  Despite being a well-established operational approach to quantify
entanglement, R\'enyi entropy calculations have been plagued by their
computational complexity. We introduce here a theoretical framework based on an
optimal thermodynamic integration scheme, where the R\'enyi entropy can be
efficiently evaluated using regularizing paths. This approach avoids slowly
convergent fluctuating contributions and leads to low-variance estimates. In
this way, large system sizes and high levels of entanglement in model or
first-principles Hamiltonians are within our reach. We demonstrate it in the
one-dimensional quantum Ising model and perform the evaluation of entanglement
entropy in the formic acid dimer, by discovering that its two shared protons
are entangled even above room temperature.
