1. Minimization Techniques for Reversible Logic Synthesis

2. Outline • Why reversible logic? • The building blocks • The synthesis problem • Some solutions • Optimization • Finding identities • Remaining problems Reversible Logic Synthesis

3. Reversible Logic • from the output you can determine the input (bijection) • Why would we want this? • Landauer's principle: every bit of information lost consumes energy • k is the Boltzman constant • T is the temperature • energy loss is small circuit network outputs inputs Reversible Logic Synthesis

4. Applications • Quantum Computing • necessarily reversible • Low power CMOS • In adiabatic circuits, current is restricted to flow across devices with low voltage drop and the energy stored on their capacitors is recycled • Optical Computing • Nano-technologies • Billiard Ball Model (BBM) Reversible Logic Synthesis

5. Standard Gates • which ones are reversible? and Embedding a non-reversible function into a reversible one not How many ways? xor or Reversible Logic Synthesis

6. Reversible Gates • Feynman gate (controlled not) x x y x y • Toffoli gate x x y y z xy z • Generalized Toffoli gate  more control lines Reversible Logic Synthesis

7. Reversible Gates • Fredkin gate (controlled swap) x x y yx’ + zx z zx’ + yx • Generalized Fredkin gate  more control lines Reversible Logic Synthesis

8. Restrictions • No fan-out • No back-feeds Reversible Logic Synthesis

9. Restrictions • No fan-out • No back-feeds Cascade of Gates g1 g2 gi gi+1 gn-1 gn … … output inputs Reversible Logic Synthesis

10. Function Representation • How do we represent a reversible function? • truth table • BDDs • Is there an easy check to see that the function is reversible? Reversible Logic Synthesis

11. Synthesis • Given a reversible function find a network of gates that realize the function • Cost should be near minimal • Possible cost assumptions: • Each gate has the same cost • The cost of each gate reflects its actual implementation cost Reversible Logic Synthesis

12. Approach 1 Reversible transformation outputs inputs Reversible Logic Synthesis

13. Approach 1 Reversible Transformation T1 outputs inputs G A T E Reversible Transformation T2 inputs outputs Reversible Logic Synthesis

14. Conditions • T2 should be “simpler” than T1 • How do you measure simplicity? • Hamming distance • Spectral techniques • Questions: • Will it converge? • How good is the result? • Improvements • Look ahead • Backtracking Reversible Logic Synthesis

15. Approach 2 Reversible Logic Synthesis

16. Approach 2 These are correct Reversible Logic Synthesis

17. Approach 2 These are correct Step 1: move up 100 to 110 Reversible Logic Synthesis

18. Approach 2 These are correct Step 1: move up 100 to 110 a a b b c c Reversible Logic Synthesis

19. Approach 2 These are correct Step 1: move up 100 to 110 a a b b c c Reversible Logic Synthesis

20. Approach 2 These are correct Step 1: move up 100 to 110 a a b b c c Reversible Logic Synthesis

21. Approach 2 These are correct Step 2: move down 110 to 010 Reversible Logic Synthesis

22. Approach 2 These are correct Step 2: move down 110 to 010 a a b b c c Reversible Logic Synthesis

23. Approach 2 These are correct Step 2: move down 110 to 010 a a b b c c Reversible Logic Synthesis

24. Approach 2 Step 2: move down 110 to 010 a a b b c c Reversible Logic Synthesis

25. Approach 2 Reversible Logic Synthesis

26. Approach 2 c c b b a a Reversible Logic Synthesis

27. Advantages/Disadvantages • Always converges • Fast • No look ahead • Several optimizations are possible • May not be optimal • Worst case? Reversible Logic Synthesis

28. Approach 3 • Local transformation • Replace a sequence of gates with another • For example replace 3 gates with 2 Reversible Logic Synthesis

29. Reversible Logic Synthesis

30. Approach 3 • Use Identity circuits for local transformations g1 g2 gi gi+1 gn-1 gn … … identity inputs Reversible Logic Synthesis

31. Approach 3 g1 g2 gi gi+1 gn-1 gn … … identity inputs F(X) F-1(X) Reversible Logic Synthesis

32. Approach 3 g1 g2 gi gi+1 gn-1 gn … … identity inputs F(X) F-1(X) Replace this sequence of gates Reversible Logic Synthesis

33. Approach 3 g1 g2 gi gi+1 gn-1 gn … … identity inputs F(X) F-1(X) with this sequence of gates Replace this sequence of gates Reversible Logic Synthesis

34. Template Description • As a class - an identity that is not reducible by another template • A template may be rotated • A template may be applied in reverse order • Size 2 ==> Duplicate gates (may be deleted) • Size 3 ==> There are none • Size 4 ==> Passing rule Reversible Logic Synthesis

35. Template Class — Size 5 A line may be duplicated A line may be removed Reversible Logic Synthesis

36. How many classes of identities are there? Reversible Logic Synthesis

37. The Templates: Application Reversible Logic Synthesis

38. The quest for identities • Exhaustive searches are not feasible • A feasible approach (to find size n identities): • Find all functions of size n/2 • With limited number of lines • With some canonical order • Pair each function with its inverse • If no reduction is possible ==> we found a new template Reversible Logic Synthesis

39. A new approach • Start with an identity that has target lines only • For any subset of gates that preserve the identity a new control line may be added • Example: Reversible Logic Synthesis

40. Recent Advances • Automatic template encoding • Application of iterative minimization heuristics • Progress in calculating “real quantum cost” • Using gates NOT, CNOT, V, and V+ • Quantum templates • Using SAT to find exact results Reversible Logic Synthesis

41. Future Work • Handling don’t cares (can be done with SAT) • Billiard ball implementation of reversible circuits (via cellular arrays) • Better representation for reversible functions • Truth table is not adequate • Some form of BDD (possibly Davio) • Minimization with aid of group theory Reversible Logic Synthesis