I will discuss a practical approach to the problem of tracking a ground state that is based on the quantum Zeno effect. By performing randomised dephasing operations at intervals determined by a Poisson process, we are able to track the eigenspace associated to a particular eigenvalue. A simple differential equation for the fidelity can then be derived, leading to general theorems bounding the time complexity of a whole class of algorithms. In many cases the bounds given by these theorems are optimal, giving a time complexity of O(1/Δ) with Δ the minimum of the gap. Applications include the Grover problem, the Quantum Linear System Problem and the Ising problem.