Then

We will give two lectures providing some basic notation, mathematics and physics background needed for the subsequent discussion in this summer school. Most material are adapted from [3, Chapter 2] and [1] (see also [2, Chapter 1]). 1 Hilbert spaces The mathematical platform of quantum mechanics/computing is Hilbert space (complete inner product space) V. We mainly focus on finite dimensional complex inner product space Cn, the set of n × 1 column vectors. Let Cn ∗ be the dual vector space of Cn consisting of 1 × n row vectors In physics, we use the bra and ket vector notation. (Dirac notation.) Let |x 〉 = (x1,..., xn)t = x1... xn ∈ Cn