Is Logic Done on Superpositional Bit Values Useful?

Let's say I have a way to represent $N$ bits such that those bits are in a superposition of the $2^N$ possible states those bits can have and that I can do XOR and AND on those superpositional bits to be able to map $N$ superpositional input bits to $M$ superpositional output bits using any logical circuit (since XOR and AND are supported).

Let's say that I am also able to include bits in those logic circuits which are not superpositional but have specific values of either 0 or 1. This way I can do a NOT by XORing a superpositional bit by 1 for example.

At the end of the logic circuit when I have the superpositional result, I can then decide what values the input bits had, and get a non superpositional result out for those inputs, without evaluating the circuit again. I can interpret the result as many times as I want for the entire permutation of input bits if I want to.

I'm wondering, would a superpositional logic technique like this be useful, and does anyone know if any work has been done in this area?

I know this is similar to how quantum computing works, except for a few key differences like not having interference, nor having probabilities for states, but it does allow cloning of the superpositional result, unlike quantum computing which only allows one question to be asked of the result.

I can think of a couple usage cases, but am wondering if there are others?

1. Using a superpositional result as a way of letting an untrusted person specify how inputs should map to outputs (basically, "script" some process), without havign to sandbox the logic, worry about division by zero, infinite loops, and the like.
2. I have yet to experiment deeply with this, but if you need to know several values of some function $f(x)$, it may be more computationally efficient to evaluate it superpositionally and then re-interpret the superpositional result several times, versus just calculating the function several times.
• I'm curious about the motivation. Separately from this, I can tell you that 'At the end of the logic circuit when I have the superpositional result, I can then decide what values the input bits had, and get a non superpositional result out for those inputs' is very different from quantum computation, but sounds a little bit like a variation of a 'signalling' variant of the so-called boxworld model of computation, if I interpret it in the way that makes the most sense to me. To be sure, you may have to more formally define how you intend to compute with the "superpositional bits". – Niel de Beaudrap Oct 9 '15 at 15:47
• Also: if you want to avoid your question from being mistaken for a misunderstanding of quantum computation by hasty downvoters, you may want to change your title. "Superposition" is a term borrowed from physics, and specifically wave-mechanics; unless there is already a model of computation called "superpositional logic" in the literature that I'm not aware of (cite it, if there is!) then you should probably be wary of using this term as though it is already well-defined (as otherwise you give the strong impression of intending to describe quantum computation). – Niel de Beaudrap Oct 9 '15 at 15:51
• The motivation is that I stumbled on a solution looking for a problem, and I'm curious if other people have gone down this road before hehe. You wouldn't happen to have a link to some boxworld model of computation information would you? Too many unrelated results in google. Title updated too btw, thanks Niel! – Alan Wolfe Oct 9 '15 at 16:47
• It seems like you could use this to (a) preprocess the question and (b) compute the value for as many inputs as you want in constant time for each. This would be extremely useful in practice in many scenarios. (Consider databases, which essentially do this for a special case without using magic superpositions.) – Peter Shor Oct 11 '15 at 13:52
• Good point about constant time evaluation, thank you for adding that! Also, I'm a bit starstruck having had an online interaction with a legend such as yourself (: – Alan Wolfe Oct 11 '15 at 15:02