Research Highlight: Mathematical Analysis of Many-Body Quantum Simulation with Coulomb Potentials
IPAM - March 2026
Research Highlight: Mathematical Analysis of Many-Body Quantum Simulation with Coulomb Potentials
By
Di Fang
Department of Mathematics
Duke Quantum Center, Duke University
1. Introduction
Efficient simulation of many-body quantum systems, originally proposed as a primary motivation for building quantum computers, remains one of the most fundamental tasks in quantum computing.
At a theoretical level, the central objective is to approximate the time evolution generated by a Hamiltonian operator—the mathematical object governing the dynamics of a quantum system—using properly designed quantum algorithms and circuits with provable complexity guarantees. A simulation is considered efficient if its computational cost grows only polynomially in the number of particles (i.e., the system size); exponential scaling would render even quantum computation impractical for large systems. Polynomial dependence on system size is therefore closely connected to the broader concept of quantum advantage. This leads to a fundamental question:
Can one rigorously demonstrate that a (quantum) algorithm can simulate realistic many-body quantum systems with cost scaling polynomially in the system size?
Recent years have witnessed substantial progress in the theory of many-body quantum simulation, particularly for systems whose Hamiltonians are bounded or whose potential terms satisfy suitable regularity assumptions. These advances have established polynomial scaling for a variety of important applications.
However, some physically fundamental systems fall outside this framework. A canonical example is the many-body Schrödinger operator with Coulomb interactions—the first-principles description of electronic and molecular dynamics. Mathematically, such Hamiltonians consist of a kinetic term (a Laplacian operator) and an interaction potential of the form 1/|x|, both of which are unbounded operators. Moreover, the Coulomb interaction is singular, long-ranged, non-smooth, and itself unbounded, violating the regularity assumptions underlying prior many-body simulation analyses.
Why does this pose a fundamental difficulty? Most existing quantum simulation algorithms exhibit cost dependence on some form of operator norm of the Hamiltonian. For post-Trotter–type methods—such as quantum signal processing (QSP), quantum singular value transformation (QSVT), and truncated Dyson or Magnus series—the circuit complexity typically scales polynomially with the operator norm of the Hamiltonian. For Trotter formulas, the error analysis instead depends on operator norms of commutators. In the Coulomb case, however, these operator norms are not bounded in the ambient Hilbert space. Consequently, the widely used finite-dimensional analysis techniques and the associated intuition break down.
2. About Trotterization
A central tool in dynamics simulation is the Trotter product formula – also known as Trotterization, product formulas, or operator splitting in different communities. Its origins trace back to Lie’s product formula and Trotter’s work on semigroups, and it was later developed into a practical computational method in numerical analysis through Strang’s second-order splitting and Suzuki’s higher-order generalizations. In the quantum setting, Lloyd proposed its use as a quantum algorithm for simulating quantum dynamics.
At its core, the method approximates the evolution generated by a Hamiltonian by alternately evolving under each component for short time intervals. For a Hamiltonian H = A + B and a total evolution time t divided into L steps, the first-order Trotter formula takes the form
By
Di Fang
Department of Mathematics
Duke Quantum Center, Duke University
1. Introduction
Efficient simulation of many-body quantum systems, originally proposed as a primary motivation for building quantum computers, remains one of the most fundamental tasks in quantum computing.
At a theoretical level, the central objective is to approximate the time evolution generated by a Hamiltonian operator—the mathematical object governing the dynamics of a quantum system—using properly designed quantum algorithms and circuits with provable complexity guarantees. A simulation is considered efficient if its computational cost grows only polynomially in the number of particles (i.e., the system size); exponential scaling would render even quantum computation impractical for large systems. Polynomial dependence on system size is therefore closely connected to the broader concept of quantum advantage. This leads to a fundamental question:
Can one rigorously demonstrate that a (quantum) algorithm can simulate realistic many-body quantum systems with cost scaling polynomially in the system size?
Recent years have witnessed substantial progress in the theory of many-body quantum simulation, particularly for systems whose Hamiltonians are bounded or whose potential terms satisfy suitable regularity assumptions. These advances have established polynomial scaling for a variety of important applications.
However, some physically fundamental systems fall outside this framework. A canonical example is the many-body Schrödinger operator with Coulomb interactions—the first-principles description of electronic and molecular dynamics. Mathematically, such Hamiltonians consist of a kinetic term (a Laplacian operator) and an interaction potential of the form 1/|x|, both of which are unbounded operators. Moreover, the Coulomb interaction is singular, long-ranged, non-smooth, and itself unbounded, violating the regularity assumptions underlying prior many-body simulation analyses.
Why does this pose a fundamental difficulty? Most existing quantum simulation algorithms exhibit cost dependence on some form of operator norm of the Hamiltonian. For post-Trotter–type methods—such as quantum signal processing (QSP), quantum singular value transformation (QSVT), and truncated Dyson or Magnus series—the circuit complexity typically scales polynomially with the operator norm of the Hamiltonian. For Trotter formulas, the error analysis instead depends on operator norms of commutators. In the Coulomb case, however, these operator norms are not bounded in the ambient Hilbert space. Consequently, the widely used finite-dimensional analysis techniques and the associated intuition break down.
2. About Trotterization
A central tool in dynamics simulation is the Trotter product formula – also known as Trotterization, product formulas, or operator splitting in different communities. Its origins trace back to Lie’s product formula and Trotter’s work on semigroups, and it was later developed into a practical computational method in numerical analysis through Strang’s second-order splitting and Suzuki’s higher-order generalizations. In the quantum setting, Lloyd proposed its use as a quantum algorithm for simulating quantum dynamics.
At its core, the method approximates the evolution generated by a Hamiltonian by alternately evolving under each component for short time intervals. For a Hamiltonian H = A + B and a total evolution time t divided into L steps, the first-order Trotter formula takes the form
\(
e^{-iHt} \approx \left( e^{-iAt/L} e^{-iBt/L} \right)^L
\)
For bounded operators, classical commutator estimates yield clean convergence guarantees. In the unbounded Coulomb setting, however, these classical arguments no longer apply.
3. Our contributions
In our recent work ([1] Fang–Wu–Soffer, 2025), we provide a rigorous analysis of first-order Trotterization for many-body Coulomb Hamiltonians, treating the singular interaction directly and without modification. Our formulation allows Coulomb interactions that are attractive, repulsive, or any combination thereof, thereby covering both electronic and molecular systems. Moreover, the result holds for all wavefunctions in the domain of the Hamiltonian—that is, for all states for which the Schrödinger operator is well defined. The main result establishes two findings.
1. Polynomial system-size dependence.
We prove that the Trotter error grows only polynomially in the number of particles, despite the unbounded and singular nature of the Hamiltonian. In particular, the convergence remains quantumly efficient. Our analysis treats the Coulomb potential as an unbounded operator and does not rely on spatial discretization, making it compatible with various circuit constructions.
2. Sharp 1/4-order convergence rate.
We show that the time-step convergence rate is exactly of order 1/4, and that this rate is optimal. Remarkably, it is saturated by the ground state of the hydrogen atom in [2], demonstrating its physical relevance. Our theoretical result also matches prior numerical findings.
Ongoing work ([3] Fang–Wu, in preparation) extends the analysis to second-order Trotter formulas, where a similar 1/4 convergence rate with polynomial dependence on the system size persists in the general conditions. At the same time, improved first- and second-order convergence can be recovered for certain physically meaningful classes of initial states related to excitation.
4. Mathematical ideas
The proof departs fundamentally from standard commutator-based arguments widely used in quantum information community. Since the relevant operators are unbounded and defined only on proper domains, classical norm estimates fail. Our approach introduces three new mathematical ingredients, all of which may be of independent interest:
• an alternative exact error representation suited to unbounded generators,
• Sobolev-space estimates that track system-size dependence,
• and refined counting arguments for Coulomb singularities related to Hardy–Littlewood–Sobolev–type structures.
5. Why it matters mathematically?
Beyond its implications for quantum simulation, this work clarifies how product formulas behave for singular differential operators in infinite-dimensional settings. It shows that unboundedness does not preclude rigorous convergence — but it may fundamentally alter the rate and structure of approximation. More broadly, it illustrates how mathematical tools from PDE and functional analysis can play a foundational role in understanding the capabilities and limitations of quantum simulation algorithms.
References:
1. On the Trotter Error in Many-body Quantum Dynamics with Coulomb Potentials, D. Fang, X. Wu, A. Soffer, arXiv:2507.22707, Comm. Math. Phys., to appear.
2. Strong error bounds for Trotter and Strang-splittings and their implications for quantum chemistry D. Burgarth, P. Facchi, A. Hahn, M. Johnsson, K. Yuasa, Physical Review Research 6 (4), 043155 and QIP2024.
3. Trotterization with Many-body Coulomb Interactions: Convergence for General Initial Conditions and State-Dependent Improvements, D. Fang, X. Wu, in prep.
3. Our contributions
In our recent work ([1] Fang–Wu–Soffer, 2025), we provide a rigorous analysis of first-order Trotterization for many-body Coulomb Hamiltonians, treating the singular interaction directly and without modification. Our formulation allows Coulomb interactions that are attractive, repulsive, or any combination thereof, thereby covering both electronic and molecular systems. Moreover, the result holds for all wavefunctions in the domain of the Hamiltonian—that is, for all states for which the Schrödinger operator is well defined. The main result establishes two findings.
1. Polynomial system-size dependence.
We prove that the Trotter error grows only polynomially in the number of particles, despite the unbounded and singular nature of the Hamiltonian. In particular, the convergence remains quantumly efficient. Our analysis treats the Coulomb potential as an unbounded operator and does not rely on spatial discretization, making it compatible with various circuit constructions.
2. Sharp 1/4-order convergence rate.
We show that the time-step convergence rate is exactly of order 1/4, and that this rate is optimal. Remarkably, it is saturated by the ground state of the hydrogen atom in [2], demonstrating its physical relevance. Our theoretical result also matches prior numerical findings.
Ongoing work ([3] Fang–Wu, in preparation) extends the analysis to second-order Trotter formulas, where a similar 1/4 convergence rate with polynomial dependence on the system size persists in the general conditions. At the same time, improved first- and second-order convergence can be recovered for certain physically meaningful classes of initial states related to excitation.
4. Mathematical ideas
The proof departs fundamentally from standard commutator-based arguments widely used in quantum information community. Since the relevant operators are unbounded and defined only on proper domains, classical norm estimates fail. Our approach introduces three new mathematical ingredients, all of which may be of independent interest:
• an alternative exact error representation suited to unbounded generators,
• Sobolev-space estimates that track system-size dependence,
• and refined counting arguments for Coulomb singularities related to Hardy–Littlewood–Sobolev–type structures.
5. Why it matters mathematically?
Beyond its implications for quantum simulation, this work clarifies how product formulas behave for singular differential operators in infinite-dimensional settings. It shows that unboundedness does not preclude rigorous convergence — but it may fundamentally alter the rate and structure of approximation. More broadly, it illustrates how mathematical tools from PDE and functional analysis can play a foundational role in understanding the capabilities and limitations of quantum simulation algorithms.
References:
1. On the Trotter Error in Many-body Quantum Dynamics with Coulomb Potentials, D. Fang, X. Wu, A. Soffer, arXiv:2507.22707, Comm. Math. Phys., to appear.
2. Strong error bounds for Trotter and Strang-splittings and their implications for quantum chemistry D. Burgarth, P. Facchi, A. Hahn, M. Johnsson, K. Yuasa, Physical Review Research 6 (4), 043155 and QIP2024.
3. Trotterization with Many-body Coulomb Interactions: Convergence for General Initial Conditions and State-Dependent Improvements, D. Fang, X. Wu, in prep.