Basis-set error bounds for first-quantised quantum simulation of chemistry

Abstract

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 $\mathcal O(N^{-1})$ for a basis of size $N$. 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 $\mathcal O(N^{-1})$ 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 $\mathcal O(N^{-1})$ 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 $\mathcal O(N^{-1})$, but we show that the error in observables scales as $\widetilde{\mathcal O}(N^{-1})$ in important cases. In particular, we may consider a smoothed observable that describes the position of particles. We show that this error scales as $\mathcal O(N^{-5/3}+tN^{-1})$ for time $t$, 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 $\mathcal O((t+\log N)/N)$ for the plane-wave and grid bases. Together, these results show that $\mathcal O(N^{-1})$ scaling of the error holds for many of the relevant cases for quantum algorithms (with a trivial log factor for kinetic energy).

The paper asks when the commonly assumed $\mathcal O(N^{-1})$ basis-set error is valid for first-quantised simulations of interacting Coulomb systems, where $N$ counts spatial basis functions per particle. It treats plane-wave Galerkin projection and a spatial grid with a specified Fourier-truncated interaction, distinguishing discretisation errors from algorithmic implementation errors. At fixed periodic cell size, it proves ground-energy error $\mathcal O(N^{-1})$, including spin and clamped or quantum nuclei. For initial states in $D(H^2)$ with cutoff-independent Hamiltonian graph norms, smooth one-particle position expectations have error $\mathcal O(N^{-5/3}+tN^{-1})$, and individual-particle kinetic energies have error $\mathcal O((t+\log N)/N)$. These dynamical rates hold for both discretisations and also yield finite-window stopping-power error bounds. The energy proof combines Coulomb Fourier-tail estimates with the eigenvalue equation and a variational cancellation, giving discarded-state norm $\mathcal O(K^{-5/2})$ and energy error $\mathcal O(K^{-3})$, with $N=\Theta(K^3)$. Observable estimates use backward-evolved test states and duality; grid estimates additionally control omitted transfers and momentum aliasing. The kinetic-energy argument uses low-Hamiltonian-energy projections and dyadic spectral comparisons. For isolated systems, a range-truncated interaction separates resolution error from finite-cell error. Under free-space and finite-cell localisation assumptions, clamped-nucleus ground energies in a specified scalar exchange sector require sufficient basis size $\mathcal O(\epsilon^{-1}\log^3(1/\epsilon))$. Dynamical comparisons require exact-solution wavefunction and shell-gradient leakage bounds; regularity alone does not imply localisation. Constants are not optimised, sharpness is not generally established, and whether the kinetic-energy logarithm is necessary remains open.