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

1
GF(4) Based Synthesis of Quaternary
Reversible/Quantum Logic Circuits

• Mozammel H. A. Khan
• East West University, Dhaka, Bangladesh
• mhakhan_at_ewubd.edu
• Marek A. Perkowski
• Portland State University, Portland, OR, USA
• mperkows_at_ee.pdx.edu

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

• 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 ? x1) ?2 (2 ? 2) ? x (1 ?
2) 3 ? x 2
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Quaternary Galois field sum of products expression

• Table 2. Basic quaternary reversible-literals

Input x2 x21 x22 x23
0 1 2 3 0 1 3 2 1 0 2 3 2 3 1 0 3 2 0 1
Input 2x2 2x21 2x22 2x23
0 1 2 3 0 2 1 3 1 3 0 2 2 0 3 1 3 1 2 0
Input 3x2 3x21 3x22 3x23
0 1 2 3 0 3 2 1 1 2 3 0 2 1 0 3 3 0 1 2
Input x x1 x2 x3
0 1 2 3 0 1 2 3 1 0 3 2 2 3 0 1 3 2 1 0
Input 2x 2x1 2x2 2x3
0 1 2 3 0 2 3 1 1 3 2 0 2 0 1 3 3 1 0 2
Input 3x 3x1 3x2 3x3
0 1 2 3 0 3 1 2 1 2 0 3 2 1 3 0 3 0 2 1
Example of one-qutrit gate
3x21
Example of one-qutrit gate
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Quaternary Galois field sum of products
expression (contd)

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

literal x x1 x2 x3
2(literal) 2x 2x2 2x3 2x1
literal 2x 2x1 2x2 2x3
2(literal) 3x 3x2 3x3 3x1
literal 3x 3x1 3x2 3x3
2(literal) x x2 x3 x1
literal x2 x21 x22 x23
2(literal) 2x2 2x22 2x23 2x21
literal 2x2 2x21 2x22 2x23
2(literal) 3x2 3x22 3x23 3x21
literal 3x2 3x21 3x22 3x23
2(literal) x2 x22 x23 x21
Example (2 ? x1) ?2 (2 ? 2) ? x (1 ?
2) 3 ? x 2
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Quaternary Galois field sum of products
expression (contd)

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

literal x x1 x2 x3
3(literal) 3x 3x3 3x1 3x2
literal 2x 2x1 2x2 2x3
3(literal) x x3 x1 x2
literal 3x 3x1 3x2 3x3
3(literal) 2x 2x3 2x1 2x2
literal x2 x21 x22 x23
3(literal) 3x2 3x23 3x21 3x22
literal 2x2 2x21 2x22 2x23
3(literal) x2 x23 x21 x22
literal 3x2 3x21 3x22 3x23
3(literal) 2x2 2x23 2x21 2x22
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Quaternary Galois field sum of products
expression (contd)

• Product of two or more basic quaternary
  reversible-literals is called a QGFP.
• (2x2)(3x22)(2x2)
• Sum of two or more QGFP is called a QGFSOP
• (2x2)(3x22) (3x1)(2x) x

These may be functions of one or more variables
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Quaternary Galois field expansions

• Cofactors

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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

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Quaternary Galois field expansions (contd)

• QGFE 5

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Quaternary Galois field expansions (contd)

• QGFE 6

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Quaternary Galois field expansions (contd)

• QGFE 7

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Quaternary Galois field expansions (contd)

• QGFE 8
• QGFE 9

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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) 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




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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

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Quaternary Galois field decision diagrams (contd)
x




y

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

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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.

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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

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Quaternary 2-qudit Muthukrishnan-Stroud gate
family

• Figure 4. Quaternary Muthukrishnan-Stroud gate
  family

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Quaternary Feynman gate

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

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Quaternary Toffoli gate

• Figure 7. Quaternary Toffoli gate
• Figure 8. Realization of quaternary Toffoli gate

One ancilla bit

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Quaternary Toffoli gate (contd)

• Figure 9. Four-input quaternary Toffoli gate

Two ancilla bits

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Synthesis of QGFSOP expressions

• Figure 10. Realization of QGFSOP expression

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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

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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.

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Oracle for Quantum Map of Europe Coloring
Germany
France
Switzerland
Spain
quaternary
Spain
France
Germany
Switzerland
?
?
?
?
Good coloring
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Oracle for Quantum Map of Europe Coloring
0 1 2 3
0 1 2
3
0 1 2 3
1 0 3 2
2 3 0 1
3 2 1 0
0 1 2 3
011 110 213 312
000 101 202 303
033 132 231 330
022 123 220 321
0 1 2 3
A
1
1 when A B
0 1 2 3
0 1 2 3
B
1
2
3
3
2
Quaternary Feynman
Quaternary input/binary output comparator of
equality
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Oracle for Quantum Map of Europe Coloring
Comparator for each frontier
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
Binary signal 1 when all frontiers well colored
0
Quaternary controlled binary target gate
Binary Toffoli
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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

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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 gt 3)
  Toffoli gates using 3-qudit Toffoli gates

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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

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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

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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.

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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).

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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.

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Research Questions

1. 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?

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• Thanks
• Questions?