Trotter error bounds and dynamic multi-product formulas for Hamiltonian simulation

TL;DR

This work advances Hamiltonian simulation on quantum devices by rigorously bounding MPF errors and establishing a practical, robust framework. It extends commutator-based Trotter error analysis to MPFs, showing quadratic error reduction without deeper circuits, and introduces dynamic and minimax MPFs to optimize, stabilize, and harden coefficient selection against noise. The Minimax MPF method yields well-conditioned, time-dependent coefficients that tolerate algorithmic and sampling errors, with provable worst-case error bounds. Numerical experiments on spin-chain models demonstrate substantial accuracy gains and depth reductions, underscoring MPFs' potential for near-term quantum simulations.

Abstract

Multi-product formulas (MPF) are linear combinations of Trotter circuits offering high-quality simulation of Hamiltonian time evolution with fewer Trotter steps. Here we report two contributions aimed at making multi-product formulas more viable for near-term quantum simulations. First, we extend the theory of Trotter error with commutator scaling developed by Childs, Su, Tran et al. to multi-product formulas. Our result implies that multi-product formulas can achieve a quadratic reduction of Trotter error in 1-norm (nuclear norm) on arbitrary time intervals compared with the regular product formulas without increasing the required circuit depth or qubit connectivity. The number of circuit repetitions grows only by a constant factor. Second, we introduce dynamic multi-product formulas with time-dependent coefficients chosen to minimize a certain efficiently computable proxy for the Trotter error. We use a minimax estimation method to make dynamic multi-product formulas robust to uncertainty from algorithmic errors, sampling and hardware noise. We call this method Minimax MPF and we provide a rigorous bound on its error.

Figures (7)

• Figure 1: Examples of multi-product formulas (MPFs) of order $p=2,4,6$. Such MPFs have a form $\mu(t)=\sum_{i=1}^{p+1} c_i \rho_{k_i}(t)$ where $\rho_{k_i}(t)$ is an approximation of the exact time evolved state obtained by applying $k_i$ steps of the base product formula ${\cal S }(t/k_i)$. The sequence of time steps $(k_1,\ldots,k_{p+1})$ reported in the table was obtained by minimizing the factor $\sum_{i=1}^{p+1} |c_i|/k_i^{2p}$ in the upper bound of Theorem \ref{['thm:MPF']} over all tuples $(k_1,\ldots,k_{p+1})$ with $1\le k_i\le k_{max}$ and solving the linear system Eq. (\ref{['ZNE']}) with $r=p+1$ to obtain the coefficients $c_i$. To avoid clutter, we round the condition number and coefficients in the error scaling to the nearest integer. Note that the MPF approximation error can be reduced by performing a rescaling $k_i\gets \lambda k_i$ for all $i$, where $\lambda\ge 1$ is any integer. The reduces the upper bound of Theorem \ref{['thm:MPF']} by the factor $1/\lambda^{2p}$ without changing the coefficients $c_i$ and the condition number. Meanwhile, for the regular order-$p$ product formula, the rescaling $k\gets \lambda k$ reduces the approximation error only by the factor $1/\lambda^p$, see Lemma \ref{['fact:trotter_error']}.
• Figure 2: Approximation error achieved by the second-order Trotter circuit with $k_3=850$ time steps (blue) and MPF with $(k_1,k_2,k_3)=(200,650,850)$ (orange) for the Heisenberg spin chain Hamiltonian Eq. (\ref{['spin_chain']}) with $n=14$ qubits. Green line shows the fitting formula Eq. (\ref{['order2fit']}).
• Figure 3: Comparison between the true MPF approximation error $\|\mu(t)-\rho(t)\|_1$ (circles) and fitting formula Eq. (\ref{['order2fit']}) (solid lines) for the second-order product formulas with $(k_1,k_2,k_3)=\lambda \cdot (4,13,17)$. Top panel: $n=10$, Bottom panel: $n=14$.
• Figure 4: Approximation error achieved by the fourth-order Trotter circuit with $k_5=250$ time steps (blue) and MPF with $(k_1,k_2,k_3,k_4,k_5)=(20, 90, 170, 230, 250)$ (orange) for the Heisenberg spin chain Hamiltonian Eq. (\ref{['spin_chain']}) with $n=14$ qubits. Green line shows the fitting formula Eq. (\ref{['order4fit']}).
• Figure 5: Comparison between the true MPF approximation error $\|\mu(t)-\rho(t)\|_1$ (circles) and fitting formula Eq. (\ref{['order4fit']}) (solid lines) for the fourth-order product formulas with $(k_1,k_2,k_3,k_4,k_5)=\lambda(2, 9, 17, 23, 25)$. Top panel: $n=10$, Bottom panel: $n=14$. For small evolution time $t$ the fitting formula underestimates the error since it neglects corrections $a_2t^{2p+1}$ and $a_3t^{2p}$ in the upper bound of Theorem \ref{['thm:MPF']}.

Theorems & Definitions (14)

• Lemma 1: Trotter error
• Theorem 1: MPF Trotter error
• Lemma 2: Euler–Maclaurin formula
• Lemma 3
• Lemma 4: Taylor's theorem
• proof
• proof : Proof of Lemma \ref{['lemma:remainder']}
• Lemma 5
• proof
• Proposition 1