Symmetry-guided algorithms for variational quantum optimization
Quantum numbers associated with symmetries classify quantum states and identify the sectors relevant to a given simulation. Generic hardware-efficient ansatzes do not necessarily preserve the quantum numbers and may therefore explore states outside the target sector. We develop symmetry-guided algorithms based on the quantum-accessible Hilbert-Schmidt asymmetry $\mathrm{A}_\mathrm{HS}$, which quantifies the deviation of a quantum state from invariance under a specified symmetry group. This quantity guides the optimization toward a highly reduced symmetry-based target sector, without enforcing the symmetry on the preparation circuit. We demonstrate the approach as a proof of principle in variational quantum eigensolver calculations targeting singlet ground states with collective SU(2) symmetry. The applications include a five-qubit $S$-wave $p+{}^3\mathrm{He}$ two-cluster Hamiltonian, deduced from $\textit{ab initio}$ many-body calculations, and collective spin models with up to 12 particles, which provide a many-particle benchmark of asymmetry-guided optimization. Across these applications, $\mathrm{A}_\mathrm{HS}$-based guidance increases the singlet probability. Overall, the proposed algorithms improve energy accuracy during optimization and ground-state fidelity compared with direct energy minimization, with substantial fidelity improvements in the larger spin systems. These results demonstrate the utility of an asymmetry objective determined by the group action, independently of the Hamiltonian energy, for improving the reliability of variational quantum optimization.