ternary balanced (01) = (1 a) + + a a2a2 + a a2a2 (1+a+a 2 )=0 (1+a 2 +a)=0 3a+3a 2 +3a 2 a= 0 00 01 02 12 22 10 11 20 21

17. 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a Butterfly for ternary Chrestenson 1aa21aa2 0 (1+a 2 +a 2 a 2 )=0 1+aa 2 +aa 2 =3 0 0 3 0 0 0 3 0 9 0 Ternary variable b + + a a2a2 + a a2a2 0 3 0 1aa21aa2 1aa21aa2 00 01 02 12 22 10 11 20 21

18. 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a Butterfly for ternary Chrestenson 1a2a1a2a (1+a+a 2 )=0 1+a 2 a 2 +aa=0 0 0 3 0 0 0 3 0 9 0 Ternary single variable function (01) (b) + + a a2a2 + a a2a2 3 0 0 1a2a1a2a 1a2a1a2a (1+aa 2 +aa 2 )=3 0 0 0

19. 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1 1 1 1 a a 2 1 a 2 a 1aa21aa2 (1+aa+a 2 a 2 )=0 1+aa 2 +a 2 a=3 0 0 0 3 3a 3a 2 000000 + + a a2a2 + a a2a2 0 aa21aa21 a21aa21a 0 0 0 012 120 201 0 1 2 0 1 20 1 2 1a1a 00 01 02 12 22 10 11 20 21 (1+a+a 2 )=0 a+aa 2 +a 2 1=0 a+a 2 a 2 +a1=3a a 2 +a+a 2 a=0 a 2 +a 2 +a 2 =3a 2 3(1+a+a 2 )=0 3(1+a 2 +a 2 a 2 )=0 3(1+aa 2 +a 2 a)=9

20. Short Review Next time we will show that the best FPRM can be found using the general approach of quantum computational intelligence – Grover algorithm. The set of all FPRM transforms will be calculated in a classical reversible circuit. The only creative part of this approach will be to build the oracle and how to combine it with Grover search. This is a representative of many unpublished problems that I solved while in Korea: –A) graph coloring –B) Petrick function –C) Satisfiabilty (many variants) –D) Exact ESOP minimization (Using Helliwell Function) –H) Hamiltonian and Eulerian paths in a graph –I) Maximum clique in a graph Any NP hard problem can be solved like this if you know how to build the oracle – which is an exercise in reversible logic synthesis.

21. New spectral quantum ideas Now we will discuss new methods based on combining quantum ideas and classical spectral theory: –1. Direct measuring of some spectral coefficients –A) deterministic solutions –B) probabilistic solutions –C) quantum games –2. Calculating various classical parameters of Boolean and Multiple- Valued functions using quantum counting. –We count certain minterms in certain cofactors. –3. Using exact correlation transform for certain coefficients and using certain tree strategy to gain information. –This has several applications: Boolean decomposition Ashenhurst-Curtis Boolean decomposition Bidecomposition Finding symmetry of boolean functions Finding generalized symmetry Finding Primes and coverings EXOR logic

22. Tasks for ECE students (math volunteers are welcome) 1.Reformulate classical binary Deutsch algorithm for ternary logic using Chrestenson gates 2.Use all methods that I have shown for binary 3.Try to modify to other Chrestenson gates. The so-called new Chrestenson gates above. 4.Generalize to functions that are ternary affine for two variables. 5.Generalize to n-variable Deutsch 6.Generalize to n-variable affine function separation. 7.You must analyze what is the order of spectral coefficients in outputs in each case to be able to derive formulas for n variables. This may be not trivial. All these problems can be done by generalization of binary but are not published and not completely trivial. Thy will be good exercises for you in ternary logic, ternary transforms, ternary functions and the very idea of quantum separation of functions.