Strong Error Bounds for Trotter & Strang-Splittings and Their Implications for Quantum Chemistry

Abstract

Efficient error estimates for the Trotter product formula are central in quantum computing, mathematical physics, and numerical simulations. However, the Trotter error's dependency on the input state and its application to unbounded operators remains unclear. Here, we present a general theory for error estimation, including higher-order product formulas, with explicit input state dependency. Our approach overcomes two limitations of the existing operator-norm estimates in the literature. First, previous bounds are too pessimistic as they quantify the worst-case scenario. Second, previous bounds become trivial for unbounded operators and cannot be applied to a wide class of Trotter scenarios, including atomic and molecular Hamiltonians. Our method enables analytical treatment of Trotter errors in chemistry simulations, illustrated through a case study on the hydrogen atom. Our findings reveal: (i) for states with fat-tailed energy distribution, such as low-angular-momentum states of the hydrogen atom, the Trotter error scales worse than expected (sublinearly) in the number of Trotter steps; (ii) certain states do not admit an advantage in the scaling from higher-order Trotterization, and thus, the higher-order Trotter hierarchy breaks down for these states, including the hydrogen atom's ground state; (iii) the scaling of higher-order Trotter bounds might depend on the order of the Hamiltonians in the Trotter product for states with fat-tailed energy distribution. Physically, the enlarged Trotter error is caused by the atom's ionization due to the Trotter dynamics. Mathematically, we find that certain domain conditions are not satisfied by some states so higher moments of the potential and kinetic energies diverge. Our analytical error analysis agrees with numerical simulations, indicating that we can estimate the state-dependent Trotter error scaling genuinely.

Figures (9)

• Figure 1: The Trotter error at time $t=1$ in the Hartree atomic units $\hbar=m_\mathrm{e}=a_0=1$, as a function of the Trotter steps for the ground state $\Psi_{100}$ of the hydrogen atom. The radial cutoff in the simulations is $R=30$. We show five different levels of discretization characterized by the number of Bessel modes. For reference, we show the slopes of $N^{-1}$ (grey dashed line) and our bound (brown solid line), which scales as $N^{-1/4}$. See Eqs. \ref{['eqErrorBoundhydrogenGround1']}--\ref{['eqErrorBoundhydrogenGround2']}. We see that the asymptotics for any finite discretization initially start in a consistent way with our bound, and eventually go as $N^{-1}$, but $N$ at which this transition happens becomes larger with increasing number of modes. This provides an evidence that the scaling in the infinite-mode limit is indeed slower than $N^{-1}$. To indicate a potential curve for the infinite-mode limit, we show a slope (gray solid line) with the same scaling as our analytic bound.
• Figure 2: The Trotter error at time $t=1$ in the Hartree atomic units $\hbar=m_\mathrm{e}=a_0=1$, as a function of the Trotter steps for the state $\Psi_{210}$ of the hydrogen atom. The radial cutoff in the simulations is $R=30$. We show five different levels of discretization characterized by the number of Bessel modes; however, beyond 200 modes (red) the results are indistinguishable. The grey dashed line shows the slope of the $N^{-1}$ scaling, and we see that the Trotter error scales as $N^{-1}$ in all cases, even though the state $\Psi_{210}$ does not satisfy the domain conditions for Thm. \ref{['thm:trotter_thm']}. Our analytic bound, which scales as $N^{-3/4}$, is depicted in brown.
• Figure 3: The Trotter error at time $t=1$ in the Hartree atomic units $\hbar=m_\mathrm{e}=a_0=1$, as a function of the Trotter steps for the state $\Psi_{320}$ of the hydrogen atom. The radial cutoff in the simulations is $R=40$. We use four different levels of discretization characterized by the number of Bessel modes; however, beyond 50 modes (green) the results are indistinguishable. The grey dashed line shows the slope of the $N^{-1}$ scaling, and we see that the Trotter error scales as $N^{-1}$ in all cases as expected. For reference, we plot our analytic bound in brown. It also shows an $N^{-1}$ behavior.
• Figure 4: Numerical simulations of the ionization probability as a function of the number of Trotter steps. The initial state $\varphi$ is prepared in various energy eigenstates $\Psi_{n\ell m}$ of the hydrogen atom, and then we let it evolve to $\varphi(t)=U_N(t)\varphi$ by the Trotter evolution $U_N(t)$, with the Hamiltonians $H_1$ and $H_2$ given by Eqs. \ref{['eq:hamiltonian_kin']} and \ref{['eq:hamiltonian_pot']}, for a total time $t=1$ in the Hartree atomic units $\hbar=m_\mathrm{e}=a_0=1$. The ionization probability is given by $1-\|P_{\mathrm{bd}}\varphi(t)\|^2$, where $P_{\mathrm{bd}}=\sum_{n\ell m} |\Psi_{n\ell m}\rangle \langle\Psi_{n\ell m}|$ is the projection on the space of bound states of the hydrogen atom. For the evolution under the non-Trotterized Hamiltonian $H = H_1 + H_2$, the ionization fraction was zero up to numerical errors $(<10^{-10})$ as expected. For the Trotterized evolution, $\varphi(t)$ acquires a nonzero component out of the space spanned by the bound states, resulting in a nonzero ionization probability, as shown in the figure. The $\ell=0$ eigenstates ionize much more heavily than the $\ell=2$ eigenstates, but in both cases, the ionization rate decreases with the number of Trotter steps as the Trotter approximation approaches the true evolution.
• Figure 5: The Trotter error at time $t=1$ in the Hartree atomic units $\hbar=m_\mathrm{e}=a_0=1$, as a function of the Trotter steps for an approximation $\Psi_{100}^\text{STO-3G}(r)=0.44\,\mathrm{e}^{-0.11 r^2} + 0.53\,\mathrm{e}^{-0.41 r^2} + 0.15\, \mathrm{e}^{-2.23 r^2}$ of the ground state of the hydrogen atom in the STO-3G representation. The radial cutoff in the simulations is $R=40$. We show five different levels of discretization characterized by the number of Bessel modes. For reference, we show the slopes of $N^{-1}$ (grey dashed line) and $N^{-1/4}$ (grey line). We see that the asymptotics for any finite discretization initially start in a consistent way with the grey $N^{-1/4}$ line, and eventually go as $N^{-1}$, but $N$ at which this transition happens becomes larger with increasing number of modes. This provides an evidence that the scaling in the infinite-mode limit is indeed slower than $N^{-1}$.

Theorems & Definitions (34)

• Theorem 1
• proof
• Theorem 2
• proof
• Corollary 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof