arXiv:2601.18898 · International Journal of Quantum Chemistry
Fault-tolerant quantum simulation of the Pauli-Breit Hamiltonian for ab initio hybrid quantum-classical molecular design with applications to photodynamic therapy
Emil Zak
quant-phphysics.chem-phSubmitted Jan 26, 2026• Announced Jan 28, 2026• 0 citations
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.
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.
Photodynamic therapy (PDT) is a targeted cancer treatment that uses light-activated photosensitizers to generate reactive oxygen species that selectively destroy tumor cells, generally causing less collateral damage than conventional treatments. However, its clinical success hinges on the availability of photosensitizers with strong optical sensitivity and high efficiency in generating reactive oxygen species. While classical computational methods have provided useful insights into photosensitizer design, they struggle to scale and often lack the accuracy needed for these simulations. In this work, we show how fault-tolerant quantum algorithms can be used to identify promising photosensitizer candidates for PDT. To predict photosensitizer performance, we assess two computational properties. First, we quantify light sensitivity by calculating the cumulative absorption in the therapeutic window with a threshold projection algorithm. Second, we determine the efficiency of reactive oxygen generation by estimating intersystem crossing (ISC) rates using the evolution-proxy approach, complemented by a vibronic dynamic treatment where appropriate. We apply these algorithms to a clinically relevant and actively pursued class of photosensitizers, BODIPY derivatives, including heavy-atom and transition-metal-substituted systems that are challenging for classical methods. Our resource estimates, obtained with PennyLane, suggest that systems with active spaces ranging from 11 to 45 spatial orbitals can be simulated using 180-350 logical qubits and Toffoli gate depths between 107 and 109, placing our algorithms within reach of realistic fault-tolerant quantum devices. This paves the way to an efficient quantum-based workflow for designing photosensitizers that can accelerate the discovery of new PDT agents.
Abstract We develop and analyze a fault-tolerant quantum algorithm for non-linear response properties of molecular and condensed phase systems. We consider a semi-classical description in which the electronic degrees of freedom are treated quantum mechanically and the light is treated as a classical field interacting via the electric dipole approximation. We use the technique of eigenstate filtering, to efficiently resolve excitation energies for dominant dipole transitions. When applied to the electronic structure Hamiltonian with double factorized representation, each significant spectral line can be approximated to a width of ±γ, and to a height of ±ϵ with O(N6η2γ−1ϵ−1log(1/ϵ)) O N 6 η 2 γ − 1 ϵ − 1 log ( 1 / ϵ ) queries to the block encoding of the unperturbed electronic structure Hamiltonian for η electrons over N basis functions. These quantities can be used to compute the nth order response functions for non-linear spectroscopies under limited assumptions using O(N5n+1ηn+1/γnϵ) O ̃ N 5 n + 1 η n + 1 / γ n ϵ queries to the block encoding of the Hamiltonian.
In this work, we present a quantum algorithm for direct first-principles simulation of electron-nuclear dynamics on a first-quantized real-space grid. Our algorithm achieves best-in-class efficiency for block-encoding the pre-Born-Oppenheimer molecular Hamiltonian by harnessing the linear scaling of swap networks for implementing the quadratic number of particle interactions, while using a novel alternating sign implementation of the Coulomb interaction that exploits highly optimized arithmetic routines. We benchmark our approach for a series of scientifically and industrially relevant chemical reactions. We demonstrate over an order-of-magnitude reduction in costs compared to previous state-of-the-art for the NH3+BF3 reaction, achieving a Toffoli cost of 8.7×109 per femtosecond using 1362 logical qubits (system + ancillas). Our results significantly lower the resources required for fault-tolerant simulations of photochemical reactions, while providing a suite of algorithmic primitives that are expected to serve as foundational building blocks for a broader class of quantum algorithms.
We develop and analyze a fault-tolerant quantum algorithm for computing n-th order response properties necessary for analysis of non-linear spectroscopies of molecular and condensed phase systems. We use a semi-classical description in which the electronic degrees of freedom are treated quantum mechanically and the light is treated as a classical field. The algorithm we present can be viewed as an implementation of standard perturbation theory techniques, focused on {\it ab initio} calculation of n-th order response functions. We provide cost estimates in terms of the number of queries to the block encoding of the unperturbed Hamiltonian, as well as the block encodings of the perturbing dipole operators. Using the technique of eigenstate filtering, we provide an algorithm to extract excitation energies to resolution γ, and the corresponding linear response amplitude to accuracy ε using O(N6η2γ−1ε−1log(1/ε)) queries to the block encoding of the unperturbed Hamiltonian H0, in double factorized representation. Thus, our approach saturates the Heisenberg O(γ−1) limit for energy estimation and allows for the approximation of relevant transition dipole moments. These quantities, combined with sum-over-states formulation of polarizabilities, can be used to compute the n-th order susceptibilities and response functions for non-linear spectroscopies under limited assumptions using O(N5n+1ηn+1/γnε) queries to the block encoding of H0.
While quantum computers promise to solve classically intractable problems, identifying the point at which fault-tolerant quantum computation outperforms the best classical algorithms for practical applications remains an outstanding challenge. Here we establish a concrete quantum-classical crossover for quantum many-body dynamics under realistic hardware conditions. We introduce a scalable fault-tolerant framework that combines coherent observable estimation with a space-time-efficient implementation of non-Clifford rotations, suppressing the residual logical errors that limit existing partially fault-tolerant approaches. A benchmark against state-of-the-art tensor-network and variational Monte Carlo algorithms reveals a concrete crossover for mixed-field Ising dynamics at modest system sizes. For a physical error rate of p=10−3, fault-tolerant simulation requires approximately 2 hours and 3.7×105 physical qubits for a 100-site 1D system, whereas tensor network approaches would require about 100 years. For 2D models, where rapid entanglement growth limits the classical evolution time, we project quantum runtimes within minutes. A physical error rate of p=10−4 leads to at least an order of magnitude reduction in qubit count (3.1×104 physical qubits) and runtime (minutes for 1D and seconds for 2D). The reduction in quantum runtime arises from our improved rotation-state injection and co-design of quantum error correction and observable-estimation protocols, which jointly suppress logical-error accumulation and reduce sampling overhead. Our results establish a scalable route towards practical quantum advantage and identify quantitative engineering targets for future fault-tolerant architectures.
The first-principles quantum simulation of the full electronic Hamiltonian of a material model of realistic size appears daunting even in a future era of fully fault-tolerant quantum computers. In contrast, classical simulations of material excited states rely on heuristic quasiparticle frameworks, such as the GW approximation and Bethe-Salpeter equation, where only a few quasiparticles are explicitly considered. Here we give a detailed description of quasiparticle quantum simulation for multi-excitonic states within the BSE. We develop an explicit first-quantized, orbital-based block encoding of the multi-exciton BSE Hamiltonian for qubitization, and reduce its cost using integral factorization and crystal symmetries. For a materials model of size N, and a fixed number of quasiparticles, the resulting algorithms require O(N) Toffoli gates and O(logN) logical qubits, or, using space-time tradeoffs, O(N) Toffoli gates and O(N) logical qubits, achieving up to a sixth power reduction in Toffoli count, or up to exponential reduction in qubit count, versus standard full Hamiltonian simulation techniques that have been explored in materials simulation. Resource estimates for the classically challenging problem of triexcitonic eigenvalue estimation across a range of semiconductor models of a realistic physical size of O(103) atoms, show that our Toffoli-optimized implementation reduces the Toffoli count by 106−107 relative to full Hamiltonian simulation estimates, while a qubit-optimized implementation requires only a few hundred logical qubits. More broadly, these results show that classical heuristic frameworks will be essential in the design of future fault-tolerant quantum algorithms for materials.