The experiments in this chapter show that a quantum mechanical architecture can achieve miniature natural language processing tasks quite successfully. One should bear in mind, however, that the models and problems proposed in this chapter are highly simplified. This immediately raises the question of the scalability of a quantum mechanical framework. As far as the simulation of a quantum process on a conventional computer is concerned, the efficiency is probably not feasible for NLP of very large scale. This can be seen from the complexity of the optimization procedure. For one thing, the number of free parameters in a Hamiltonian operator is proportional to the square of the number of eigenstates (or the size of the vocabulary). Therefore the complexity will grow very fast as the vocabulary grows. Secondly, in order to calculate the unitary reasoning operator
(using the Jacobi algorithm [67], for example). This is a tedious task, and very resource-hungry. But we should bear in mind that for these simulations we used a conventional computer, so the criticism on efficiency is not fair. If we can build a quantum computer in the near future (this is not unlikely) [28,68,69], the lack-of-efficiency argument may not stand.
This is not to say that a quantum mechanical framework cannot have practical value at the present time. For instance, one may build a hybrid model in which classical symbolic computation and/or statistic-connectionist modules can be implemented to work with a quantum mechanical ``arbitration'' module that takes care of crucial decisions. This is similar to a scenario in which a human is assisted by computer programs (such as database or number-crunching programs) to make decisions. After all, if the picture described in Chapter 4 and 5 is correct, the human brain must be working in a classical-quantum hybrid mode.
A closer look at the errors in this chapter shows that most of them result from the last combinatorial procedure, for the pre-measurement states of affairs are indeed quite ``well-behaved.'' (This can be seen from the absolute squares/phases graphics of the erroneous instances.) A full scale time-dependent quantum encoder/decoder might be able to solve this problem.
A question (aside from lack-of-efficiency) arises when we notice that the experiments done in this chapter are all simulations on conventional computers (this is because we have a powerful mathematical formalism). Can simulations on conventional computers tackle all the problems a quantum mechanical NLP system can solve? This is probably not the case. In fact, we only simulate the formalism, not the experiment itself. There are many experimental problems that cannot be dealt with in classical computation. Therefore they also cannot be solved with simulations. For instance, genuine random numbers and counterfactual computation in quantum mechanics cannot be carried out on a conventional computer.
A further remark is that the models proposed here are all ``first-order,'' by which we mean that the Hamiltonian is fixed and given. In fact, the Hamiltonian used in this chapter are calculated using classical optimization algorithms, but the brain may work very differently. In a genuine quantum computational environment, the Hamiltonian must itself be the result of a chain of quantum computations. It may be the source of active thinking and creativity in the brain.
The activeness is twofold: for one thing, the brain actively compiles a string of symbols (eigenstates) into a state of affairs; for the other, the brain should be able to actively arrange the quantum experiment. In general, Hamiltonians are set up by way of other quantum measurement, in which the experimental arrangement is setup by yet another quantum mechanical arrangement. This may go from one level to another ad infinitum. In this regard, thought can be seen as a continuous process of ``preparing'' and ``measuring'' -- a constant ``enfolding'' and ``unfolding,'' as David Bohm put it [1,36].
Generally speaking, if the brain is indeed a quantum computer, an adequate simulation of it has to be a higher-order quantum computer, in order that it may exhibit similar behavior of our brains. Indeed, our superior mental ability may be closely related to these higher order quantum computations, especially for creative tasks in language and mathematics7.9. Today, we know that there are algorithms believed intractable on a classical computer that can be solved elegantly and quickly by a quantum computer [30]. Although this is only in theory, it is encouraging to see quantum computation able to solve some seemingly intractable human cognition problems. For a full scale application of this approach, of course, we need a genuine quantum computer.