Languages that lack contraction, weakening or exchange
As other commenters have mentioned what you are actually asking for is a language for planar string diagrams, aka monoidal categories.

But let me address your question purely from the point of view of quantum computation.  Clearly in any reasonable system you would want to have an introduction rule for the tensor product -- your proposed weakening rule is just a special case of this.  Note that if you have any interesting (i.e. entanglement creating) constants in your language it's not really true to say that your can just forget $A$ in the output, unless the $A$ and $B$ parts never interact.

It also not quite right to say that contraction can't be admitted.  There are (many, inequivalent) maps of type $A \to A\otimes A$ which are quantum-realisable. However none of them are natural transformations, which probably you want.

There were many attempts in the early 2000s to shoe-horn quantum computing into linear logic, which were largely unsuccessful:  too much of the LL structure becomes degenerate, and too little of what makes quantum work is present in bare LL.

However you asked for an example of a system which lacks these rules; an early example is Selinger's QPL http://dx.doi.org/10.1017/S0960129504004256.  More or less every other proposed quantum programming language that actually has a type system will disallow contraction and weakening for quantum data.

If you want to get rid of the exchange rule you might be interested in the field of topological quantum computing.  In this setting the unitary operations are constructed from braidings; so the exchange rule is replaced by the braiding.