Exact Factorization of Unitary Transformations with Spin-Adapted Generators

TL;DR

This paper introduces an exact, efficient method to factorize spin-adapted unitaries into Pauli operator exponentials, enabling symmetry-preserving quantum circuits for electronic structure simulations.

Contribution

It presents a novel Lie algebra-based factorization approach that reformulates the problem as a low-dimensional nonlinear optimization, improving quantum circuit design for spin symmetry preservation.

Findings

Provides a practical strategy for constructing symmetry-conserving quantum circuits.

Enables precise numerical reparametrization of unitaries without symbolic manipulation.

Reduces implementation cost while maintaining accurate electronic state representation.

Abstract

Preserving spin symmetry in variational quantum algorithms is essential for producing physically meaningful electronic wavefunctions. Implementing spin-adapted transformations on quantum hardware, however, is challenging because the corresponding fermionic generators translate into noncommuting Pauli operators. In this work, we introduce an exact and computationally efficient factorization of spin-adapted unitaries derived from fermionic double excitation and deexcitation rotations. These unitaries are expressed as ordered products of exponentials of Pauli operators. Our method exploits the fact that the elementary operators in these generators form small Lie algebras. By working in the adjoint representation of these algebras, we reformulate the factorization problem as a low-dimensional nonlinear optimization over matrix exponentials. This approach enables precise numerical reparametrization of the unitaries without relying on symbolic manipulations. The proposed factorization provides a practical strategy for constructing symmetry-conserving quantum circuits within variational algorithms. It preserves spin symmetry by design, reduces implementation cost, and ensures the accurate representation of electronic states in quantum simulations of molecular systems.