1. Templates for Toffoli Network Synthesis by Dmitri Maslov Gerhard W. Dueck Michael D. Miller

2. Outline - Basic definitions. - Templates as a simplification tool. - The templates: definition; classification; application. - Conclusion. IWLS 2003 Laguna Beach, CA page 1/14

3. Basic Definitions Definition. Multiple output Boolean function is called reversible iff: 1. 2. - is a bijection. Examples. 1. - (NOT) is reversible. 2. - (Feynman gate) is reversible. 3. - is not reversible. IWLS 2003 Laguna Beach, CA page 2/14

4. Basic Definitions NOT CNOT (Feynman) … Generalized Toffoli Toffoli In reversible logic fan-outs and feed-back conventionally are not allowed, thus any network is a cascade. IWLS 2003 Laguna Beach, CA page 3/14

5. Templates as a Simplification Tool As 3 gates can be rewritten by a sequence of 2, some s gates can be rewritten as a sequence of k gates (k<s). Network simplification approach. - preprocessing: find as many rewriting rules as possible. - simplify by matching rewriting rules and rewriting the circuit. IWLS 2003 Laguna Beach, CA page 4/14

6. Templates as a Simplification Tool Problems in such naive approach. 1.The number of rewriting rules is very large: a. For s=3, k=2 and n=3 (number of lines) the number of rewriting rules is 180. b. Many rewriting rules are redundant. c. The number of non-redundant rewriting rules only grows exponentially on n. 2. Very often a rewriting can be applied only when certain gates are moved. IWLS 2003 Laguna Beach, CA page 5/14

7. The Templates: Definition Observation 1. If one has a rewriting rule then the gates in it satisfy equation Observation 3. If, then Observation 2. If we have an identity then for any parameter p, is a valid rewriting rule. IWLS 2003 Laguna Beach, CA page 6/14

8. The Templates: Definition A size m template is a cascade of m gates which realizes the identity function. Any template of size m should be independent of smaller size templates, i.e. application of smaller templates does not decrease the number of gates in a size m template. Given G 0 G 1 …G m-1, a template of size m, its application for parameter p, is: - for IWLS 2003 Laguna Beach, CA page 7/14

9. The Templates: Definition How many rewriting rules are there in one template? Parameter p: m/2 choices. Starting gate i: m choices. Directions of application: 2 (forward, backward). m2m2 A B C D E F G Example. Template ABCDEFG. p=4. Starting gate B. Direction: backward. B B A A G G F F C C D D E E IWLS 2003 Laguna Beach, CA page 8/14

10. The Templates: Classification A class of templates can be described as one picture (in terms of the defined pictorial representation), where: - t i – always a single line; - C i – a (finite) set of lines. The following is the complete classification of templates size m=1,2,…,7 for any number of horizontal lines, n. IWLS 2003 Laguna Beach, CA page 9/14

11. The Templates: Classification IWLS 2003 Laguna Beach, CA page 10/14

12. The Templates: Classification IWLS 2003 Laguna Beach, CA page 11/14

13. The Templates: Application In a program realization: - use size 4 templates to move gates – moving rule. - for other templates create a hierarchy where priority is proportional to the inverse of size. - try to apply templates with higher priority first. - given a template, match it by trying both directions, starting with any gate and trying to move other gates by the moving rule. IWLS 2003 Laguna Beach, CA page 12/14

14. The Templates: Application IWLS 2003 Laguna Beach, CA page 13/14

15. Conclusion Templates are a useful simplification tool. IWLS 2003 Laguna Beach, CA page 14/14 Sometimes, it is even possible to prove optimality of the Toffoli circuit.

16. Templates for Toffoli Network Synthesis by Dmitri Maslov Gerhard W. Dueck Michael D. Miller