In this section, a quantum computational solution of the XOR-problem is presented. The reader should bear in mind, however, that the XOR-problem is presented here in a bottom-up fashion. That is, the architecture is not designed to perform XOR operation, as they usual are performed in the quantum computation literature [39], but to learn the behavior of XOR from training data. As far as the underlying structure is concerned, the quantum structure is reversible (without being measured) while classical XOR is not.
It is well known that the common classical logic operators (
, 
, 
, and 
), called logical primitives, are in fact redundant. For instance, from Negation-AND - NAND (or equivalently Negation-OR - NOR) alone all four logical operators can be derived. However, NAND and NOR have very obscure intuitive content. It is quite implausible that our cognitive process is built on NAND or NOR. On the other hand, it seems that XOR and AND are intuitively more ``primitive'' operations on which our cognition is based6.1. Furthermore, XOR is somewhat ``intimate'' to 
-measurement in the sense that every quantum measurement manifests itself in an either ... or ... (XOR) scheme. This seems to be a good incentive for us to begin with the discussion of a quantum computational realization of the XOR function.
Indeed, XOR is a classical problem that cannot be solved with a simple linear model. In a classical connectionist treatment of XOR, it is well-known that one needs at least one hidden layer as well as processing units (neurons) with a non-linear threshold function [40]. While a quantum mechanical system is linear, the problems a quantum system can successfully solve are not necessarily linear. We shall see that the dynamics and characteristics of this implementation are much richer than that of a classical logic gate or a connectionist architecture. It also suggests that the ``fine-structure'' is more realistic for our everyday reasoning and natural language. This will be discussed in the following sections.