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My current approach to investigating reversible quantum gates requires the solution of Boolean functional equations. For example,

$$f(x,y,z) = f(x,y \oplus f(x,y,z), z \oplus f(x, y, z)),$$

where $f\colon B^3 \to B$, $B = \{0, 1\}$, and $\oplus$ is xor.

I have solved this one, and a number of similar types, but I do not know of an approach to solving similar equations with a higher number of variables. Is there a general algebraic approach to the solution of Boolean functional equations of this, or similar, types?

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    $\begingroup$ I do not know the theory of (methods for) finding specific solutions to functional equations in the domain of Boolean algebras. There is a general theory starting with Abel for functions on the reals which focuses on roots (f^n=f) and some variations. There is another general theory (clone theory, a part of universal algebra) which might suggest when to approach solving such equations in other domains such as Boolean Algebras, but to my knowledge is not developed enough to give you a direct and concise answer to your posted question. Gerhard "Needs A Clone Theory Review" Paseman, 2019.10.22. $\endgroup$ Oct 22, 2019 at 17:10
  • $\begingroup$ The question is too open-ended. Can you describe more precisely what is the general shape of functional equations that you consider? $\endgroup$ Oct 22, 2019 at 19:39
  • $\begingroup$ Thank you for the replies. I need to pursue my work on invertible gates a little further before I am able to specify a general class of interest. My current method of solution is fine for gates (or functions) from B^2 to B^2 and B^3 to B^3, However I'm facing a combinatorial explosion with my method as I move to the B^4 case and beyond. $\endgroup$
    – Helmut Bez
    Oct 23, 2019 at 14:07

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