Curator's Take
This research tackles one of the most challenging problems in quantum simulation: efficiently handling the notorious Coulomb interactions that govern electrons in atoms and molecules. The authors prove that Trotterization, a fundamental quantum algorithm for time evolution, can achieve rigorous convergence guarantees even when dealing with the mathematically unruly Coulomb potential - which is singular, long-ranged, and violates many standard assumptions used in quantum simulation analysis. Most importantly, they show the algorithm remains "quantumly efficient" with polynomial scaling in system size, suggesting that quantum computers could eventually simulate realistic chemical and materials systems without needing artificial smoothing of the problematic Coulomb singularities. This represents a crucial theoretical foundation for quantum chemistry applications, providing the mathematical rigor needed to confidently apply Trotterization to real molecular systems where Coulomb interactions dominate.
— Mark Eatherly
Summary
Efficiently simulating many-body quantum systems with Coulomb interactions is a fundamental question in quantum physics, quantum chemistry, and quantum computing, yet it presents unique challenges: the Hamiltonian is an unbounded operator (both kinetic and potential parts are unbounded); its Hilbert space dimension grows exponentially with particle number; and the Coulomb potential is singular, long-ranged, non-smooth, and unbounded, violating the regularity assumptions of many prior state-of-the-art many-body simulation analyses. In this work, we establish rigorous error bounds for Trotter formulas applied to many-body quantum systems with Coulomb interactions. Our first main result shows that for general initial conditions in the domain of the Hamiltonian, second-order Trotter achieves a sharp $1/4$ convergence rate with explicit polynomial dependence of the error prefactor on the particle number. The polynomial dependence on system size suggests that the algorithm remains quantumly efficient, even without introducing any regularization of the Coulomb singularity. Notably, although the result under general conditions constitutes a worst-case bound, this rate has been observed in prior work for the hydrogen ground state, demonstrating its relevance to physically and practically important initial conditions. Our second main result identifies a set of physically meaningful conditions on the initial state under which the convergence rate improves to first and second order. For hydrogenic systems, these conditions are connected to excited states with sufficiently high angular momentum. Our theoretical findings are consistent with prior numerical observations.