  We show that from the point of view of the generalized pairing Hamiltonian,
the atomic nucleus is a system with small entanglement and can thus be
described efficiently using a 1D tensor network (matrix-product state) despite
the presence of long-range interactions. The ground state can be obtained using
the density-matrix renormalization group (DMRG) algorithm, which is accurate up
to machine precision even for large nuclei, is numerically as cheap as the
widely used BCS (Bardeen-Cooper-Schrieffer) approach, and does not suffer from
any mean-field artifacts. We apply this framework to compute the even-odd mass
differences of all known lead isotopes from $^{178}$Pb to $^{220}$Pb in a very
large configuration space of 13 shells between the neutron magic numbers 82 and
184 (i.e., two major shells) and find good agreement with the experiment. We
also treat pairing with non-zero angular momentum and determine the lowest
excited states in the full configuration space of one major shell, which we
demonstrate for the $N=126$, $Z\geq 82$ isotones. To demonstrate the
capabilities of the method beyond low-lying excitations, we calculate the first
100 excited states of $^{208}$Pb with singlet pairing and the two-neutron
removal spectral function of $^{210}$Pb, which relates to a two-neutron pickup
experiment.
