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

TL;DR

This work presents a fault-tolerant quantum algorithm for simulating the Pauli-Breit Hamiltonian, which augments the nonrelativistic electronic Hamiltonian with one- and two-electron spin-orbit and spin-spin interactions. It introduces a symmetry-adapted, doubly factorized Majorana representation that enables efficient block-encoding and qubitization, with Spin-SWAP networks decoupling spin and orbital control to reduce overhead. Resource analyses show that explicit spin degrees of freedom do not worsen asymptotic scaling and yield up to a twofold prefactor reduction relative to direct LCU approaches. The proposed hybrid quantum–classical workflow targets photodynamic therapy design by providing first-principles calculations of spin-orbit effects, intersystem crossing rates, and related photophysical properties, which are then propagated into ML surrogates for high-throughput screening. Together, these advances create a scalable pathway for accurate relativistic molecular design in PDT and related spin-dependent photochemistry.

Abstract

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.

Figures (18)

• Figure 1: Spin-orbit induced electronic excitations between spin-orbitals and the respective coupling matrix elements.
• Figure 2: Addition of block-encoding of $\hat{H}^{(1)}$ represented by $U^{(0)}$ and $\hat{H}^{(2)}$ represented by $U^{(1)}$. Here $V$ is the Hadamard gate.
• Figure 3: Quantum circuit preparing a two-qubit state with amplitudes given by the corresponding square roots of Pauli matrix elements $\sqrt{P^{(\mu)}_{\sigma\rho}}$, $\mu=0,X,Y,Z$.
• Figure 4: Circuit primitives preparing two-qubit states with amplitudes given by the corresponding elements of the Pauli matrices $\mu=0,X,Y,Z$.
• Figure 5: a) Quantum circuit directly block encoding the factorized Hamiltonian given in eq. \ref{['eq:one-body-majorana-factorized1']}; b) Block-encoding circuit for the same Hamiltonian employing spin-swapping networks implemented via $\hat{S}^{(\pm)}_{\sigma}$.