Symmetry-Adapted State Preparation for Quantum Chemistry on Fault-Tolerant Quantum Computers

Abstract: We present systematic and resource-efficient constructions of continuous symmetry projectors, particularly $U(1)$ particle number and $SU(2)$ total spin, tailored for fault-tolerant quantum computations. Our approach employs a linear combination of unitaries (LCU) as well as generalized quantum signal processing (GQSP and GQSVT) to implement projectors. These projectors can then be coherently applied as state filters prior to quantum phase estimation (QPE). We analyze their asymptotic gate complexities for explicit circuit realizations. For the particle number and $S_z$ symmetries, GQSP offers favorable resource usage features owing to its low ancilla qubit requirements and robustness to finite precision rotation gate synthesis. For the total spin projection, the structured decomposition of $\hat{P}_{S,M_S}$ reduces the projector T gate count. Numerical simulations show that symmetry filtering substantially increases the QPE success probability, leading to a lower overall cost compared to that of unfiltered approaches across representative molecular systems. Resource estimates further indicate that the cost of symmetry filtering is $3$ to $4$ orders of magnitude lower than that of the subsequent phase estimation step This advantage is especially relevant in large, strongly correlated systems, such as FeMoco, a standard strongly correlated open-shell benchmark. For FeMoco, the QPE cost is estimated at ${\sim}10^{10}$ T gates, while our symmetry projector requires only ${\sim}10^{6}$--$10^{7}$ T gates. These results establish continuous-symmetry projectors as practical and scalable tools for state preparation in quantum chemistry and provide a pathway toward realizing more efficient fault-tolerant quantum simulations.