  Quantum algorithms for quantum chemistry map the dynamics of electrons in a
molecule to the dynamics of a coupled spin system. To reach chemical accuracy
for interesting molecules, a large number of quantum gates must be applied
which implies the need for quantum error correction and fault-tolerant quantum
computation. Arbitrary fault-tolerant operations can be constructed from a
small, universal set of fault-tolerant operations by gate compilation. Quantum
chemistry algorithms are compiled by decomposing the dynamics of the coupled
spin-system using a Trotter formula, synthesizing the decomposed dynamics using
Clifford operations and single-qubit rotations, and finally approximating the
single-qubit rotations by a sequence of fault-tolerant single-qubit gates.
Certain fault-tolerant gates rely on the preparation of specific single-qubit
states referred to as magic states. As a result, gate compilation and magic
state distillation are critical for solving quantum chemistry problems on a
quantum computer. We review recent progress that has improved the efficiency of
gate compilation and magic state distillation by orders of magnitude.
