A quantum circuit for first-quantization-based Hamiltonian simulation, capable of running on a small-scale quantum computer emulator with fewer than 30 qubits, was constructed and demonstrated for one- and two-dimensional hydrogen-atom models, the simplest settings sufficient to check the proposed circuits without the confounding effects of electron-electron interaction and correlation. For these systems, Uniformly Controlled Rotation Z (UCRZ) circuits were found valuable for constructing a Coulomb potential circuit compact enough to fit on a small-scale emulator. Although this approach does not scale to larger models, we compare it against a conventional arithmetic-circuit implementation to identify the regime in which UCRZ is preferable. To deal with the Coulomb singularity, we propose regularizing it via a geometric average of the potential over a small circular region around the pole, avoiding divergence and minimizing the resulting energy error. To select parameters minimizing error for a given number of qubits, we determine the optimal spatial grid spacing from the trade-off between spatial discretization error and domain truncation error, and determine the time evolution step size not from the formal worst-case Trotter error bound, but from a sampling-theorem-based bound on the wave function’s maximum frequency component, avoiding unnecessary circuit depth without sacrificing accuracy. While the validity of a UCRZ-based circuit structure is restricted to the small-scale regime, the singularity treatment and parameter-selection procedure address the grid discretization itself and apply broadly to grid-based first-quantization simulations.
A quantum circuit for first-quantization-based Hamiltonian simulation, capable of running on a small-scale quantum computer emulator with fewer than 30 qubits, was constructed and demonstrated for one- and two-dimensional hydrogen-atom models, the simplest settings sufficient to check the proposed circuits without the confounding effects of electron-electron interaction and correlation. For these systems, Uniformly Controlled Rotation Z (UCRZ) circuits were found valuable for constructing a Coulomb potential circuit compact enough to fit on a small-scale emulator. Although this approach does not scale to larger models, we compare it against a conventional arithmetic-circuit implementation to identify the regime in which UCRZ is preferable. To deal with the Coulomb singularity, we propose regularizing it via a geometric average of the potential over a small circular region around the pole, avoiding divergence and minimizing the resulting energy error. To select parameters minimizing error for a given number of qubits, we determine the optimal spatial grid spacing from the trade-off between spatial discretization error and domain truncation error, and determine the time evolution step size not from the formal worst-case Trotter error bound, but from a sampling-theorem-based bound on the wave function’s maximum frequency component, avoiding unnecessary circuit depth without sacrificing accuracy. While the validity of a UCRZ-based circuit structure is restricted to the small-scale regime, the singularity treatment and parameter-selection procedure address the grid discretization itself and apply broadly to grid-based first-quantization simulations.
Accurate time-dependent quantum dynamics of Coulombic systems on grid-based representations remains computationally demanding due to the singularity of the Coulomb potential, which necessitates extremely fine spatial grids to mitigate discretisation errors. We propose two complementary correction schemes that, under identical resource budgets, consistently outperform the uncorrected counterparts. The first scheme modifies the potential operator to incorporate grid-basis structure into its representation, while the second introduces a corrected initial wavefunction inspired by analytical solutions of softened Coulomb potentials. Applied to hydrogenic systems, these corrections deliver improved energy accuracy and time fidelity across long evolutions. Beyond classical simulations, the proposed framework aligns naturally with quantum computing architectures, where the corrected operators and states can be encoded through truncated Walsh and Fourier series expansions. A resource analysis for the representative 2D hydrogen system yields a circuit depth of 1.5×108 gates over 6,000 Trotter steps. This study thus establishes practical strategies toward high-accuracy Coulombic dynamics on both classical and emerging quantum platforms.
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 H2 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.
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 1/4 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 5/4 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 1/2, rather than the general 1/4 rate.
High Resolution Image Download MS PowerPoint Slide Chemical simulation is a key application area that can leverage the power of quantum computers. A chemical simulator that implements a grid-based first quantization method has promising characteristics, but an implementation fully in quantum circuits seems to have not been published. Here, we present “crsQ” (chemical reaction simulator Q), which is a quantum circuit generator that generates such a chemical simulator. The generated simulator is capable of antisymmetrization of the initial wave function and time-evolution of the wave function based on the Suzuki–Trotter decomposition. The potential energy term of the Hamiltonian is implemented using arithmetic gates, such as adders, subtractors, multipliers, dividers, and square roots. Circuit diagrams and output samples are shown. The number of qubits in the circuits scales on the order of O (η log η), where η is the number of electrons. Each component of the generated circuit was verified in unit tests. Along with this development, we designed frameworks to ease the development of large-scale circuits, namely, a temporary qubit allocation framework and an abstract syntax tree framework for arithmetic formulas. These frameworks are expected to be useful in large-scale quantum circuit generators.
Simulation of electronic structure is one of the most promising applications on noisy intermediate-scale quantum (NISQ) era devices. However, NISQ devices suffer from a number of challenges like limited qubit connectivity, short coherence times, and sizable gate error rates. Thus, desired quantum algorithms should require shallow circuit depths and low qubit counts to take advantage of these devices. Here, we attempt to reduce quantum resource requirements for molecular simulations on a quantum computer while maintaining the desired accuracy with the help of classical quantum chemical theories of canonical transformation and explicit correlation. In this work, compact ab initio Hamiltonians are generated classically, in the second quantized form, through an approximate similarity transformation of the Hamiltonian with (a) an explicitly correlated two-body unitary operator with generalized pair excitations that remove the Coulombic electron-electron singularities from the Hamiltonian and (b) a unitary one-body operator to efficiently capture the orbital relaxation effects required for accurate description of the excited states. The resulting transcorrelated Hamiltonians are able to describe both the ground and the excited states of molecular systems in a balanced manner. Using the variational quantum eigensolver (VQE) method based on the unitary coupled cluster with singles and doubles (UCCSD) ansatz and only a minimal basis set (ANO-RCC-MB), we demonstrate that the transcorrelated Hamiltonians can produce ground state energies comparable to the reference CCSD energies with the much larger cc-pVTZ basis set. This leads to a reduction in the number of required CNOT gates by more than 3 orders of magnitude for the chemical species studied in this work. Furthermore, using the quantum equation of motion (qEOM) formalism in conjunction with the transcorrelated Hamiltonian, we are able to reduce the deviations in the excitation energies from the reference EOM-CCSD/cc-pVTZ values by an order of magnitude. The transcorrelated Hamiltonians developed here are Hermitian and contain only one- and two-body interaction terms and thus can be easily combined with any quantum algorithm for accurate electronic structure simulations.
Discretizing electronic degrees of freedom into atomic orbitals is the default choice to accurately model electrons in molecules in a compact and flexible way. We consider the task of simulating such electronic structure models on a quantum computer for N orbitals, keeping the number of orbitals per atom constant. Atomic orbitals give rise to Hamiltonians with potentially up to O(N4) terms and little structure, which made it challenging to match the complexity of methods based on plane waves and real-space grids. In this work, we show that a Trotter step for atomic orbitals can be implemented in depth O(polylog(N)) without ancillas by combining a rigorous treatment of orbital localization, a hierarchical Hamiltonian decomposition based on the Fast Multipole Method, and shallow quantum Fourier arithmetic circuits. By bounding the Trotter error, we obtain sublinear simulation depth, and our total gate count N5/3+o(1) matches the best scaling known for any basis.