ternary deutsch’s, deutsch-jozsa and affine functions problems all those problems are not...

Ternary Deutsch’s, Ternary Deutsch’s, Deutsch-Jozsa and Deutsch-Jozsa and Affine functions Affine functions ProblemsProblems

All those problems are not published yet

Ternary Deutsch’s ProblemTernary Deutsch’s ProblemDetermine whether ternary function f(x) of single variable is constant or balanced using as few queries to the oracle as possible.

Ternary Deutsch-Jozsa ProblemTernary Deutsch-Jozsa ProblemDetermine whether ternary function f(x1,..xn) of n variables is constant or balanced using as few queries to the oracle as possible.

Ternary Affine function separation ProblemTernary Affine function separation Problem

Determine for ternary affine function f(x1,..xn) 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 ProblemM-valued Affine function separation Problem

Determine for M-valued affine function f(x1,..xn) of n variables what is the affine function with accuracy to adding a ternary constant

New problems to solve

Ternary DeutschTernary Deutsch

Classically we need to query the oracle two times to solve ternary Deutsch’s Problem

f

ff(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

Balanced Functions of single variable2

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

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.

Butterfly for ternary Chrestenson

1 1 1 1 a a2

1 a2 a

1 1 1 1 a a2

1 a2 a

1 1 1 1 a a2

1 a2 a

1 1 1 1 a a2

1 a2 a

1 1 1 1 a a2

1 a2 a

1 1 1 1 a a2

1 a2 a

1 1 1 1 a a2

1 a2 a

1 1 1 1 a a2

1 a2 a

1 1 1 1 a a2

1 a2 a

1 1 1 1 a a2

1 a2 a

1 1 1 1 a a2

1 a2 a

1

1

1

1 1

a

a

a2

a2

a = e i 2/3Chrestenson generalizes Hadamard

Classical ternary Chrestenson

1 1 1 1 a a2

1 a2 a

a 1 a2 1 1 1

a2 1 a

a a2 1 a2 a 1 1 1 1

First new ternary Chrestenson

Second new ternary Chrestenson

Butterfly for ternary Chrestenson

1 1 1 1 a a2

1 a2 a

1 1 1 1 a a2

1 a2 a

1 1 1 1 a a2

1 a2 a

a a a a a2 1 a 1 a2

a2 a2 a2

a2 1 a a2 a 1

1 1 1 1 a a2

1 a2 a

1 1 1 1 a a2

1 a2 a

a a a a a2 1 a 1 a2

a2 a2 a2

a2 1 a a2 a 1

From Kronecker product we obtain this unitary matrix for a parallel connection of two Chrestensons:

Affine Ternary functions0a2ab2ba+ba+2b2a+b2a+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

Constant functions

1 1 1 1 1 1

1 1 1

a a a a a a a a a

a a a a2 a2 a2

1 1 1

a2 a2 a2 1 1 1

a a a

1 1 1 a a a

a2 a2 a2

a a2 1 a a2 1

a a2 1

1 a a2

a a2 1 a2 1 a

1 a a2 1 a a2

1 a a2

a2 a2 a2 a2 a2 a2

a2 a2 a2

Examples of maps of functions of two ternary variables.

1=0 a=1 a2=2

0 1 2 X Y X Y X Y

• 0• a• 2a• b• 2b• a+b• a+2b• 2a+b• 2a+2b

a 2a

0 1 2

+

+a

a2

+a

a2

1 1 1

1 a a2

1 a2 a

Butterfly for ternary Chrestenson

1 1 1 1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

Butterfly for ternary Chrestenson

+

+a

a2

+a

a2

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

Butterfly for ternary Chrestenson

1 1 1

1 1 1

1 1 1

3

1+a+a2=0

1+a+a2=0

3

0

0

3

0

0

9

3(1+a+a2)=0

3(1+a+a2)=0

0

0

0

0

0

0

0

0

0

0

0

0

Ternary constant 1

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

Butterfly for ternary Chrestenson

a a a

a a a

a a a

3a

a(1+a+a2)=0

a(1+a+a2)=0

3a

0

0

3a

0

0

9a

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Ternary constant a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

Butterfly for ternary Chrestenson

a2 a2 a2

3a2

a2(1+a+a2)=0

a2(1+a+a2)=0

3a2

0

0

3a2

0

0

9a2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Ternary constant a2

a2 a2 a2

a2 a2 a2

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

Butterfly for ternary Chrestenson

a a a

1 1 1

a2 a2 a2

3a

a(1+a+a2)=0

a(1+a+a2)=0

3

3a2

0

0

3(a+1+a2)=0

3a+3a+3a2a2=9a

0

0

0

0

0

0

0

0

0

0

0

0

Ternary balanced (01) = (1 a)

+

+aa2

+aa2

(1+a+a2)=0

(1+a2+a)=0

3a+3a2+3a2a=0

00

01

02

12

22

10

11

20

21

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

Butterfly for ternary Chrestenson

1 a a2

0

(1+a2+a2a2)=0

1+aa2+aa2=3

0

0

0

3

0

0

0

0

0

3

3

3

0

0

0

9

0

0

Ternary variable b

+

+aa2

+aa2

0

3

0

1 a a2

1 a a2

00

01

02

12

22

10

11

20

21

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

Butterfly for ternary Chrestenson

1 a2 a

(1+a+a2)=0

1+a2a2+aa=0

0

0

3

0

0

0

3

3

3

0

0

0

9

0

0

0

0

0

Ternary single variable function (01) (b)

+

+aa2

+aa2

3

0

0

1 a2 a

1 a2 a

(1+aa2+aa2)=3

0

0

0

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 1 1

1 a a2

1 a2 a

1 a a2

(1+aa+a2a2)=0

1+aa2+a2a=3

0

0

0

0

0

3

3a

3a2

0

0

0

+

+aa2

+aa2

0

a a2 1

a2 1 a

0

0

0

0 1 2

1 2 0

2 0 1

0 1 2

0 1 2

1 a

00

01

02

12

22

10

11

20

21

(1+a+a2)=0

(1+a+a2)=0

(1+a+a2)=0

a+aa2+a21=0

a+a2a2+a1=3a

a2+a+a2a=0

a2+a2+a2=3a2

3(1+a+a2)=0

3(1+a2+a2a2)=0

3(1+aa2+a2a)=9

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.

New spectral quantum ideasNew 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

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 binary3. 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 Deutsch6. 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.