The quantal version of Boolean algebra means that the basic logical functions have quantum inverses. The inverse of C=A ∧ B represents the quantum superposition of all pairs A and B for which A∧ B=C hols true. Same is true for ∨. How could these additional quantum logical functions with no classical counterparts extend the capacities of logician?
What comes in mind is logical reverse engineering. Consider the standard problem solving situation repeatedly encountered by my hero Hercule Poirot. Someone has been murdered. Who could have done it? Who did it? Actually scientists who want to explain instead of just applying the method to get additional items to the CVC, meet this kind of problem repeatedly. One has something which looks like an experimental anomaly and one has to explain it. Is this anomaly genuine or is it due to a systematic error in the information processing? Could the interpretation of data be somehow wrong? Is the model behind experiments based on existing theory really correct or has something very delicate been neglected? If a genuine anomaly is in question (someone has been really murdered- this is always obvious in the tales about the deeds of Hercule Poirot since the mere presence of Hercule guarantees the murder unless it has been already done) one encounters what might be called Poirot problem in honor of my hero. As a matter fact, from the point of view of Boolean algebra, one has the same reverse Boolean engineering problem irrespective of whether it was a genuine anomaly or not.
This brings in my mind the enormously simplified problem. The logical statement C is found to be true. Which pairs A,B could have implied C as C=A∧ B (or A∨ B). Of course, much more complex situations can be considered where C corresponds to some logical function C=f(A1,A2,...,An). Quantum Poirot could use quantum computer able to realize the co-gates for gates AND and OR (essentially time reversals) and write a quantum computer program solving the problem by constructing the Boolean co-function of Boolean function f.
What would happen in TGD Universe obeying zero energy ontology is following.
To conclude, I am a Boolean dilettante and know practically nothing about what quantum computer theorists have done- in particular I do not know whether they have considered quantum inverse gages. My feeling is that only the gates with bits replaced with qubits are considered: very natural when one thinks in terms of Boolean logic. If this is really the case, quantal co-AND and co-OR having no classical counterparts would bring a totally new aspect to quantum computation in solving problems in which one cannot do without (quantum) Poirot and his little gray (quantum) brain cells.
For details and background the reader can consult either to the chapter Physics as Generalized Number Theory: Infinite Primes or to the article How infinite primes relate to other views about mathematical infinity?.