/ 99 10 06 9 v 1 1 5 O ct 1 99 9 Note on Coherent States and Adiabatic Connections

We give a possible generalization to the example in the paper of Zanardi and Rasetti (quant–ph 9904011). For this generalized one explicit forms of adiabatic connection, curvature and etc. are given. ∗E-mail address : [email protected] This is a comment paper to Zanardi and Rasetti[1] and the aim is to give a mathematical inforcement to [1]. After the breakthrough by P. Shor[2] there has been remarkable progress in Quantum Computation (or Computer)(QC briefly). See [3] in outline. On the other hand, Gauge Theories are widely recognized as the basis in quantum field theories. Therefore it is very natural to intend to include gauge theories in QC · · · a construction of “gauge theoretic” quantum computation or of “geometric” quantum computation in our terminology. Zanardi and Rasetti proposed in [1] and [4] such an idea using non-abelian Berry phase (quantum holonomy), see also [5]. In their model a Hamiltonian (including some parameters) must be degenerated because an adiabatic connection is introduced using this degeneracy [6]. They gave a simple example to explain their idea. However there are many misprints in their calculations, so it is not easy to follow their idea. We believe that this example will become important in the near future. Therefore we deal with it once more and give a possible generalization. For the generalized model explicit forms of adiabatic (Berry) connection, curvature and etc are given. It is not easy to predict the future of gauge theoretic quantum computation. However it is an arena worth challenging for mathematical physicists. We start with mathematical preliminaries. Let H be a separable Hilbert space over C. For m ∈ N, we set Stm(H) ≡ { V = (v1, · · · , vm) ∈ H× · · · × H|V †V = 1m } , (1) where 1m is a unit matrix in M(m,C). This is called a (universal) Stiefel manifold. Note that the unitary group U(m) acts on Stm(H) from the right: Stm(H)× U(m) → Stm(H) : (V, a) 7→ V a. (2) 1 Next we define a (universal) Grassmann manifold Grm(H) ≡ { X ∈ M(H)|X = X,X† = X and trX = m } , (3) where M(H) denotes a space of all bounded linear operators on H. Then we have a projection π : Stm(H) → Grm(H) , π(V ) ≡ V V † , (4) compatible with the action (2) (π(V a) = V a(V a)† = V aa†V † = V V † = π(V )). Now the set {U(m), Stm(H), π, Grm(H)} , (5) is called a (universal) principal U(m) bundle, see [7] and [8]. Next let M be a n-dimensional differentiable manifold and the map P : M → Grm(H) be given. For this P the pull-back bundle over M is defined as follows[7]: (U(m), E, πE,M) ≡ P ∗ (U(m), Stm(H), π, Grm(H)) , E = {(x, V ) ∈ M × Stm(H)|P (x) = π(V )} , πE : E → M , πE((x, V )) = x . U(m) U(m)