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arXiv:2602.00555

Entanglement-Dependent Error Bounds for Hamiltonian Simulation

Prateek P. Kulkarni

quant-phcs.CCSubmitted Jan 31, 2026• Announced Feb 3, 2026• 0 citations
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Abstract

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 $S_\text{max}$ across all bipartitions, we prove that the first-order Trotter error scales as $\mathcal{O}(t^2 S_\text{max} \operatorname{polylog}(n)/r)$ rather than the worst-case $\mathcal{O}(t^2 n/r)$, where $n$ is the system size and $r$ is the number of Trotter steps. This yields improvements of $\tildeΩ(n^2)$ for one-dimensional area-law systems and $\tildeΩ(n^{3/2})$ for two-dimensional systems. We extend these bounds to higher-order Suzuki formulas, where the improvement factor involves $2^{pS^*/2}$ for the $p$-th order formula. We further establish a separation result demonstrating that volume-law entangled systems fundamentally require $\tildeΩ(n)$ 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.

Comments
35 pages, 2 tables, 1 figure

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Product formulas are a basic tool for simulating quantum many-body dynamics, but a fundamental degree of freedom in their application---the ordering of Hamiltonian terms---has remained largely unexplored. Here we systematically study such ordering effects, quantifying their impact on simulation accuracy and exploiting them to design more accurate product-formula simulations. We develop an efficiently computable, ordering-sensitive bound on the finite-step-size Trotter error for geometrically local Hamiltonians for arbitrarily high-order product formulas. For translation-invariant one-dimensional chains, our bounds are close to exact second-, fourth-, and sixth-order Trotter errors in small-system tests, while on large-scale systems they improve on previous higher-order analytical bounds by orders of magnitude. We further use the local structure of the error bounds and estimates to optimize ordering efficiently, reducing the long-chain problem to a minimum-mean-weight-cycle problem on a weighted de Bruijn graph. For random translation-invariant chains, sequential (staircase-like) orderings consistently outperform standard brickwall circuits; when circuit depth is restricted, the optimizer instead favors wave-like parallel-sequential orderings, which preserve much of the accuracy advantage over brickwall circuits. These results establish ordering as a useful degree of freedom for both understanding and improving product-formula simulation.
Abstract Quantum simulation is a promising application of future quantum computers. Product formulas, or Trotterization, are the oldest and still remain an appealing method to simulate quantum systems. For an accurate product formula approximation, the state-of-the-art gate complexity depends on the number of terms in the Hamiltonian and a local energy estimate. In this work, we give evidence that product formulas, in practice, may work much better than expected. We prove that the Trotter error exhibits a qualitatively better scaling for the vast majority of input states, while the existing estimate is for the worst states. For general k-local Hamiltonians and higher-order product formulas, we obtain gate count estimates for input states drawn from any orthogonal basis. The gate complexity significantly improves over the worst case for systems with large connectivity. Our typical-case results generalize to Hamiltonians with Fermionic terms, with input states drawn from a fixed-particle number subspace, and with Gaussian coefficients (e.g., the SYK models). Technically, we employ a family of simple but versatile inequalities from non-commutative martingales called uniform smoothness, which leads to Hypercontractivity, namely p-norm estimates for k-local operators. This delivers concentration bounds via Markov’s inequality. For optimality, we give analytic and numerical examples that simultaneously match our typical-case estimates and the existing worst-case estimates. Therefore, our improvement is due to asking a qualitatively different question, and our results open doors to the study of quantum algorithms in the average case.
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)\mathcal{O}(\varepsilon^{-2})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 KKK-th order product formula applied to a kkk-local Hamiltonian on NNN qubits with ΓΓΓ Pauli terms and local interaction strength g0g_0g0​, HNCC estimates tr⁡[Oe−itHρeitH]\operatorname{tr}[Oe^{-i tH}ρe^{i tH}]tr[Oe−itHρeitH] to additive precision ε∥O∥\varepsilon\|O\|ε∥O∥ using O(ε−2)\mathcal{O}(\varepsilon^{-2})O(ε−2) repetitions. Its maximum gate count per circuit is O(kN12K+1Γ1−12K+1max⁡{Γ,Nlog⁡(1/ε)}12K+1(kg0tlog⁡(1/ε))1+12K+1)\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)O(kN2K+11​Γ1−2K+11​max{Γ,Nlog(1/ε)}2K+11​(kg0​tlog(1/ε))1+2K+11​). Finite-size resource estimates for the periodic Heisenberg chain indicate that HNCC has the lowest estimated TTT-gate count per circuit among the product-formula-based methods considered.
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.
Suppressing the Trotter error in dynamical quantum simulation typically requires running deeper circuits, posing a great challenge for noisy near-term quantum devices. Studies have shown that the empirical error is usually much smaller than the one suggested by existing bounds, implying the actual circuit cost required is much less than the ones based on those bounds. Here, we improve the estimate of the Trotter error over existing bounds, by introducing a general framework of commutant decomposition that separates disjoint error components that have fundamentally different scaling with time. In particular we identify two error components that each scale as $O({\ensuremath{\tau}}^{p}t)$ and $O({\ensuremath{\tau}}^{p})$ for a pthp\mathrm{th}pth-order product formula evolving to time ttt using a fixed step size $\ensuremath{\tau}$, implying one would scale linearly with time ttt and the other would be a constant of ttt. We show that this formalism not only straightforwardly reproduces previous results but also provides a better error estimate for higher-order product formulas. We demonstrate the improvement both analytically and numerically. We also apply the analysis to observable error relating to the heating in Floquet dynamics and thermalization, which is of independent interest.