The question seems to be groping for a fancy, specific answer when, in my view, the most important connection is relatively basic and general.

In mathematics, as you say, you have symmetric monoidal categories. A word in a symmetric monoidal category has a set of inputs and a set of outputs and a certain type of labelled graph in between them. Symmetric monoidal categories are used in many situations. They were inspired by multilinear algebra with tensors; I gather that they are also important in linear logic.

In theoretical computer science, there is the notion of a circuit, and the closely related notion of a straight-line program. A circuit is also a labelled acyclic graph with inputs and outputs. Computer scientists began with boolean circuits, but these days there are a lot of study of arithmetic circuits over any ring, as well as quantum circuits composed of unitary gates. A precursor to quantum circuits is the clever definition of reversible and conservative circuits of Toffoli and Fredkin. In the paper cited, they have in mind a dynamical interpretation of graphs that don't have to be acyclic. However, the case of what they did that has had the most influence is finite, acyclic graphs. In this case, their essential insight is that you can have a perfectly good model of boolean circuits based on permutations of $\{0,1\}^n$ rather than functions from $\{0,1\}^a$ to $\{0,1\}^b$.

So the point is that the math concept of monoidal categories, which are broadly important, is basically the same as the CS concept of circuits, which are also broadly important. Every monoidal category, maybe together with some distinguished generators, gives you a new type of circuit, so that you can then ask circuit complexity questions. A particular monoidal category may or may not yield interesting circuit questions, and two different monoidal categories might yield equivalent circuit questions. Nonetheless, there are many interesting different cases.

Actually, not-necessarily-acyclic graphs can also be interpreted as words in symmetric monoidal categories that are also "closed" or "pivotal". There are several ways to interpret tensor words of this type as circuits; one of these ways (for some types of words) is as dynamical circuits.

Linear logic and conservative logic are both called "logic", and they both use monoidal categories. (Also, as Peter Shor mentioned, linear logic is partly inspired by quantum probability, while conservative logic is used in quantum computation.) Other than that, they don't look particularly related to me.