This paper considers right-invariant and controllable quantum systems withminputsu= (u1,… ,um) and stateX(t) evolving on the unitary Lie group U(n). The ε-steering problem is introduced and solved for systems with drift: given any initial conditionX0at the initial time instantt0≥ 0, any goal stateU(n) and ε > 0, find a control law such that, whereis big enough and dist(X1,X2) is a convenient right-invariant notion of distance between two elementsX1,X2∈ U(n). The purpose is to approximately generate arbitrary quantum gates corresponding toXgoal. This is achieved by solving a tracking problem for a special kind of reference trajectories: [t0, ∞) → U(n), which are here calledc-universal reference trajectories. It is shown that, for this special kind of trajectories, the tracking problem can be solved up to an error ε for any reference trajectorywhich is a right-translation of, at least whenis finite. Furthermore, it is shown thatconverges uniformly exponentially to zero in the sense that the rate of convergence is independent oft0,RandX0. The approach considered here for showing such convergence is a generalisation of the results of a previous paper of the authors, which is mainly based on the central ideas of Jurdjevic and Quinn and Coron's return method. Taking a right-translationRsuch that, one may solve the ε-steering problem by solving the tracking problem for the reference trajectory, at least when. When, it is shown that the ε-steering problem can be globally solved in a two-iteration procedure. The underlying algorithmic complexity to get the steering control is essentially equivalent to the numerical integration of the Cauchy problem governingX(t). A numerical example considering aToffoliquantum gate on U(8) for a chain of three coupled qubits that are controlled only locally is presented. [ABSTRACT FROM AUTHOR]