Trotterization with Many-body Coulomb Interactions: Convergence for General Initial Conditions and State-Dependent Improvements
Di Fang, Xiaoxu Wu
Abstract
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 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.
AI Impact Assessments
(3 models)Scientific Impact Assessment
Core Contribution
This paper establishes rigorous error bounds for Trotter product formulas applied to many-body quantum systems with Coulomb interactions — a setting where the Hamiltonian is unbounded, the potential is singular, long-ranged, and non-smooth, violating regularity assumptions of most prior Trotter error analyses. The paper makes two main contributions:
1. Sharp 1/4 convergence rate for second-order Trotter (Strang splitting) with explicit polynomial dependence (N^4.5) on particle number, for general initial states in the domain of the Hamiltonian. This extends the authors' prior work on first-order Trotter to the second-order case and demonstrates that increasing the Trotter order does not improve convergence rates for Coulomb systems under general initial conditions — a fundamentally different behavior from bounded-operator settings.
2. State-dependent improved convergence rates: Under physically motivated regularity conditions on the initial state (related to angular momentum and behavior near particle coalescence), the convergence rates recover to first order (for Lie-Trotter) and second order (for Strang splitting). For hydrogen eigenstates, these conditions connect to excited states with sufficiently high angular momentum quantum numbers.
Methodological Rigor
The mathematical analysis is highly rigorous and technically sophisticated. Key methodological strengths include:
The consistency of theoretical results with prior numerical observations (particularly the 1/4 rate observed for the hydrogen ground state in [38]) provides strong validation.
Potential Impact
Quantum computing and simulation: The polynomial N-dependence of the error prefactor (N^4.5) demonstrates that Trotterization remains quantumly efficient for Coulomb systems without requiring regularization of the singularity. This is practically important for quantum simulation of molecular and electronic structure problems.
Quantum chemistry: The state-dependent analysis provides actionable guidance — for states with high angular momentum components, Trotter methods perform significantly better. This could inform algorithm design and resource estimation for quantum chemistry applications.
Mathematical physics: The Sobolev regularity preservation result (Theorem 14) may have independent mathematical interest beyond quantum simulation. The paper also advances understanding of how unbounded operators fundamentally differ from bounded ones in approximation theory.
Numerical analysis: The exact error representations and the smooth cutoff technique contribute to the broader toolkit for analyzing product formulas with singular potentials.
Timeliness & Relevance
This work addresses a current bottleneck in quantum simulation: rigorously understanding Trotter error for physically realistic Hamiltonians. Most prior Trotter error analyses assume bounded operators or smooth potentials, yet the Coulomb interaction — the most important interaction in chemistry and materials science — violates these assumptions. As quantum hardware advances toward practical quantum chemistry applications, rigorous resource estimates for realistic systems become increasingly important.
The work builds naturally on the authors' prior first-order analysis [37] and fills a clear gap by extending to second-order Trotter and establishing state-dependent improvements.
Strengths & Limitations
Key Strengths:
Notable Limitations:
Overall Assessment
This is a mathematically deep and physically well-motivated paper that significantly advances our rigorous understanding of quantum simulation for realistic Coulomb systems. It provides the first proof that the 1/4 rate degradation persists even for higher-order Trotter formulas, while simultaneously identifying conditions under which this limitation can be overcome. The work occupies an important niche at the intersection of quantum computing theory, PDE analysis, and quantum chemistry, and its results should influence both theoretical complexity analysis and practical algorithm design for quantum simulation.
Generated Apr 10, 2026
Comparison History (40)
While Paper 1 provides rigorous mathematical bounds for quantum simulation, Paper 2 addresses a critical bottleneck in fault-tolerant quantum computing. By significantly reducing space and time overhead in lattice surgery, Paper 2 directly accelerates the practical realization of scalable, error-corrected quantum computers, offering broader and more immediate technological impact.
Paper 2 likely has higher impact due to strong methodological rigor (rigorous, explicit Trotter error bounds for unbounded, singular Coulomb Hamiltonians), high timeliness for quantum simulation/quantum chemistry, and broad applicability across physics, chemistry, and quantum computing. It addresses a key bottleneck—provable convergence without Coulomb regularization and with polynomial system-size dependence—directly relevant to near-term and fault-tolerant quantum algorithms. Paper 1 is novel and useful for generating tunable-entanglement random states for classical simulation, but its immediate real-world algorithmic/experimental leverage and cross-field reach appear narrower.
Paper 1 addresses a fundamental and notoriously difficult theoretical problem in quantum simulation: handling the unbounded and singular nature of Coulomb interactions. By providing rigorous error bounds and polynomial scaling for Trotterization, it significantly advances quantum chemistry and physics applications, which are considered the most promising near-term use cases for quantum computers. While Paper 2 offers a practical systems-level solution for modular QPUs, Paper 1 provides foundational mathematical guarantees that will durably impact the core algorithmic theory of quantum simulation across multiple disciplines.
Paper 1 addresses a fundamental computational challenge in quantum simulation—rigorously bounding Trotter errors for many-body Coulomb systems with unbounded, singular potentials. This has broad impact across quantum computing, quantum chemistry, and condensed matter physics, providing the first sharp convergence guarantees without regularization. The methodological rigor and practical relevance to quantum algorithm efficiency give it high impact potential. Paper 2 presents an interesting extension of the HOM effect to non-local settings, but is more incremental in scope, primarily advancing understanding within quantum optics rather than opening broadly impactful new directions.
Paper 2 addresses a fundamental problem in quantum computing—efficient simulation of many-body Coulomb systems—with rigorous mathematical results that bridge quantum chemistry, physics, and computer science. It tackles technically challenging unbounded operators with singular potentials, establishes sharp convergence rates with explicit system-size dependence, and demonstrates quantum efficiency without regularization. This has broad implications for practical quantum simulation algorithms. Paper 1, while providing elegant insights connecting subradiance and Anderson localization in waveguide QED, addresses a more specialized topic with narrower cross-disciplinary impact.
Paper 1 addresses a fundamental bottleneck in quantum computing and chemistry: efficiently simulating many-body Coulomb interactions. Providing rigorous error bounds and proving polynomial scaling for such complex systems has profound implications for demonstrating practical quantum advantage. Paper 2 proposes an interesting quantum thermal device, but its scope and applicability are much narrower. Paper 1's rigorous theoretical contributions are therefore more likely to achieve broad, cross-disciplinary scientific impact.
Paper 1 addresses a fundamental problem in quantum computing—rigorous error bounds for Trotterization with Coulomb interactions—which is central to quantum simulation of real physical/chemical systems. It resolves mathematical challenges (unbounded operators, singular potentials) that have limited prior analyses, establishing sharp convergence rates with explicit particle-number dependence. This has broad implications for quantum chemistry and many-body physics simulations. Paper 2 demonstrates a proof-of-concept quantum optical classifier, but it is limited to simple perceptron/shallow networks on benchmark datasets, with unclear scalability advantages over classical methods for practical problems.
Paper 2 addresses a fundamental problem in quantum computing and quantum chemistry—rigorous Trotter error bounds for many-body Coulomb systems—with broad implications for quantum simulation algorithms. It tackles long-standing mathematical challenges (unbounded operators, singular potentials) and provides sharp convergence rates with explicit particle-number dependence, directly relevant to practical quantum computing applications. Paper 1 contributes interesting results on quantum battery charging via dissipation, but addresses a narrower topic with less immediate practical impact. Paper 2's methodological rigor and relevance to quantum algorithm development give it broader cross-disciplinary significance.
Paper 2 addresses a fundamental and practically important problem in quantum computing and quantum chemistry—rigorous Trotter error bounds for many-body Coulomb systems with unbounded, singular potentials. It provides sharp convergence rates with explicit particle-number dependence and state-dependent improvements, filling a significant gap in quantum simulation theory. Its mathematical rigor, direct relevance to quantum algorithm design, and broad applicability across quantum physics, chemistry, and computing give it higher potential impact. Paper 1, while conceptually interesting in deriving classical-quantum dynamics from open quantum theory, uses a toy model and addresses a more niche foundational question.
While Paper 1 makes rigorous and important contributions to quantum simulation theory (Trotter error bounds for Coulomb systems), it addresses a relatively specialized technical problem. Paper 2 demonstrates a fundamentally counterintuitive result—ghost imaging using zero photons—which challenges conventional understanding of imaging and quantum/classical correlations. Its conceptual novelty, accessibility, potential to resolve debates in quantum optics, and broader appeal across quantum information, imaging, and foundations of physics give it higher potential for scientific impact and cross-disciplinary influence.