7 Fixed anyons Mobile  anyon w$_{1}$ w$_{2}$ P !  (rad) ! FIG. 6: (Color online) The figure illustrates a configuration in which all anyons are kept fixed except one, which is moved on the sphere. To compute different quantities at an arbitrary point P of the sphere, we introduce a local coordinate system ( w$_{1}$, w$_{2}$ ) as described in the text. FIG. 5: (Color online) The spectrum of H ( w$_{1}$, . . . , w$_{4}$ ) ⇂$_{H}$$_{0}$ as a function of the probes’ position. The probes were moved along the for meridians: ( θ, φ = 0), ( θ, φ = π ), ( π − θ, φ = π/ 2), ( π − θ, φ = − π/ 2) (see inset). The results correspond to N$_{el}$ = 14 and N$_{φ}$ = 27. in the adiabatic limit, i.e. in the limit when the motion along the path is infinitely slow. If we denote by P$_{x}$ the projector onto the zero modes space of H ( x ), then the classic Adiabatic Theorem says that: 27–29 energy level, separated by an energy gap from the rest of the spectrum (when compared with other energy separa- tions visible in the graph). The gap becomes independent of the positions of the probes when the probe separation becomes larger than approximately 4 magnetic lengths. As the probes come closer to each other, the gap de- creases and, at the intersection points θ =0 and π , the gap is completely closed, leading to an abrupt increase in the level degeneracy. where U$_{a}$ ( t ) is the adiabatic propagator, i.e. the unique solution of the following system of equations: 28 U ( t ) P$_{x}$$_{0}$ = U$_{a}$ ( t ) P$_{x}$$_{0}$ + o ( v, t ) , (22) i∂$_{t}$U$_{a}$ ( t ) = ( H ( t ) + i [ ∂$_{t}$P$_{x}$, P$_{x}$ ]) U$_{a}$ ( t ) , U$_{a}$ (0) = I. (23) As long as the energy gap between the zero energy of the anyons and the energies of the excited states remains open, the adiabatic evolution is a good approximation of the true evolution, when restricted to the zero modes space. In fact, the two become identical if the velocity v is infinitely small. We want to point out that there are rigorous estimates (see for example Refs. 27–29) of the errors o ( v, t ) that are made when one approximates U ( t ) by U$_{a}$ ( t ), i.e. of the non-adibatic effects. Since the braids corresponding to the elementary TQC logic gates are quite long, 13 these estimates may play an important role in the field. Let us consider the general situation when the number of flux has been increased by n units, N$_{φ}$ = N 0 $_{φ}$+ n . We use the pinning Hamiltonian H ( w$_{1}$, . . . , w$_{2}$$_{n}$ ) ⇂$_{H}$$_{0}$ given in Eq. (18) to control the position of the anyons. Given that each w$_{i}$ is a 2-dimensional variable, the Hamiltonian depends, parametrically, on 4 n coordinates. We use the symbol x to denote these coordinates and the shorthand H ( x ) for H ( w$_{1}$, . . . , w$_{2}$$_{n}$ ) ⇂$_{H}$$_{0}$ . The adiabatic evolution satisfies the following funda- mental relation: V. BRAIDING THE MOORE-READ STATES A. General considerations. P$_{x}$ = U$_{a}$ ( x ) P$_{x}$$_{0}$U$_{a}$ ( x ) − $^{1}$. (24) Let us consider an arbitrary closed path in the parame- ter space: ℓ → x ( ℓ ) ( x$_{0}$ = x (0)), where we use the length ℓ to parametrize the loop. Now imagine that we move along this path with constant velocity v . One would like to study the time evolution U ( t ) generated by the time dependent Hamiltonian H ( t ) ≡ H ( x ( vt )): This equation tells us that the zero modes space is mapped into itself by the adiabatic evolution. This map- ping can be computed by solving Eq. (23), which tells us how the zero modes space is transported during the adiabatic evolution. One can put all these into a geo- metric perspective, by defining the following 1-form (the i∂$_{t}$U ( t ) = H ( t ) U ( t ) , U (0) = I, (21)