State-dependent Trotter Limits and their approximations

Abstract

The Trotter product formula is a key instrument in numerical simulations of quantum systems. However, computers cannot deal with continuous degrees of freedom, such as the position of particles in molecules, or the amplitude of electromagnetic fields. It is therefore necessary to discretize these variables to make them amenable to digital simulations. Here, we give sufficient conditions to conclude the validity of this approximate discretized physics. Essentially, it depends on the state-dependent Trotter error, for which we establish explicit bounds that are also of independent interest.

Figures (5)

• Figure 1: In the simulation of a quantum system one starts with two simple continuous-degree Hamiltonians $H^{(1)}$ and $H^{(2)}$. Then, the goal is to simulate the dynamics under $H=H^{(1)}+H^{(2)}$ through the Trotter product formula (blue path). However, due to its continuous nature, it is not possible to implement this strategy on digital simulators such as computers. Therefore, it is common practice to follow the green path instead. This is, one first truncates $H^{(1)}$ and $H^{(2)}$ to a finite level $d$. Afterwards, one implements the dynamics under the sum of the truncated Hamiltonians $H_d=H_d^{(1)}+H_d^{(2)}$ via the Trotter product formula. Unfortunately, $H_d$ does not necessarily cover the physics of the full model $H$ correctly, so that this method can lead to deceptive results. Here, we establish a method to close this gap, which is indicated by the dashed green arrow.
• Figure 2: State-dependent Trotter error for the operators $H^{(1)}=\frac{1}{2}\left(Q^2+P^2\right)$ and $H^{(2)}=\frac{1}{2}\left(QP+PQ\right)$. We consider the first five Fock-basis states $\Set{|m\rangle}=\Set{|0\rangle,\dots,|4\rangle}$ for different dimensions of truncations $d=1,\dots,300$. The total evolution time is fixed to $t=3$ and the number of Trotter steps is $n=1000$. The saturation of the error indicates the convergence of this Trotter problem.
• Figure 3: State-dependent error for the operators $H^{(1)}=Q^3$ and $H^{(2)}=P^2$. We consider the first five Fock-basis states $\Set{|m\rangle}=\Set{|0\rangle,\dots,|4\rangle}$ for different dimensions of truncations $d=1,\dots,300$. The total evolution time is fixed to $t=1$ and the number of Trotter steps is $n=1000$. The Trotter error does not saturate, which shows that this Trotter problem does not converge.
• Figure 4: State-dependent error for the operators $H^{(1)}=\frac{1}{2}Q^2$ and $H^{(2)}=\frac{1}{2}P^2$. We consider the first five Fock-basis states $\Set{|m\rangle}=\Set{|0\rangle,\dots,|4\rangle}$ for different dimensions of truncations $d=1,\dots,50$. The total evolution time is fixed to $t=1$ and the number of Trotter steps is $n=1000$. The dots are a numerical simulation, whereas the lines show the explicit error bounds from Eq. \ref{['Trotter_X2P2']}. Since the Trotter error can be bounded independently of the truncation dimension, this Trotter problem converges.
• Figure 5: Uniform Trotter error for $H^{(1)}=\frac{1}{2}\left(Q^2+P^2\right)$ and $H^{(2)}=\frac{1}{2}\left(QP+PQ\right)$ with $Q$ the position and $P$ the momentum operator. Parameters: $n=20$ Trotter steps, total evolution time of $t=2$. The error reaches the maximal norm distance of $2$ showing that the Trotter formula does not converge in norm.

Theorems & Definitions (14)

• Proposition 3
• proof
• Lemma 4
• proof
• proof
• Theorem 5
• proof
• Lemma 6
• proof
• proof : Proof of Prop. \ref{['thm:uniformTrotter']}