1. A Transformation Based Algorithm for Reversible Logic Synthesis D. Michael Miller Dmitri Maslov Gerhard W. Dueck Design Automation Conference, 2003

2. Objectives Synthesis of Toffoli cascades Synthesis of Toffoli cascades reversible logic reversible logic application in application in quantum computing quantum computing low power CMOS low power CMOS optical computing optical computing Approach Approach use a naive algorithm to find a solution use a naive algorithm to find a solution apply transformations to simplify the network apply transformations to simplify the network

3. Toffoli Gate Family We use such gates for any number of inputs

4. The Basic Algorithm Reversible function is given as a truth table Reversible function is given as a truth table For each row in the truth table For each row in the truth table Introduce Toffoli gates such that the output equals the input Introduce Toffoli gates such that the output equals the input Make sure that previous outputs are not affected Make sure that previous outputs are not affected

5. The Basic Algorithm in 000 001 010 011 100 101 110 111 out 001 000 011 010 101 111 100 110 S1 000 001 010 011 100 110 101 111 S1 000 001 010 011 100 110 101 111 S1 000 001 010 011 100 110 101 111 S2 000 001 010 011 100 111 101 110 S3 000 001 010 011 100 101 111 110 S3 000 001 010 011 100 101 111 110 S3 000 001 010 011 100 101 110 111 Final circuit c b a Read quantum array in inverse order to applying gates Observe that after applying S3 wires are as in input

6. Bidirectional Algorithm in 000 001 010 011 100 101 110 111 out 111 000 001 010 011 100 101 110 need three gates need only one gate

7. Bidirectional Algorithm in 000 001 010 011 100 101 110 111 out 111 000 001 010 011 100 101 110 S1 000 111 010 001 100 011 110 101 S2 000 001 010 111 100 101 110 011 S3 000 001 010 011 100 101 110 111 Final circuit c b a S1 After applying S3 my wires are the same as inputs inputs ab

8. Improvements Output permutations Output permutations via swap gates via swap gates Control input reduction Control input reduction there may be more than on possible assignment of control variables there may be more than on possible assignment of control variables select the one that makes the function “simple” select the one that makes the function “simple” Template matching Template matching

9. Templates Idea: replace a sequence of gates with an equivalent shorter sequence Idea: replace a sequence of gates with an equivalent shorter sequence example: example:

10. Example

11. Template Classification

12. Larger Example rd53

13. Number of reversible functions using a specified number of gates gates 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 Average 1 14 92 380 1113 2468 4311 6083 7044 6754 5379 3549 1922 839 286 72 12 1 8.67 naive 4 72 477 1759 4179 6912 8389 7766 5615 3183 1391 453 104 15 1 7.65 permutation 5 87 550 1901 4267 6828 8221 7670 5610 3204 1402 455 104 15 1 7.67 input control 3 86 493 2312 6944 11206 10169 5945 2375 650 121 15 1 6.53 bidirectional 5 110 792 4726 11199 12076 7518 2981 767 130 15 1 6.18 templates 12 6817 17531 11194 3752 844 134 15 1 5.63 optimal

14. Future Work Start from a non-reversible function Start from a non-reversible function Verify the completeness of the templates Verify the completeness of the templates Expand the algorithm to include Fredkin gates Expand the algorithm to include Fredkin gates Select the best permutation (not by exhaustive search) Select the best permutation (not by exhaustive search)