Derivatives of Perkowski

1
Derivatives of Perkowskis Gate
Generalized multi-input multiplexer
f2
De Vos gates
?
A
P
Feynman gates
Many other gate families
f 1
Q
B
P
A
?
B
Q
f 2
R
C
01
CMOS gates
S
D
Kerntopf gates
0
1
Generalized Maitra gates
Maitra gates
Fredkin gates
2
Structure of Wave Cascade

• The definitions presented in this section are
  based on and , with some modifications.
• Definition. A complex Maitra term is recursively
  defined as follows
• (1) Constant 0 (1) Boolean function is a Maitra
  term.
• (2) A literal is a Maitra term.
• (3) If Mi is a Maitra term, a is a literal, and G
  is an arbitrary two-input Boolean function, then
  Mi1  G( a, Mi1 ) is a Maitra term.
• Additionally, it is required that each variable
  appears in each Maitra term only once and that
  the same variable ordering is used to represent
  all Maitra terms.
• Previous authors restricted the two-input
  functions used in the Maitra terms to only
  functions AND, OR, and EXOR.
• For the purposes of reversible logic synthesis,
  on the other hand, it is better to use the above
  more general definition.
• In a variation of our algorithm targeting
  low-power CMOS implementation, G cannot be an
  EXOR function and its complement, NEXOR.

3
Generalized Maitra Gate, Maitra Gate or CMOS Gate
Reversible Wave Cascade
4
Material for exam

1. Definition of reversible functions.
2. Balanced and Conservative functions. Interaction
   and other optically-realizable reversible gates
   that are not nn gates.
3. Toffoli, Feynman, Peres, Fredkin, Miller,
   Margolus and Kerntopf gates. Properties. Be able
   to synthesize one from another one. SWAP gates.
4. Convertion of irreversible circuit to reversible
   quantum array.
5. Concepts of garbage, input constants and ancilla
   bits.
6. Mirrors, local mirrors, spies, folding of
   constants.
7. Generalized controlled gates (so-called
   Perkowskis gates).
8. Transposition vector, vs set of cycles, vs Kmap
   vs truth table vs BDD notations. Go from one to
   another.
9. MMD algorithm and synthesis without ancilla bits.
10. EXOR Lattice for symmetric and non-symmetric
    functions. Understanding of symmetry and its
    uses.
11. Naïve methods of converting from irreversible
    netlist to reversible quantum cascade.
12. Billard Ball model.
13. KFDD and function graphs with linear and
    nonlinear preprocessors. Davio versus Shannon
    expansions.
14. Synthesis of linear reversible circuits.
15. Nets versus lattices, single-output versus
    multi-output
16. Multiple-valued reversible logic. Fundamentals.
    (no advanced methods).

5
Material for exam

• 17. PPRM, FPRM and ESOP. Synthesis and use in
  reversible logic.
• 18. Kronecker Products and matrix representation
  of logic.
• 19. QUIDDs.
• 20. Analyze a circuit with CV, CV and CNOT
  gates.
• 21. Analyze Mitra cascades or other complex
  realizations. No synthesis.
• 22. Being able to use symmetry or other
  properties such as balancedness, or NPNP
  classification classes in problems formulated in
  natural language.
• 23. Being able to combine several methods to
  design a practical quantum oracle, such as a
  spectral transform or arithmetic circuit.
• 24. Being able to combine several methods, or at
  least select one good method to analyze a
  reversible circuit. Binary of multi-valued.
• 25. Being able to analyze a reversible fuzzy
  circuit.
• 26. Being able to solve linear equations to solve
  some problem in reversible logic. (1) expansions
  of non-symmetric functions. (2) inverse gate. (3)
  find inputs states from output states.
• 27. Testability of simple cascades. EXOR cascade.
  Cascade from PPRM.