The Distribution of Reversible Functions is Normal

TL;DR

This paper analyzes the distribution of reversible functions, showing it tends to a normal distribution with implications for quantum computing and genetic programming, and demonstrates evolvability in certain circuit designs.

Contribution

It introduces the limiting distribution of reversible programs, explores fitness landscapes for evolutionary search, and extends findings to quantum computing contexts.

Findings

Distribution of reversible functions approaches normal as size increases

Sufficiently good circuits are more common than perfect solutions

Larger circuits exhibit greater evolvability in genetic programming

Abstract

The distribution of reversible programs tends to a limit as their size increases. For problems with a Hamming distance fitness function the limiting distribution is binomial with an exponentially small chance (but non~zero) chance of perfect solution. Sufficiently good reversible circuits are more common. Expected RMS error is also calculated. Random unitary matrices may suggest possible extension to quantum computing. Using the genetic programming (GP) benchmark, the six multiplexor, circuits of Toffoli gates are shown to give a fitness landscape amenable to evolutionary search. Minimal CCNOT solutions to the six multiplexer are found but larger circuits are more evolvable.