Variational quantum eigensolvers and quantum phase estimation prepare electronic wavefunctions using parameterized unitary transformations. To obtain physically meaningful results, these unitaries must preserve molecular symmetries, especially spin. In practice, however, the generators of spin-preserving transformations are sums of non-commuting terms, so naive decompositions can break the very symmetry they are meant to enforce.

In this talk, I will show that spin-adapted fermionic operators generate small compact Lie algebras, and that this structure leads to exact decompositions of the corresponding unitaries into simple implementable rotations. The key tool is the adjoint representation, which turns the problem into a low-dimensional matrix calculation rather than one in an exponentially large operator space. For some algebras, the decomposition is fully analytic and requires no numerical optimization. This framework gives a systematic way to build exact, symmetry-preserving quantum circuits.