Why is Heisenberg picture used if Schrödinger picture is simpler? In Heisenberg picture the operators evolve in time instead of states, which evolve in time in Schrödinger picture. But this also means that to solve the equations (at least numerically) one has to work with $N^2$ numbers instead of Schrödinger's $N$ (i.e. Hamiltonian matrix vs state vector).
So, why, despite being dramatically more inefficient for calculations, is Heisenberg picture still used?
 A: There are a couple of types of situation in which the Schrodinger picture can be problematic. If you are working in a relativistic context you might not want to be tied down to one particular reference frame in which the states are a function of time and extend across all of space.
The other situation in which the Schrodinger picture is problematic is any situation in which you don't understand what's going on. It is almost always easier to tell how a system has evolved in the Heisenberg picture than in the Schrodinger picture. In the Heisenberg picture you can pick a set of representative observables that span the vector space of operators that can act on the system under consideration, including the unitary operators. From how that set of observables changes you can usually reconstruct what unitary operator acted on them. So the Heisenberg picture observables contain more information than the Schrodinger picture state. So, for example, entangled quantum systems instantiate locally inaccessible information in that the observables can be dependent on a parameter but their expectation values are not. This helps explain  correlations between the results of measurement on entangled quantum systems
http://xxx.lanl.gov/abs/quant-ph/9906007
http://arxiv.org/abs/1109.6223.
