  On a quantum superconducting processor we observe partial and
infinite-temperature thermalization induced by a sequence of repeated quantum
projective measurements, interspersed by a unitary (Hamiltonian) evolution.
Specifically, on a qubit and two-qubit systems, we test the state convergence
of a monitored quantum system in the limit of a large number of quantum
measurements, depending on the non-commutativity of the Hamiltonian and the
measurement observable. When the Hamiltonian and observable do not commute, the
convergence is uniform towards the infinite-temperature state. Conversely,
whenever the two operators have one or more eigenvectors in common in their
spectral decomposition, the state of the monitored system converges differently
in the subspaces spanned by the measurement observable eigenstates. As a
result, we show that the convergence does not tend to a completely mixed
(infinite-temperature) state, but to a block-diagonal state in the observable
basis, with a finite effective temperature in each measurement subspace.
Finally, we quantify the effects of the quantum hardware noise on the data by
modelling them by means of depolarizing quantum channels.
