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arXiv:2408.14385 · PRX Quantum

Exponentially Reduced Circuit Depths Using Trotter Error Mitigation

James D. Watson, Jacob Watkins

Published Jul 2, 2025• 12 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/kw39-yxq5

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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.
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 10−210^{-2}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 ααα, 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(Δt)^{2}(Δt)2 algorithmic contribution and steepens the Trotter-error scaling with the two-qubit circuit depth ddd from d−2d^{-2}d−2 to d−4d^{-4}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.
Quantum metrology allows for measuring properties of a quantum system at the optimal Heisenberg limit. However, when the relevant quantum states are prepared using digital Hamiltonian simulation, the accrued algorithmic errors will cause deviations from this fundamental limit. In this work, we show how algorithmic errors due to Trotterized time evolution can be mitigated through the use of standard polynomial interpolation techniques. Our approach is to extrapolate to zero Trotter step size, akin to zero-noise extrapolation techniques for mitigating hardware errors. We perform a rigorous error analysis of the interpolation approach for estimating eigenvalues and time-evolved expectation values, and show that the Heisenberg limit is achieved up to polylogarithmic factors in the error. Our work suggests that accuracies approaching those of state-of-the-art simulation algorithms may be achieved using Trotter and classical resources alone for a number of relevant algorithmic tasks.
Product-formula (PF) based quantum simulation is a promising approach for simulating quantum systems on near-term quantum computers. Achieving a desired simulation precision typically requires a polynomially increasing number of Trotter steps, which remains challenging due to the limited performance of current quantum hardware. To alleviate this issue, post-processing techniques such as the multi-product formula (MPF) have been introduced to suppress algorithmic errors within restricted hardware resources. In this work, we propose a dual-channel multi-product formula that achieves a two-fold improvement in Trotter error scaling. As a result, our method enables the target simulation precision to be reached with approximately half the circuit depth compared to conventional MPF schemes. Importantly, the reduced circuit depth directly translates into lower physical error mitigation overhead when implemented on real quantum hardware. We demonstrate that, for a fixed CNOT count as a measure of quantum circuit, our proposal yields significantly smaller algorithmic errors, while the sampling error remains essentially unchanged.
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.
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.
Understanding the dynamics of quantum systems is crucial in many areas of physics, but simulating many-body systems presents significant challenges due to the large Hilbert space to navigate and the exponential growth of computational overhead. Quantum computers offer a promising platform to overcome these challenges, particularly for simulating the time evolution with Hamiltonians. Trotterization is a widely used approach among available algorithms in this regard, and well suited for near-term quantum devices. However, it introduces algorithmic Trotter errors due to the non-commutativity of Hamiltonian components. Several techniques such as multi-product formulas have been developed to mitigate Trotter errors, but often require deep quantum circuits, which can introduce additional physical errors. In this work, we propose a resource-efficient scheme to reduce the algorithmic Trotter error with relatively shallow circuit depth. We develop a profiling method by introducing an auxiliary parameter to estimate the error effects in expectation values, enabling significant error suppression with a fixed number of Trotter steps. Our approach offers an efficient way of quantum simulation on near-term quantum processors with shallow circuits.