  A number of exciting recent results have been seen in the field of quantum
error correction. These include initial demonstrations of error correction on
current quantum hardware, and resource estimates which improve understanding of
the requirements to run large-scale quantum algorithms for real-world
applications. In this work, we bridge the gap between these two developments by
performing careful estimation of the resources required to fault-tolerantly
perform quantum phase estimation (QPE) on a minimal chemical example.
Specifically, we describe a detailed compilation of the QPE circuit to lattice
surgery operations for the rotated surface code, for a hydrogen molecule in a
minimal basis set. We describe a number of optimisations at both the
algorithmic and error correction levels. We find that implementing even a
simple chemistry circuit requires 1,000 qubits and 2,300 quantum error
correction rounds, emphasising the need for improved error correction
techniques specifically targeting the early fault-tolerant regime.
