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arXiv:2607.11856

Trotter error compensation with polylogarithmic precision and nested-commutator scaling without ancillas

Xinzhao Wang, Shuo Zhou, Ziruo Wang, Pei Zeng, Jinzhao Sun, Qi Zhao, Tom Gur, Tongyang Li

quant-phSubmitted Jul 13, 2026• Announced Jul 14, 2026• 0 citations• Revised Jul 23, 2026 (v2)
v1 · Jul 13, 2026v2 · Jul 23, 2026
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Abstract

Product formulas are among the most practical approaches to Hamiltonian simulation, requiring no ancillary qubits and exhibiting error bounds governed by nested commutators rather than only by Hamiltonian norms. Their circuit size, however, scales polynomially with the inverse precision. We develop a high-order nested-commutator compensation (HNCC) algorithm that preserves the main advantages of product formulas while achieving polylogarithmic precision dependence in the circuit size and the standard $\mathcal{O}(\varepsilon^{-2})$ sampling cost. HNCC uses a truncated Baker--Campbell--Hausdorff expansion to represent high-order Trotter errors by products of nested commutators and compensates these errors at the channel level through randomly sampled Pauli-rotation channels, avoiding Hadamard tests and ancillary qubits. For a fixed $K$-th order product formula applied to a $k$-local Hamiltonian on $N$ qubits with $Γ$ Pauli terms and local interaction strength $g_0$, HNCC estimates $\operatorname{tr}[Oe^{-i tH}ρe^{i tH}]$ to additive precision $\varepsilon\|O\|$ using $\mathcal{O}(\varepsilon^{-2})$ repetitions. Its maximum gate count per circuit is $\mathcal{O}\bigl( kN^{\frac{1}{2K+1}} Γ^{1-\frac{1}{2K+1}} \max\{Γ,N\log(1/\varepsilon)\}^{\frac{1}{2K+1}} (kg_0t\log(1/\varepsilon))^{1+\frac{1}{2K+1}} \bigr)$. Finite-size resource estimates for the periodic Heisenberg chain indicate that HNCC has the lowest estimated $T$-gate count per circuit among the product-formula-based methods considered.

Comments
23 pages, 4 figures, 2 tables

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kN^{\frac{1}{2K+1}}
Γ^{1-\frac{1}{2K+1}}
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(kg_0t\log(1/\varepsilon))^{1+\frac{1}{2K+1}} \bigr).Finite−sizeresourceestimatesfortheperiodicHeisenbergchainindicatethatHNCChasthelowestestimated. Finite-size resource estimates for the periodic Heisenberg chain indicate that HNCC has the lowest estimated .Finite−sizeresourceestimatesfortheperiodicHeisenbergchainindicatethatHNCChasthelowestestimatedT$-gate count per circuit among the product-formula-based methods considered.
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(n3)\mathcal{O}(n^3)O(n3) to O(n)\mathcal{O}(n)O(n) for second-order, and from O(n5)\mathcal{O}(n^5)O(n5) to O(n2)\mathcal{O}(n^2)O(n2) for fourth-order formulas on nnn 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 (Li4_44​Mn2_22​O) 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.
Trotter decomposition provides a simple approach to simulating open quantum systems by decomposing the Lindbladian into a sum of individual terms. While it is established that Trotter errors in Hamiltonian simulation depend on nested commutators of the summands, such a relationship remains poorly understood for Lindbladian dynamics. In this Letter, we derive commutator-based Trotter error bounds for Lindbladian simulation, yielding an O(N)O(\sqrt{N})O(N​) scaling in the number of Trotter steps for locally interacting systems on NNN sites. When estimating observable averages, we apply Richardson extrapolation to achieve polylogarithmic precision while maintaining the commutator scaling. To bound the extrapolation remainder, we develop a general truncation bound for the Baker-Campbell-Hausdorff expansion that bypasses common convergence issues in physically relevant systems. For local Lindbladians, our results demonstrate that the Trotter-based methods outperform prior simulation techniques in system-size scaling while requiring only O(1)O(1)O(1) ancillas. Numerical simulations further validate the predicted system-size and precision scaling.
Product-formula Hamiltonian simulation is naturally suited to near-term quantum processors, but its accuracy is set by two competing errors: finite-step Trotter bias and physical hardware noise. We introduce a joint extrapolation strategy that ties the tunable per-layer noise strength to the Trotter step size, λ(s)=c(sT)p+1λ(s)=c(sT)^{p+1}λ(s)=c(sT)p+1 for a ppp-th order product formula. Along this one-dimensional path, the leading physical-noise and Trotter corrections over the full evolution both enter at order sps^psp and can be canceled by a single Richardson extrapolation. Building on a previously established finite-order Baker--Campbell--Hausdorff truncation bound, we derive a provably commutator-scaling resource guarantee for the joint extrapolation. For local Hamiltonians with local Lindbladian noise, the protocol mitigates physical noise together with Trotter error with only a constant asymptotic overhead relative to noiseless Trotter extrapolation, provided the required noise strengths lie above the intrinsic device-noise floor. We experimentally demonstrate the protocol for Ising dynamics on a superconducting quantum computer using learned noise amplification. Complementary 100-qubit Sparse Pauli Dynamics (SPD) simulations achieve comparable accuracy to two-dimensional Richardson extrapolation with fewer circuit settings.
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Trotter and linear combination of unitary (LCU) operations are two popular Hamiltonian simulation methods. The Trotter method is easy to implement and enjoys good system-size dependence endowed by commutator scaling, while the LCU method admits high-accuracy simulation with a smaller gate cost. We propose Hamiltonian simulation algorithms using LCU to compensate Trotter error, which enjoy both of their advantages. By adding few gates after the K th -order Trotter formula, we realize a better time scaling than 2 K th -order Trotter. Our first algorithm exponentially improves the accuracy scaling of the K th -order Trotter formula. For a generic Hamiltonian, the estimated gate counts of the first algorithm can be 2 orders of magnitude smaller than the best analytical bound of fourth-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 K th -order Trotter. For the lattice system, the second algorithm can achieve 3 to 4 orders of magnitude higher accuracy with the same gate costs as the optimal Trotter algorithm. These algorithms provide an easy-to-implement approach to achieve a low-cost and high-precision Hamiltonian simulation.
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