Ternary Deutsch

1
All those problems are not published yet
Ternary Deutschs, Deutsch-Jozsa and Affine
functions Problems
2
New problems to solve
Ternary 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 XY, XY1 and XY2 are
in the same category. Addition is modulo 3.
M-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
3
Ternary Deutsch
Three constant functions F(x)0, F(x)1, F(x)2
Six balanced functions F(x)x, F(x)x1,
F(x)x2, F(x) (01)(x) F(x)(02)(x). F(x)(12)(
x)
y mod3 F(x)
0,1,2
Classically we need to query the oracle two
times to solve ternary Deutschs Problem
equivalence
f
0
?
f(0) ? f(1)
f
1
1 for balanced, 0 for constants
4
Balanced Functions of single variable
0
1
2
2
0
1
0
2
1
1
0
2
1
2
0
2
1
0
Constant Functions of single variable
0
0
0
2
2
2
1
1
1
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 a2 1 a2 a
1 1 1 1 a a2 1 a2 a
?
Chrestenson generalizes Hadamard
a e i 2?/3
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 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
a2
1
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
7
1 1 1 1 a a2 1 a2 a
Classical ternary Chrestenson
a 1 a2 1 1 1 a2 1 a
First new ternary Chrestenson
a a2 1 a2 a 1 1 1 1
Second new ternary Chrestenson
8
Butterfly for ternary Chrestenson
From Kronecker product we obtain this unitary
matrix for a parallel connection of two
Chrestensons
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
9
Affine Ternary functions

• 0
• a
• 2a
• b
• 2b
• ab
• a2b
• 2ab
• 2a2b

10 1a 12a 1b 12b 1ab 1a2b 12ab 12a2b
20 2a 22a 2b 22b 2ab 2a2b 22ab 22a2b
Binary function of 2 variables has 2 2 4
spectral coefficients
Binary function has 3 2 9 coefficients
10
0 1 2
X Y
X Y
X Y
a a a a a a a a a
a2 a2 a2 a2 a2 a2 a2 a2 a2
Constant functions
1 1 1 1 1 1 1 1 1
10 a1 a22
0
1
2
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
2a
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

• 0
• a
• 2a
• b
• 2b
• ab
• a2b
• 2ab
• 2a2b

Examples of maps of functions of two ternary
variables.
11
Butterfly for ternary Chrestenson


a
a2

a2
1 1 1 1 a a2 1 a2 a
a
12
Butterfly for ternary Chrestenson
13
Butterfly for ternary Chrestenson
Ternary constant 1
3 1aa20
9
1 1 1
3(1aa2)0
1aa20
3(1aa2)0
3 0 0
0 0 0
0 0 0
1 1 1
3 0 0
0 0 0
1 1 1
0 0 0
14
Butterfly for ternary Chrestenson
Ternary constant a
9a
3a a(1aa2)0
a a a
0
a(1aa2)0
0
3a 0 0
0 0 0
0 0 0
a a a
3a 0 0
0 0 0
a a a
0 0 0
15
Butterfly for ternary Chrestenson
Ternary constant a2
9a2
3a2 a2(1aa2)0
a2 a2 a2
0
a2(1aa2)0
0
3a2 0 0
0 0 0
0 0 0
a2 a2 a2
3a2 0 0
a2 a2 a2
0 0 0
0 0 0
16
Butterfly for ternary Chrestenson
Ternary balanced (01) (1 a)
3(a1a2)0
3a a(1aa2)0
a a a
00
3a3a3a2a29a
01
a(1aa2)0
3a3a23a2a0
02
3
0 0 0
0 0 0
1 1 1
10
(1aa2)0
11
(1a2a)0
12
3a2 0 0
a2 a2 a2
0 0 0
0 0 0
20
21
22
17
Butterfly for ternary Chrestenson
Ternary variable b
0
0 (1a2a2a2)0
1 a a2
00
0
01
0
02
1aa2aa23
0
0 0 0
0 0 0
1 a a2
10
0
11
3
12
0 0 3
1 a a2
3 3 3
9 0 0
20
21
22
18
Butterfly for ternary Chrestenson
Ternary single variable function (01) (b)
0
0
(1aa2)0
1 a2 a
(1aa2aa2)3
0
0
1a2a2aa0
0
0
0
3 3 3
9 0 0
1 a2 a
3
0
0 3 0
0 0 0
0 0 0
1 a2 a
19
0 1 2
(1aa2)0
0
0
0 1 2
1 2 0
2 0 1
0 1 2
1 a a2
1 a
00
0
(1aaa2a2)0
0
01
1aa2a2a3
0
0
02
(1aa2)0
0 0 0
0 0 0
a a2 1
10
aaa2a210
11
aa2a2a13a
12
(1aa2)0
3 3a 3a2
3(1aa2)0
a2 1 a
20
a2aa2a0
3(1a2a2a2)0
21
a2a2a23a2
3(1aa2a2a)9
22
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.