  The next-generation lepton colliders, such as CEPC, FCC-ee, and ILC will make
precision measurement of the Higgs boson properties. We first extract the Higgs
coupling precision from Higgs observables at CEPC to illustrate the potential
of future lepton colliders. Depending on the related event rates, the precision
can reach percentage level for most couplings. Then, we try to estimate the new
physics scales that can be indirectly probed with Higgs and electroweak
precision observables. The Higgs observables, together with the existing
electroweak precision observables, can probe new physics up to 10TeV (40TeV for
the gluon-related operator $\mathcal O_g$) at 95% C.L. Including the $Z/W$ mass
measurements and $Z$-pole observables at CEPC further pushes the limit up to
35TeV. Although $Z$-pole running is originally for the purpose of machine
calibration, it can be as important as the Higgs observables for probing the
new physics scales indirectly. The indirect probe of new physics scales at
lepton colliders can mainly cover the energy range to be explored by the
following hadron colliders of pp (50-100TeV), such as SPPC and FCC-hh.
