Talk 1 abstract:

How to Build Exact, Symmetry-Preserving Quantum Circuits (Paarth Jain)

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.

Talk 2 abstract:

Exact Unitary Transformations via Adjoint Representations (Praveen Jayakumar)

Exact unitary transformations are widely used in physics and chemistry, but they are often derived case by case, making it hard to tell when exact transformations exist and when they can be implemented efficiently. In this talk, I will present a general criterion for exact transformations based on the adjoint action of a unitary on a finite operator space. This viewpoint unifies familiar examples, including unitaries generated by Lie-algebra elements and by finite-spectrum operators, and it provides a systematic way to simplify the transformation itself.

I will then apply this framework to the transformation of fermionic strings and Pauli products under unitaries generated by fermionic excitations. This yields an exact and number-symmetry-preserving alternative to standard Pauli propagation methods. I will conclude by showing how the same perspective can be used to design generators whose transformations are known in closed form, with potential applications to variational quantum algorithms.