A Monte Carlo approach to bound Trotter error

TL;DR

The paper addresses the difficulty of bounding Trotter error in quantum simulation, where conventional bounds based on nested commutator norms are hard to compute and often loose. It proposes a practical approach by proving $||A|| \le ||\mathrm{abs}(A)||$ and estimating $||\mathrm{abs}(A)||$ via sign-problem-free projector Monte Carlo (FCIQMC), focusing on the second-order product formula $S_2(t)=e^{-iVt/2}e^{-iTt}e^{-iVt/2}$ and the bounds $W_{VTV}$ and $W_{TVT}$. Across models including the PPP, extended Hubbard, and the uniform electron gas, the Monte Carlo bounds are shown to be extremely tight, with substantial reductions in estimated Trotter costs and sublinear scaling for long-range interactions (e.g., PPP). This provides a practical resource-estimation tool for fault-tolerant quantum computation and can be extended to higher-order formulas and broader Hamiltonians, with data made available publicly.

Abstract

Trotter product formulas are a natural and powerful approach to perform quantum simulation. However, the error analysis of product formulas is challenging, and their cost is often overestimated. It is established that Trotter error can be bounded in terms of spectral norms of nested commutators of the Hamiltonian partitions [Childs et al., Phys. Rev. X 11, 011020], but evaluating these expressions is challenging, often achieved by repeated application of the triangle inequality, significantly loosening the bound. Here, we show that the spectral norm of an operator can be upper bounded by the spectral norm of an equivalent sign-problem-free operator, which can be calculated efficiently to large system sizes using projector Monte Carlo simulation. For a range of Hamiltonians and considering second-order formulas, we demonstrate that this Monte Carlo-based bound is often extremely tight, and even exact in some instances. For the uniform electron gas we reduce the cost of performing Trotterization from the literature by an order of magnitude. For the Pariser-Parr-Pople model for linear acene molecules, which has $\mathcal{O}(N^2)$ long-range interaction terms, we show that it suffices to use $\mathcal{O}(N^{0.57})$ Trotter steps and circuit depth $\mathcal{O}(N^{1.57})$ to implement Hamiltonian simulation. We hope that this approach will lead to a better understanding of the potential accuracy of Trotterization in a range of important applications.

Figures (7)

• Figure 1: Comparison of commutator bounds for PPP model and extended Hubbard model examples. In (a) and (b) we consider the PPP model applied to linear acene molecules, while (c) and (d) consider an extended Hubbard model on a one-dimensional lattice. The commutators considered are $[[V,T],V]$ and $[[V,T],T]$, for which we aim to provide tight upper bounds on the spectral norm (restricting to half-filling and projected spin $s_z=0$). Results in blue show an upper bound from the L1 norm of the commutator, Eq. (\ref{['eq:l1_norm']}). Results in purple ((c) and (d) only) use a tighter triangle inequality from Ref. bay_smidt_2025. Results in red are calculated from the Monte Carlo bound. Exact results are obtained from either exact diagonalization or DMRG.
• Figure 2: Comparison of Trotter error norm upper bounds for two-dimensional systems. We consider: (a) the extended Hubbard model on a hexagonal lattice with $\tau=1$, $U=4$, $V=2$; (b) a cuprate model on a square lattice with $\tau=1$, $\tau'=0.3$, $\tau"=0.2$, $U=8$; and (c), the 2D uniform electron gas with $r_s = 10$. Half-filling and projected spin $s_z=0$ are taken in each case. We compare the Monte Carlo bound, Eq. (\ref{['eq:upper_bound']}) to the L1 norm, Eq. (\ref{['eq:l1_norm']}). We also compare to tighter triangle inequality bounds where available, using Eqs. (\ref{['eq:vtt_tighter']})--(\ref{['eq:vtv_tighter']}) for the cuprate model and results from Ref. bay_smidt_2025 for the extended Hubbard model. Results labeled "Exact" are obtained with a combination of exact diagonalization, DMRG, and exact FCIQMC (see Appendix \ref{['sec:exact_fciqmc']}). Results for the UEG are in units of Ha$^3$.
• Figure 3: FCIQMC simulations performed on the commutator $A = -[[V,T],T]$ for a napthalene molecule described by the PPP model. The dashed line is minus the spectral norm, and the dashed-dotted line is minus the spectral norm of the sign-problem-free commutator, denoted $-\lambda_{\mathrm{PF}}$. (a) FCIQMC performed on the exact commutator. This operator has a sign problem, so the shift estimator converges to a value below $- \lVert A \rVert$, and the signal from the mixed estimator becomes swamped in noise beyond iteration $\sim 250$. (b) FCIQMC performed on the sign-problem-free operator, $-\mathrm{abs}(A)$. Here the propagation is stable and both estimators sample $-\lVert \mathrm{abs}(A) \rVert$, where $\lVert \mathrm{abs}(A) \rVert \ge \lVert A \rVert$ is guaranteed. In both (a) and (b) the shift is allowed to vary once the walker population (not shown) reaches $1000$. Note that for this example we could sample the exact solution in (a) with a slightly larger walker; we use a low walker population here for demonstrative purposes.
• Figure 4: Results comparing the true worst-case Trotter error (green) of Eq. (\ref{['eq:exact_trotter_error_def']}), to the commutator bound (red) of Eq. (\ref{['eq:w2_app']}) and the bound which can be calculated efficiently by Monte Carlo (blue) of Eq. (\ref{['eq:w2_mc']}). We consider three systems from the main text, for small examples where exact Trotter error can be calculated: (a) the PPP model for a benzene molecule; (b) the extended Hubbard model on a 1D lattice with $N=6$ sites; (c) the uniform electron gas in 2D, with $r_s=10$ and a $2 \times 2$ grid.
• Figure 5: Comparison of spectral norm estimates (restricted to half-filling and projected spin $s_z=0$) for commutators $[[V,T],V]$ and $[[V,T],T]$, for (a--b) the extended Hubbard model on a hexagonal lattice with $t=1$, $U=4$, $V=2$; (c--d) a cuprate model on a square lattice with $t=1$, $t'=0.3$, $t"=0.2$, $U=8$; and (e--f), the uniform electron gas with $r_s = 10$. These results correspond to those from Fig. \ref{['fig:2d']} in the main text. Results for the UEG are in units of Ha$^3$. Dashed lines are included to guide the eye for larger systems, where finite-size effects are smaller.