Fault‐Tolerant Quantum Simulation of the Pauli‐Breit Hamiltonian for Ab Initio Hybrid Quantum‐Classical Molecular Design With Applications to Photodynamic Therapy

ABSTRACT Relativistic spin effects are driving subtle molecular processes ranging from intersystem crossing in photodynamic therapy to spin‐mediated catalysis and high‐resolution spectroscopy. These effects can be described by the Pauli‐Breit Hamiltonian, which extends the nonrelativistic electronic Hamiltonian by including explicit one‐ and two‐electron spin‐orbit and spin‐spin interactions. However, first‐principles simulations of the full Pauli‐Breit Hamiltonian quickly become intractable on classical computers due to the rapid growth of Hilbert space dimension and the complexity of two‐body spin‐dependent terms. In this work, we propose a fault‐tolerant quantum algorithm for computing molecular energy levels and properties governed by the Pauli‐Breit Hamiltonian. The central result of the paper is an explicit block‐encoding construction for the relativistic Hamiltonian in a second‐quantized, doubly factorized representation. By reformulating the Pauli‐Breit Hamiltonian in a symmetry‐adapted Majorana basis, we construct efficient linear‐combination‐of‐unitaries circuits that encode both one‐ and two‐electron spin‐orbit coupling without resorting to 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 a modest overhead relative to spin‐free electronic structure simulations. We analyze the resulting quantum resources in terms of logical qubit counts and T‐gate complexity, and show that the inclusion of spin degrees of freedom does not fundamentally worsen the asymptotic scaling. The prefactor in our approach is 2–4 lower than if the linear‐combination of unitaries technique was applied directly. The present manuscript focuses on explicit circuit constructions and scaling analysis, rather than molecule‐specific numerical resource estimates. To illustrate one prospective application, we outline a first‐principles hybrid quantum‐classical workflow for the rational design of photodynamic therapy photosensitizers, artificial photosynthesis catalysts, and other molecular systems where accurate treatment of relativistic spin effects is essential.