A discriminator variety of linear Heyting algebras with operators arising in quantum computation

Unlike classical computation, quantum computation [7] allows one to encode two atomic information bits in parallel. Here, in fact, the appropriate counterpart of a classical bit is the qubit, de…ned as a unit vector in C, i.e. j i = a j0i + b j1i, where a; b are complex numbers s.t. jaj + jbj = 1, and fj0i ; j1ig is the canonical base. Supposing that, in analogy with the classical case, j0i and j1i represent maximal and precise pieces of information, the superposition state j i corresponds to an uncertain information: as dictated by the Born rule, jaj yields the probability of the information described by the pure state j0i, while jbj yields the probability of the information described by the pure state j1i. A system of n qubits, also called a n quregister, is represented by a unit vector in the n-fold tensor product Hilbert space C. Qubits and quregisters, therefore, encode possibly uncertain, yet maximal information. Non-maximal information pieces are matched, on a mathematical level, by qumixes, i.e. density operators in C or in appropriate tensor products C of C. Similarly to the classical case, we can introduce and study the behaviour of a number of quantum logical gates operating on such information units. These gates are mathematically represented by unitary operators on the appropriate Hilbert spaces. Hereafter we mention a crucial example in the framework of quregisters: for any n 1, the square root of the negation on C is the unitary operator p Not (n) such that, for every element ja1; :::; ani of the computational basis B(n)1 , p Not (n) (ja1; :::; ani) = ja1; :::; an 1i 12 ((1 + i) jani+ (1 i) j1 ani), where i is the imaginary unit. The counterpart of p Not (n) in the framework