  Recent progress in quantum simulation and algorithms has demonstrated a rapid
expansion in capabilities. The search continues for new techniques and
applications to exploit quantum advantage. Here we propose an innovative method
to investigate directly the properties of a time-dependent density operator
$\hat{\rho}(t)$. Using generalised quantum commutation simulators, we can
directly compute the expectation value of the commutation relation and thus of
the rate of change of $\hat{\rho}(t)$. The approach can be utilised as a
quantum eigen-vector solver for the von Neumann equation and a decoherence
investigator for the Lindblad equation, by using just the statistics of
single-qubit measurements. A simple but important example is demonstrated in
the single-qubit case and we discuss extension of the method for practical
quantum simulation with many qubits, towards investigation of more realistic
quantum systems.
