Second-Order Trotter Formula

• Second-Order Trotter Formula is a symmetric product formula that approximates the exponential of a sum of noncommuting operators using boundary half-step evolutions, canceling first-order errors.
• It achieves improved error scaling of O(t³/r²) in bounded scenarios and is widely applied in quantum simulation, operator splitting, and digital circuit synthesis.
• Recent studies refine commutator-based error bounds and address challenges with unbounded operators, enhancing scalability and precision in complex simulations.

Searching arXiv for recent and foundational papers on the second-order Trotter formula. arXiv_search(query="second-order Trotter formula Strang splitting Hamiltonian simulation", max_results=10, sort_by="relevance"){"result":[{"arxiv_id":"(Childs et al., 2019)","title":"A Theory of Trotter Error","authors":"Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, Shuchen Zhu","abstract":"The Lie-Trotter formula, together with its higher-order generalizations, provides a direct approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of prior approaches based on truncating the Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Lie-algebraic structure, our approach holds in general. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo methods, including simulations of second-quantized plane-wave electronic structure, k-local Hamiltonians, rapidly decaying power-law interactions, clustered Hamiltonians, the transverse field Ising model, and quantum ferromagnets, nearly matching or even outperforming the best previous results. We obtain further speedups using the fact that product formulas can preserve the locality of the simulated system. Specifically, we show that local observables can be simulated with complexity independent of the system size for power-law interacting systems, which implies a Lieb-Robinson bound as a byproduct. Our analysis reproduces known tight bounds for first- and second-order formulas. Our higher-order bound overestimates the complexity of simulating a one-dimensional Heisenberg model with an even-odd ordering of terms by only a factor of 5, and is close to tight for power-law interactions and other orderings of terms. This suggests that our theory can accurately characterize Trotter error in terms of both asymptotic scaling and constant prefactor.","categories":"quant-ph"},{"arxiv_id":"(Layden, 2021)","title":"First-Order Trotter Error from a Second-Order Perspective","authors":"Marios C. Bañuls, Kiran N. Natarajan, M. C. Bañuls, David J. Luitz, Erez Zohar, J. Ignacio Cirac, Andrew M. Childs, Nathan Wiebe, Vicente Martín, Enrique Rico","abstract":"Simulating quantum dynamics beyond the reach of classical computers is one of the main envisioned applications of quantum computers. The most promising quantum algorithms to this end in the near-term are the simplest, which use the Trotter formula and its higher-order variants to approximate the dynamics of interest. The approximation error of these algorithms is often poorly understood, even in the most basic cases, which are particularly relevant for experiments. Recent studies have reported anomalously low approximation error with unexpected scaling in such cases, which they attribute to quantum interference between the errors from different steps of the algorithm. Here we provide a simpler picture of these effects by relating the Trotter formula to its second-order variant. Our method generalizes state-of-the-art error bounds without the technical caveats of prior studies, and elucidates how each part of the total error arises from the underlying quantum circuit. We compare our bound to the true error numerically, and find a close match over many orders of magnitude in the simulation parameters. Our findings reduce the required circuit depth for the most basic quantum simulation algorithms, and illustrate a useful method for bounding simulation error more broadly.","categories":"quant-ph"},{"arxiv_id":"(Burgarth et al., 2023)","title":"Strong Error Bounds for Trotter & Strang-Splittings and Their Implications for Quantum Chemistry","authors":"Darya Krymova, Barbara Roos, Dmitrii Basanets, Christian Lubich, Michael M. Wolf","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.","categories":"math-ph,quant-ph"},{"arxiv_id":"(Spiteri et al., 2024)","title":"A pair of Second-order complex-valued, N-split operator-splitting methods","authors":"D. M. Juárez, M. V. de Hoop, M. Schreiber","abstract":"The use of operator-splitting methods to solve differential equations is widespread, but the methods are generally only defined for a given number of operators, most commonly two. Most operator-splitting methods are not generalizable to problems with N operators for arbitrary N. In fact, there are only two known methods that can be applied to general N-split problems: the first-order Lie--Trotter (or Godunov) method and the second-order Strang (or Strang--Marchuk) method. In this paper, we derive two second-order operator-splitting methods that also generalize to N-split problems. These methods are complex valued but have positive real parts, giving them favorable stability properties, and require few sub-integrations per stage, making them computationally inexpensive. They can also be used as base methods from which to construct higher-order N-split operator-splitting methods with positive real parts. We verify the orders of accuracy of these new N-split methods and demonstrate their favorable efficiency properties against well-known real-valued operator-splitting methods on both real-valued and complex-valued differential equations.","categories":"math.NA,math.AP"},{"arxiv_id":"(Maxwell et al., 29 Jun 2026)","title":"Practical Estimation of Trotter Error for Hamiltonian Simulation","authors":"A. Roggero, I. D. Kivlichan, W. Huggins, M. Rubin, Y. Shi, J. Lee, S. B. Bravyi, M. B. Hastings, N. Wiebe, R. Babbush","abstract":"Trotter product formulas are a leading approach for Hamiltonian simulation on quantum computers, yet their practical performance has remained difficult to assess due to the challenge of accurately estimating the Trotter error. In this work, we develop new theoretical results, algorithms, and software tools that advance the state-of-the-art in Trotter error estimation by orders of magnitude in both scale and accuracy. On the theoretical side, we prove that in the asymptotic limit the error of a product formula depends on the diagonal elements of the Baker-Campbell-Hausdorff (BCH) error operator in the eigenbasis of the Hamiltonian, rather than its full spectral norm -- yielding an improved scaling for Hamiltonian simulation using product formulas. On the algorithmic side, we introduce a compact representation of the BCH expansion that reduces the number of commutators from O(n^3) to O(n) for second-order, and from O(n^5) to O(n^2) for fourth-order formulas on n fragments, complemented by an importance sampling scheme to further reduce the computational cost. We provide implementations of these techniques in software and demonstrate their power on two applications: (i) X-ray absorption spectroscopy of an electronic Hamiltonian (Li4Mn2O) at up to 56 qubits using tensor networks; and (ii) vibronic dynamics of naphthalene at over 100 qubits using ML-MCTDH, where we find that naive analytical bounds overestimate the required number of Trotter steps by nearly five orders of magnitude. Our framework enables, for the first time, the accurate estimation of Trotter error at practically relevant system sizes, providing a foundation for fair algorithmic comparisons and rational design of product formulas.","categories":"quant-ph"},{"arxiv_id":"(Liu et al., 2019)","title":"Novel Trotter formulas for digital quantum simulation","authors":"Ruiyun Liu, Yunguo Yang, Paul N. Halder, Kaden R. A. Hazzard","abstract":"Quantum simulation promises to address many challenges in fields ranging from quantum chemistry to material science, and high-energy physics, and could be implemented in noisy intermediate-scale quantum devices. A challenge in building good digital quantum simulators is the fidelity of the engineered dynamics given a finite set of elementary operations. Here we present a framework for optimizing the order of operations based on a geometric picture, thus abstracting from the operation details and achieving computational efficiency. Based on this geometric framework, we provide two alternative second-order Trotter expansions, one with optimal fidelity at a short time scale, and the second robust at a long time scale. Thanks to the improved fidelity at different time scale, the two expansions we introduce can form the basis for experimental-constrained digital quantum simulation.","categories":"quant-ph"},{"arxiv_id":"(Zeng et al., 2022)","title":"Simple and high-precision Hamiltonian simulation by compensating Trotter error with linear combination of unitary operations","authors":"Dong An, Di Fang, Lin Lin, Yu Tong","abstract":"Trotter and linear-combination-of-unitary (LCU) are two popular Hamiltonian simulation methods. We propose Hamiltonian simulation algorithms using LCU to compensate Trotter error, which enjoy both of their advantages. By adding few gates after the Kth-order Trotter, we realize a better time scaling than 2Kth-order Trotter. Our first algorithm exponentially improves the accuracy scaling of the Kth-order Trotter formula. In the second algorithm, we consider the detailed structure of Hamiltonians and construct LCU for Trotter errors with commutator scaling. Consequently, for lattice Hamiltonians, the algorithm enjoys almost linear system-size dependence and quadratically improves the accuracy of the Kth-order Trotter.","categories":"quant-ph"},{"arxiv_id":"(Wittek et al., 2012)","title":"A Second-Order Distributed Trotter-Suzuki Solver with a Hybrid Kernel","authors":"T. J. Visser, T. V. T. Driessen, H. J. M. Bastiaens, A. van den Bos, R. A. de Vries, L. D. J. F. van de Vusse, A. P. J. van Deursen, B. Koren, R. M. Verhaaren","abstract":"The Trotter-Suzuki approximation leads to an efficient algorithm for solving the time-dependent Schroedinger equation. Using existing highly optimized CPU and GPU kernels, we developed a distributed version of the algorithm that runs efficiently on a cluster. Our implementation also improves single node performance, and is able to use multiple GPUs within a node. The scaling is close to linear using the CPU kernels, whereas the efficiency of GPU kernels improve with larger matrices. We also introduce a hybrid kernel that simultaneously uses multicore CPUs and GPUs in a distributed system. This kernel is shown to be efficient when the matrix size would not fit in the GPU memory. Larger quantum systems scale especially well with a high number nodes. The code is available under an open source license.","categories":"cs.DC,physics.comp-ph"},{"arxiv_id":"(Chehade et al., 2024)","title":"Error Estimates and Higher Order Trotter Product Formulas in Jordan-Banach Algebras","authors":"Muhammad Ahsan Binyamin, J.-P. Wang, Y. Wang","abstract":"In quantum computing, Trotter estimates are critical for enabling efficient simulation of quantum systems and quantum dynamics, help implement complex quantum algorithms, and provide a systematic way to control approximate errors. In this paper, we extend the analysis of Trotter-Suzuki approximations, including third and higher orders, to Jordan-Banach algebras. We solve an open problem in our earlier paper on the existence of second-order Trotter formula error estimation in Jordan-Banach algebras. To illustrate our work, we apply our formula to simulate Trotter-factorized spins, and show improvements in the approximations. Our approach demonstrates the adaptability of Trotter product formulas and estimates to non-associative settings, which offers new insights into the applications of Jordan algebra theory to operator dynamics.","categories":"math.OA,quant-ph"}]} The second-order Trotter formula is a symmetric product formula for approximating the exponential of a sum of noncommuting generators. In quantum dynamics, for a Hamiltonian $H=\sum_{j=1}^L H_j$ and total evolution time $t$ divided into $r$ timesteps, it replaces $e^{-iHt}$ by a forward sweep of half-step exponentials followed by a reverse sweep of half-step exponentials, repeated $r$ times. Because the construction is time symmetric, it cancels the leading first-order splitting error and, in bounded settings, yields cumulative error scaling of order $O(t^3/r^2)$. The method is also known as the symmetric Suzuki formula or Strang splitting, and it occupies a central position in Hamiltonian simulation, operator splitting for differential equations, and digital quantum circuit synthesis (Layden, 2021, Childs et al., 2019, Kluber, 2023).

1. Canonical form and equivalent formulations

For $H=\sum_{j=1}^L H_j$, the second-order product formula used in quantum simulation can be written as

$$U_2=\left[\left(e^{-iH_1\frac{t}{2r}}\cdots e^{-iH_L\frac{t}{2r}}\right)\left(e^{-iH_L\frac{t}{2r}}\cdots e^{-iH_1\frac{t}{2r}}\right)\right]^r.$$

Its defining structural feature is the use of half-step evolutions at the beginning and end of each symmetric cycle. For two operators $S$ and $T$, the same construction takes the familiar Strang form

$t$0

In operator-splitting language for an $t$1-split differential equation $t$2, the corresponding Strang--Marchuk composition is

$t$3

These formulations are equivalent instances of the same symmetric splitting principle (Layden, 2021, Kluber, 2023, Spiteri et al., 2024).

Variant	Formula	Leading scaling
First-order Lie--Trotter	$t$4	$t$5
Second-order Strang	$t$6	$t$7
Multi-term symmetric Suzuki	forward half-steps, then reverse half-steps	$t$8

The formula is used both as an abstract convergence statement and as a constructive algorithm. In quantum computing, the exponentials $t$9 are typically the directly synthesizable unitaries, whereas the full propagator $r$0 is not. In numerical analysis, the same symmetry is valued because it preserves unitarity for Schrödinger evolution and improves accuracy without introducing backward asymmetry (Wittek et al., 2012, Kluber, 2023).

2. Commutator structure and sharp error bounds

The leading error of the second-order Trotter formula is governed by nested commutators. Childs, Su, Tran, Wiebe, and Zhu derived constant-prefactor, commutator-based bounds that avoid the looseness of crude Baker--Campbell--Hausdorff truncations and reproduce known tight bounds for first- and second-order formulas (Childs et al., 2019). For

$r$1

their multi-term bound is

$r$2

For two terms $r$3, this reduces to

$r$4

The symmetry of the construction is algebraically decisive. For symmetric product formulas, only nested commutators of odd order appear in the leading error operator, so the first nonvanishing term is cubic in the timestep rather than quadratic (Maxwell et al., 29 Jun 2026). This is the origin of the standard statement that second-order formulas have local error $r$5 and cumulative error $r$6.

Recent perturbative analysis sharpens the asymptotic picture further. In the large-time asymptotic regime, the error of a product formula depends on the diagonal elements of the BCH error operator in the eigenbasis of the Hamiltonian, rather than on its full spectral norm. For second-order formulas on $r$7 fragments, the same work gives a compact BCH representation that reduces the number of commutators from $r$8 to $r$9, materially changing the practicality of error estimation at scale (Maxwell et al., 29 Jun 2026).

3. Relation to the first-order formula

A notable development is the reinterpretation of first-order Trotter error from a second-order perspective. For

$e^{-iHt}$0

Bañuls and coauthors showed that when $e^{-iHt}$1, the first-order and second-order product formulas are almost identical except at the circuit start and end. Specifically,

$e^{-iHt}$2

The difference between PF1 and PF2 is therefore localized to a pair of boundary half-steps rather than distributed throughout the bulk of the circuit (Layden, 2021).

This observation yields an improved error decomposition: $e^{-iHt}$3 Using Kubo's identity, the boundary commutator can be bounded by

$e^{-iHt}$4

After symmetrizing over the two possible orderings, the resulting first-order bound becomes

$e^{-iHt}$5

with

$e^{-iHt}$6

$e^{-iHt}$7

Conceptually, this reframes previously reported “destructive interference” between timestep errors. The bulk of the first-order circuit can behave like a second-order method, while the dominant discrepancy is a boundary effect. For two-term Hamiltonians, this explains why PF1 can empirically perform far better than naïve $e^{-iHt}$8 worst-case bounds suggest, and why PF2 is often preferable without additional circuit complexity (Layden, 2021).

4. State dependence, unbounded operators, and breakdown of the naive hierarchy

The textbook $e^{-iHt}$9 picture is a bounded-operator, worst-case statement. Krymova, Roos, Basanets, Lubich, and Wolf developed explicit state-dependent bounds for Strang splitting in settings where operator-norm commutator bounds become vacuous, especially for unbounded kinetic and Coulomb terms in quantum chemistry (Burgarth et al., 2023). For an eigenstate $r$0 of $r$1, their second-order error measure

$r$2

obeys

$r$3

where $r$4 and $r$5.

The hydrogen atom case study shows that the higher-order hierarchy can fail for physically relevant states. For the second-order formula, the error scales as $r$6 for $r$7, $r$8 for $r$9, $O(t^3/r^2)$0 for $O(t^3/r^2)$1, $O(t^3/r^2)$2 or $O(t^3/r^2)$3 for $O(t^3/r^2)$4, and only recovers the expected $O(t^3/r^2)$5 scaling for $O(t^3/r^2)$6 (Burgarth et al., 2023). The paper attributes the enlarged error physically to ionization due to the Trotter dynamics, and mathematically to the failure of the necessary domain conditions: higher moments of the potential and kinetic energies diverge for some states.

The second-order framework has also been extended outside associative operator algebras. In Jordan-Banach algebras, a 2024 study solved an open problem on the existence of second-order Trotter error estimation and proved an explicit Strang-type bound: $O(t^3/r^2)$7 This demonstrates that second-order symmetric product formulas and their error analysis can be carried into non-associative settings as well (Chehade et al., 2024).

5. Optimized orderings, compensated formulas, and modern variants

The standard symmetric ordering is not the only practically relevant second-order construction. Liu, Yang, Halder, and Hazzard introduced a geometric framework in which a product formula corresponds to a discrete path, the second-order error is the signed area between that path and the diagonal, and the third-order error is a geometric moment (Liu et al., 2019). Within that framework they proposed two alternative second-order expansions. The “2-Optimal” expansion minimizes the third-order error by dynamic programming and is best at short times. The “2-Diagonal” expansion greedily stays as close as possible to the diagonal, is robust at longer times, and in their numerical study overtakes 2O beyond a crossover near $O(t^3/r^2)$8. The same work reports that 2D always outperforms the standard 2T in both second- and third-order errors for almost all parameter regimes (Liu et al., 2019).

A different line of work combines second-order Trotterization with linear-combination-of-unitaries error compensation. An, Fang, Lin, and Tong propose appending a small LCU correction after the $O(t^3/r^2)$9-th-order Trotter formula, obtaining better time scaling than the $H=\sum_{j=1}^L H_j$0-th-order Trotter formula. For second-order Trotter plus LCU, their summary gives gate complexity $H=\sum_{j=1}^L H_j$1, and for 1D lattice Hamiltonians the commutator-structured construction yields almost linear system-size dependence, reported as $H=\sum_{j=1}^L H_j$2 for the second-order case (Zeng et al., 2022).

Modern error-estimation work has made these refinements materially usable. A 2026 study introduces a compact BCH representation reducing the commutator count for second-order formulas from $H=\sum_{j=1}^L H_j$3 to $H=\sum_{j=1}^L H_j$4, together with importance sampling. It reports accurate second-order Trotter-error estimation for an electronic Hamiltonian at up to 56 qubits and for vibronic dynamics at over 100 qubits, and finds that naive analytical bounds overestimate the required number of Trotter steps by nearly five orders of magnitude in the naphthalene benchmark (Maxwell et al., 29 Jun 2026). A plausible implication is that the practical competitiveness of second-order formulas depends at least as much on error-estimation methodology as on formal asymptotic order.

6. Numerical realizations and scientific uses

Second-order Trotterization is not confined to abstract quantum algorithms. A distributed Trotter--Suzuki solver for the time-dependent Schrödinger equation implemented the second-order formulation with CPU, GPU, and hybrid kernels on a cluster, using block decomposition, halo exchange, double buffering, and MPI communication. That implementation reports scaling close to linear using the CPU kernels, improved GPU efficiency with larger matrices, and a hybrid kernel that is efficient when the matrix size would not fit in GPU memory. Larger quantum systems were reported to scale especially well with a high number of nodes, with system sizes up to $H=\sum_{j=1}^L H_j$5 in single precision (Wittek et al., 2012).

In digital quantum simulation, the second-order formula remains a workhorse because it balances gate synthesis simplicity, qubit frugality, and nontrivial error cancellation. The general motivation is that direct implementation of $H=\sum_{j=1}^L H_j$6 is typically infeasible, whereas exponentials of individual Hamiltonian fragments can often be mapped to elementary gates. Strang splitting reduces the number of Trotter steps needed for a given precision and thereby reduces circuit depth relative to first-order splitting (Kluber, 2023).

The same commutator-based theory that sharpens second-order error bounds also supports applications to second-quantized plane-wave electronic structure, $H=\sum_{j=1}^L H_j$7-local Hamiltonians, rapidly decaying power-law interactions, clustered Hamiltonians, the transverse field Ising model, and quantum ferromagnets (Childs et al., 2019). More recent large-scale estimation work applies second-order analysis to X-ray absorption spectroscopy of an electronic Hamiltonian and to vibronic dynamics, indicating that carefully designed product formulas remain viable at practically relevant system sizes (Maxwell et al., 29 Jun 2026).

The principal misconception surrounding the second-order Trotter formula is that its performance is exhausted by the slogan “error $H=\sum_{j=1}^L H_j$8.” The literature shows a more granular picture. In bounded settings with favorable commutator structure, the formula can admit tight constant-prefactor estimates and unexpectedly low empirical error (Childs et al., 2019, Layden, 2021). For unbounded operators and fat-tailed states, the expected second-order scaling can break down entirely (Burgarth et al., 2023). And in modern large-scale simulations, the dominant practical issue is often not the formal order itself but the ability to estimate the relevant error operator sharply enough to choose the timestep rationally (Maxwell et al., 29 Jun 2026).