Analyzing simulation of quantum systems in Schrödinger and Heisenberg pictures.

Computing operator expectation values is a central task in simulating quantum many-body systems. For unitarily prepared trial states, one could either compute expectation values in the Schrodinger picture (state evolution) or Heisenberg picture (operator evolution). On one hand, full knowledge of the wavefunction permits the computation of any operator, it might not be necessary to calculate certain expectation values. Physical Hamiltonians, on the other hand, are generally sparse, and only involve a polynomial number of operators. While the space of operators is much larger than that of states, there are cases where the sparsity of the operator evolution is greater than the sparsity of the state evolution. In this talk, we compare the two approaches for a few relevant examples, including molecular simulation and the recent 127 qubit IBM experimental simulation of kicked Ising dynamics.