Fault-tolerant quantum simulation of the Pauli-Breit Hamiltonian for ab initio hybrid quantum-classical molecular design with applications to photodynamic therapy

Relativistic spin effects drive subtle molecular phenomena ranging from intersystem crossing in photodynamic therapy to spin-mediated catalysis and high-resolution spectroscopy. These effects are described by the Pauli-Breit Hamiltonian, which extends the nonrelativistic electronic Hamiltonian by including one- and two-electron spin-orbit and spin-spin interactions. First-principles simulations of the full Pauli-Breit Hamiltonian rapidly become intractable on classical computers due to the exponential growth of the Hilbert space and the complexity of two-body spin-dependent terms. We propose a fault-tolerant quantum algorithm for computing molecular energy levels and properties governed by the Pauli-Breit Hamiltonian. Our approach block-encodes the relativistic Hamiltonian in a second-quantized, doubly factorized representation. By reformulating the Hamiltonian in a symmetry-adapted Majorana basis, we construct efficient linear-combination-of-unitaries circuits that encode spin-orbit interactions without effective or mean-field approximations. We introduce spin-controlled Pauli-SWAP networks that decouple spin and orbital control logic, enabling a unified treatment of relativistic spin mixing with only modest overhead relative to spin-free simulations. We analyze quantum resources in terms of logical qubits and T-gate complexity, showing that explicit spin degrees of freedom do not worsen the asymptotic scaling. The prefactor is reduced by a factor of two compared to direct linear-combination-of-unitaries approaches. Finally, we outline a hybrid quantum-classical workflow for designing photodynamic therapy photosensitizers, artificial photosynthesis catalysts, and other systems where accurate relativistic spin effects are essential.