  Analog and digital quantum simulators can efficiently simulate quantum
many-body systems that appear in natural phenomena. However, experimental
limitations of near-term devices still make it challenging to perform the
entire process of quantum simulation. The purification-based quantum simulation
methods can alleviate the limitations in experiments such as the cooling
temperature and noise from the environment, while this method has the drawback
that it requires global entangled measurement with a prohibitively large number
of measurements that scales exponentially with the system size. In this Letter,
we propose that we can overcome these problems by restricting the entangled
measurements to the vicinity of the local observables to be measured, when the
locality of the system can be exploited. We provide theoretical guarantees that
the global purification operation can be replaced with local operations under
some conditions, in particular for the task of cooling and error mitigation. We
furthermore give a numerical verification that the localized purification is
valid even when conditions are not satisfied. Our method bridges the
fundamental concept of locality with quantum simulators, and therefore expected
to open a path to unexplored quantum many-body phenomena.
