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, $λ(s)=c(sT)^{p+1}$ for a $p$-th order product formula. Along this one-dimensional path, the leading physical-noise and Trotter corrections over the full evolution both enter at order $s^p$ 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.
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+1 for a p-th order product formula. Along this one-dimensional path, the leading physical-noise and Trotter corrections over the full evolution both enter at order sp 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.
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) to O(n) for second-order, and from O(n5) to O(n2) 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.
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 O(ε−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 g0, HNCC estimates tr[Oe−itHρeitH] to additive precision ε∥O∥ using O(ε−2) repetitions. Its maximum gate count per circuit is O(kN2K+11Γ1−2K+11max{Γ,Nlog(1/ε)}2K+11(kg0tlog(1/ε))1+2K+11). 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.
Digital quantum simulation of many-body dynamics faces a tension between algorithmic Trotter error and physical noise that accumulates with circuit depth. Typical higher-order product formulas mitigate the algorithmic error at the cost of deeper circuits. Our benchmark shows that, within the limited physical error budget, the feasible advantage is confined to absolute errors well below the 10−2 scale, which vanishes under noise levels of current quantum hardware. In sharp contrast to the previous Trotter error mitigation methods, we introduce a constant-depth multi-product formula (cd-MPF) that suppresses the Trotter error by combining multiple circuits at fixed circuit depth. We identify an auxiliary parameter α, which reshapes the Trotter error terms while leaving the target evolution invariant. The classical linear combination of the measured expectation values cancels the leading (Δt)2 algorithmic contribution and steepens the Trotter-error scaling with the two-qubit circuit depth d from d−2 to d−4. Combined with physical-noise mitigation, our method can serve as a key ingredient for realizing long-time quantum dynamics simulation on near-term hardware.
We establish tight connections between entanglement entropy and the approximation error in Trotter-Suzuki product formulas for Hamiltonian simulation. Product formulas remain the workhorse of quantum simulation on near-term devices, yet standard error analyses yield worst-case bounds that can vastly overestimate the resources required for structured problems. For systems governed by geometrically local Hamiltonians with maximum entanglement entropy Smax across all bipartitions, we prove that the first-order Trotter error scales as O(t2Smaxpolylog(n)/r) rather than the worst-case O(t2n/r), where n is the system size and r is the number of Trotter steps. This yields improvements of Ω~(n2) for one-dimensional area-law systems and Ω~(n3/2) for two-dimensional systems. We extend these bounds to higher-order Suzuki formulas, where the improvement factor involves 2pS∗/2 for the p-th order formula. We further establish a separation result demonstrating that volume-law entangled systems fundamentally require Ω~(n) more Trotter steps than area-law systems to achieve the same precision. This separation is tight up to logarithmic factors. Our analysis combines Lieb-Robinson bounds for locality, tensor network representations for entanglement structure, and novel commutator-entropy inequalities that bound the expectation value of nested commutators by the Schmidt rank of the state. These results have immediate applications to quantum chemistry, condensed matter simulation, and resource estimation for fault-tolerant quantum computing.
Product formulas are a popular class of digital quantum simulation algorithms due to their conceptual simplicity, low overhead, and performance, which often exceeds theoretical expectations. Recently, Richardson extrapolation and polynomial interpolation have been proposed to mitigate the Trotter error incurred by the use of these formulas. This work provides a rigorous, general analysis of these techniques for computing time-evolved observables, simplifying the interpolation algorithm in the process, and shows that extrapolation generically improves the performance of product formulas for this task. We demonstrate that, to achieve error ϵ in a simulation of time T using a p th -order product formula with extrapolation, circuit depths of O ( T 1 + 1 / p polylog ( 1 / ϵ ) ) are sufficient—an exponential improvement in the precision over product formulas alone. Furthermore, we prove that these algorithms achieve commutator scaling, and improve the T complexity for the interpolation algorithm. By relaxing the requirement of performing exact Chebyshev interpolation, our simplified algorithm eliminates the need for fractional implementations of Trotter steps, reducing computational overhead. Finally, we show these techniques can be combined with the classical shadows method to estimate many time-evolved local observables. Taken together, our findings provide the strongest evidence yet for the utility of Trotter error-mitigation techniques in algorithmic applications.
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) scaling in the number of Trotter steps for locally interacting systems on N 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) ancillas. Numerical simulations further validate the predicted system-size and precision scaling.