Spin-Adapted Fermionic Unitaries: From Lie Algebras to Compact Quantum Circuits

Abstract: Conservation of symmetries plays a crucial role in both classical and quantum simulations of many-body systems, enabling the tracking of states with specific symmetry properties and leading to substantial reductions in the number of optimization parameters. The design of efficient quantum circuits that enforce all symmetries typically encountered in chemistry has remained elusive, mainly due to the interplay of point group and spin symmetries. By exploiting Lie algebraic techniques, we derive exact product formulas representing symmetry-adapted unitaries. These decompositions allow us to design the most efficient symmetry-preserving quantum circuits to date. Finally, we introduce a minimum universal symmetry-adapted operator pool to further reduce the required quantum resources.