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arXiv:2610.25260

Spatial Discretization Error for Quantum Chemistry Simulation

Di Fang, Jiaqi Zhang

quant-phmath-phmath.NASubmitted Oct 18, 2026• Announced Oct 20, 2026• 0 citations
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Abstract

Nonrelativistic quantum chemistry is based on the Schrödinger equation in continuous, unbounded space. Computational approaches on classical and quantum computers often truncate the spatial domain and introduce a finite-dimensional basis, reducing the continuum problem to a finite-dimensional one. There is therefore a fundamental question: to what extent does this approximation capture the original continuum problem? In this work, we develop a general Sobolev-Galerkin framework for estimating spatial discretization errors in quantum simulation with unbounded Hamiltonians. We apply this framework to many-body Coulomb systems and establish both periodization and Galerkin error estimates for wave-function dynamics and ground-state energies. For dynamics, the finite-basis projection error of the exact wave function does not capture the full basis-discretization error: an additional dynamical projection error enters our bound at a different rate, and we introduce an inner-product argument to control this contribution. Interestingly, the spatial convergence rates for wave-function dynamics and ground-state energies differ quantitatively. For ground-state energies, we establish a sharp $M^{-1}$ convergence rate at fixed cell size, where $M$ is the number of one-particle basis functions. We also show that a uniform $M^{-1}$ convergence rate is impossible for the corresponding real-time wave-function error with general $H^2$ initial data. The analysis relies on many-body Sobolev estimates with explicit polynomial dependence on system size. Combining these estimates with state-of-the-art quantum algorithms, we obtain, to our knowledge, the first rigorous end-to-end complexity bounds for quantum simulation of molecular and electronic dynamics with all spatial errors explicitly controlled. The number of basis functions is determined directly from the continuum problem, without regularization.

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Nonrelativistic quantum chemistry is based on the Schrödinger equation in continuous, unbounded space. Computational approaches on classical and quantum computers often truncate the spatial domain and introduce a finite-dimensional basis, reducing the continuum problem to a finite-dimensional one. There is therefore a fundamental question: to what extent does this approximation capture the original continuum problem? In this work, we develop a general Sobolev-Galerkin framework for estimating spatial discretization errors in quantum simulation with unbounded Hamiltonians. We apply this framework to many-body Coulomb systems and establish both periodization and Galerkin error estimates for wave-function dynamics and ground-state energies. For dynamics, the finite-basis projection error of the exact wave function does not capture the full basis-discretization error: an additional dynamical projection error enters our bound at a different rate, and we introduce an inner-product argument to control this contribution. Interestingly, the spatial convergence rates for wave-function dynamics and ground-state energies differ quantitatively. For ground-state energies, we establish a sharp M−1M^{-1}M−1 convergence rate at fixed cell size, where MMM is the number of one-particle basis functions. We also show that a uniform M−1M^{-1}M−1 convergence rate is impossible for the corresponding real-time wave-function error with general H2H^2H2 initial data. The analysis relies on many-body Sobolev estimates with explicit polynomial dependence on system size. Combining these estimates with state-of-the-art quantum algorithms, we obtain, to our knowledge, the first rigorous end-to-end complexity bounds for quantum simulation of molecular and electronic dynamics with all spatial errors explicitly controlled. The number of basis functions is determined directly from the continuum problem, without regularization.
In quantum simulation of electronic structure it is often expected that the estimate of the ground energy converges to the continuum basis limit at a rate of O(N−1)\mathcal O(N^{-1})O(N−1) for a basis of size NNN. This has been found in analytical and numerical studies, but without a rigorous treatment of the error in the quantities considered in quantum algorithms. Here we show the O(N−1)\mathcal O(N^{-1})O(N−1) scaling of the error for a wide range of cases needed for quantum simulation of chemistry. We show the error in the ground energy scales as O(N−1)\mathcal O(N^{-1})O(N−1) for plane-wave as well as spatial grid bases. For the spatial grid we consider both a periodic system and a non-periodic system with open boundaries. For time evolution, our bound on the error in the state itself is weaker than O(N−1)\mathcal O(N^{-1})O(N−1), but we show that the error in observables scales as O~(N−1)\widetilde{\mathcal O}(N^{-1})O(N−1) in important cases. In particular, we may consider a smoothed observable that describes the position of particles. We show that this error scales as O(N−5/3+tN−1)\mathcal O(N^{-5/3}+tN^{-1})O(N−5/3+tN−1) for time ttt, again both for plane-wave and spatial grid bases. For the non-periodic grid case we use a condition that the true evolution would not result in significant leakage past the boundaries, which introduces basis error unrelated to the resolution of the discretisation. We also show the error in the kinetic energy of a particle scales as O((t+log⁡N)/N)\mathcal O((t+\log N)/N)O((t+logN)/N) for the plane-wave and grid bases. Together, these results show that O(N−1)\mathcal O(N^{-1})O(N−1) scaling of the error holds for many of the relevant cases for quantum algorithms (with a trivial log factor for kinetic energy).
Quantum simulation is a prominent application of quantum computers. While there is extensive previous work on simulating finite-dimensional systems, less is known about quantum algorithms for real-space dynamics. We conduct a systematic study of such algorithms. In particular, we show that the dynamics of a d -dimensional Schrödinger equation with η particles can be simulated with gate complexity O~(ηdFpoly(log⁡(g′/ϵ))) , where ϵ is the discretization error, g′ controls the higher-order derivatives of the wave function, and F measures the time-integrated strength of the potential. Compared to the best previous results, this exponentially improves the dependence on ϵ and g′ from poly(g′/ϵ) to poly(log⁡(g′/ϵ)) and polynomially improves the dependence on T and d , while maintaining best known performance with respect to η . For the case of Coulomb interactions, we give an algorithm using η3(d+η)Tpoly(log⁡(ηdTg′/(Δϵ)))/Δ one- and two-qubit gates, and another using η3(4d)d/2Tpoly(log⁡(ηdTg′/(Δϵ)))/Δ one- and two-qubit gates and QRAM operations, where T is the evolution time and the parameter Δ regulates the unbounded Coulomb interaction. We give applications to several computational problems, including faster real-space simulation of quantum chemistry, rigorous analysis of discretization error for simulation of a uniform electron gas, and a quadratic improvement to a quantum algorithm for escaping saddle points in nonconvex optimization.
With quantum computers on the horizon, it is essential to understand what problems they can be used to solve. One potential application is simulating quantum systems, in particular, systems with many interacting electrons. However, there is a disparity between the computational resources required to study these problems and the capabilities of modern experimental quantum devices. In this thesis, we make progress toward closing this gap by reducing the cost of quantum simulation of quantum chemistry on a number of fronts. In the first half, we develop three improved quantum simulation algorithms by approximating time evolution under the Hamiltonian using a truncated Taylor series rather than the Trotter-Suzuki decomposition. We do this first for the molecular electronic structure Hamiltonian in two representations, obtaining asymptotic improvements in terms of both the number of orbitals and the required precision. We also apply the truncated Taylor series paradigm to quantum simulation in real space, where we consider simulation of general wave functions discretized on a uniform grid. In the second half, by working with Hamiltonians tailored specifically to quantum algorithms, we demonstrate that Trotter steps of the electronic structure Hamiltonian can be simulated in exactly N depth and with only linear connectivity, where N is the number of spin-orbitals. This resolves the problem of non-locality in fermion-to-qubit mappings, and has wide-ranging applications for variational and phase estimation-based quantum simulation of quantum chemistry. We further study this algorithm on an error-corrected quantum computer using the surface code. Benchmarking it alongside other product formula approaches, we show that quantum simulation of scientifically important instances of the electronic structure problem that are classically intractable can be performed with only a few hundred thousand physical qubits and fewer than one billion costly fault-tolerant gates. Finally, we study the effect of using randomized Hamiltonians in iterative phase estimation, and how this allows us to reduce the costs of simulation as well as to exploit knowledge of the ground state in how we design our simulation algorithm.
Simulations of chemical dynamics are a powerful means for understanding chemistry. However, classical computers struggle to simulate many chemical processes, especially non-adiabatic ones, where the Born-Oppenheimer approximation breaks down. Quantum computers could simulate quantum-chemical dynamics more efficiently than classical computers, but there is currently no complete quantum algorithm for calculating dynamical observables to within a known error. Here, we develop an efficient, end-to-end quantum algorithm for simulating chemical dynamics that avoids all uncontrolled approximations (including the Born-Oppenheimer approximation) and whose error is bounded subject to mild assumptions. To do so, we treat the nuclei and the electrons on an equal footing and simulate the full molecular wavefunction on a momentum-space grid in first quantization, including all algorithmic steps: initial-state preparation, time evolution using qubitization, and measurement of chemical observables such as reaction yields and rates. Our work gives the first algorithm for quantum simulation of chemistry whose end-to-end complexity achieves sublinear scaling in the size of the grid. We achieve this by developing an exponentially faster method for initial-state-preparation. Photochemistry is a likely early application of our algorithm and we estimate resources required for end-to-end simulations of non-adiabatic dynamics of atmospherically important molecules. Classically intractable photochemical computations could be performed using resources comparable to those required for other chemical applications of quantum computing.
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; its Hilbert space dimension grows exponentially with particle number; and the Coulomb potential is singular, long-ranged, and non-smooth, 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 these systems. Our first main result shows that for general initial conditions in the domain of the Hamiltonian, second-order Trotter achieves a 1/41/41/4 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. Such a worst-case rate has been observed in prior work for the hydrogen ground state, demonstrating its physical relevance. We further establish one-step local error lower bounds of order 5/45/45/4 for both formulas, showing that the local exponent is sharp. Our second main result identifies a set of physically meaningful conditions on the initial state under which the convergence rates improve, with the full first-order and second-order rates recovered at sufficiently high angular momentum. Our theoretical findings are consistent with prior numerical observations. Our third main result establishes improved convergence rates for many-body fermionic Coulomb systems with initial data of different Sobolev regularity. In particular, even without imposing any additional regularity beyond the natural domain of the Hamiltonian, both the first- and second-order Trotter formulas converge with rate 1/21/21/2, rather than the general 1/41/41/4 rate.
Finite-size errors (FSEs), the discrepancies between an observable in a finite system and in the thermodynamic limit, are ubiquitous in numerical simulations of quantum many-body systems.Although a rough estimate of these errors can be obtained from a sequence of finite-size results, a strict, quantitative bound on the magnitude of FSE is still missing.Here we derive rigorous upper bounds on the FSE of local observables in real-time quantum dynamics simulations initialized from a product state.In d-dimensional locally interacting systems with a finite local Hilbert space, our bound implies | Ŝ(t ) L -Ŝ(t ) ∞ | C(2vt/L) cL-μ , with v, C, c, μ constants independent of L and t, which we compute explicitly.For periodic boundary conditions (PBCs), the constant c is twice as large as that for open boundary conditions (OBCs), suggesting that PBCs have smaller FSEs than OBCs at early times.The bound can be generalized to a large class of correlated initial states as well.As a byproduct, we prove that the FSE of local observables in ground-state simulations decays exponentially with L under a suitable spectral gap condition.Our bounds are practically useful in determining the validity of finite-size results, as we demonstrate in simulations of the one-dimensional (1D) quantum Ising and Fermi-Hubbard models.