Trotter Extrapolation in Quantum Simulation

• Trotter extrapolation is an error-mitigation technique that systematically reduces algorithmic errors in digital quantum simulation by combining results from multiple Trotter step sizes.
• It leverages Richardson extrapolation, Vandermonde cancellation, and Chebyshev interpolation to cancel leading error terms and achieve exponential improvements in precision-resource scaling.
• The method underpins precision-guaranteed quantum simulations by lowering circuit depth requirements and adapting to state-dependent error profiles for enhanced robustness.

Trotter extrapolation is a family of error-mitigation techniques in Hamiltonian simulation that systematically reduce, or even exponentially suppress, the algorithmic error incurred by product formula (Trotter–Suzuki) approximations. The methodology leverages Richardson extrapolation, Vandermonde polynomial interpolation, and Chebyshev interpolation to combine simulation results at multiple Trotter step sizes, canceling leading error terms and thereby achieving superior precision-resource scaling relative to unmitigated approaches. This framework subsumes both traditional order-raising strategies and recent quantum error mitigation primitives, and has become a foundation for precision-guaranteed digital quantum simulation.

1. Product Formula Simulation and Trotter Error Structure

A standard approach to digital quantum simulation implements the time evolution $U(T) = e^{-i H T}$ for a time-independent Hamiltonian $H = \sum_{j=1}^\Gamma H_j$ via a $p$-th order product formula (Trotter–Suzuki expansion), denoted $\Phi_p(h)$. The simulated evolution is realized as $U_p(T;h) = [\Phi_p(h)]^n$ with step size $h = T/n$. For fixed $p$, the local (single-step) approximation error is $E_p(h) \equiv \Phi_p(h) - e^{-i H h} = O(h^{p+1})$, leading to a global error for $n$ steps scaling as $O(n h^{p+1}) = O(T h^p)$.

Importantly, the constants in the error bound are controlled by explicit sums of nested commutators of the $H = \sum_{j=1}^\Gamma H_j$0, quantifiable as $H = \sum_{j=1}^\Gamma H_j$1 (Childs et al., 2019). This commutator structure enables fine-grained error analysis and resource optimization (Watson et al., 2024).

2. Richardson Extrapolation and Vandermonde Cancellation

Richardson extrapolation cancels leading error terms by forming linear combinations of observable estimates at several step sizes. For observable evolution $H = \sum_{j=1}^\Gamma H_j$2 (where $H = \sum_{j=1}^\Gamma H_j$3, $H = \sum_{j=1}^\Gamma H_j$4), the Taylor expansion $H = \sum_{j=1}^\Gamma H_j$5 encodes the Trotter error structure (Watson et al., 2024, Childs et al., 2019, Burgarth et al., 2022).

A linear combination

$H = \sum_{j=1}^\Gamma H_j$6

with coefficients $H = \sum_{j=1}^\Gamma H_j$7 chosen so that $H = \sum_{j=1}^\Gamma H_j$8 for $H = \sum_{j=1}^\Gamma H_j$9 (solving a Vandermonde system) cancels all error terms up to $p$0. This construction is the core of Trotter extrapolation, raising the effective order without modifying the simulated circuits themselves (Watson et al., 2024).

For two-parameter extrapolation, the explicit formula

$p$1

cancels $p$2 error for a $p$3-th order formula using two runs at step sizes $p$4 and $p$5 (Chen, 2024, Watson et al., 2024).

3. Chebyshev Interpolation and Stable High-Order Extrapolation

Stability and rapid convergence of polynomial interpolation benefit from the use of optimally-chosen nodes. Chebyshev nodes are employed to mitigate the ill-conditioning associated with equispaced grids and suppress large Runge oscillations (Rendon et al., 2022, Rendon, 2023). For $p$6 even, the $p$7 Chebyshev nodes on $p$8 are $p$9.

Constructing a degree-$\Phi_p(h)$0 polynomial interpolant $\Phi_p(h)$1 and evaluating at zero (or the desired target) provides the extrapolated estimate:

$\Phi_p(h)$2

where the weights $\Phi_p(h)$3 admit closed forms controlled by the Chebyshev structure (Rendon et al., 2022). With $\Phi_p(h)$4 analytic in a neighborhood, the interpolation achieves geometric error decay in $\Phi_p(h)$5, i.e., $\Phi_p(h)$6 for $\Phi_p(h)$7 (Rendon, 2023). This allows for exponential suppression of algorithmic error with modest circuit depth overhead.

4. Circuit Depth and Precision Scaling

The circuit depth required for unmitigated Trotter simulation to reach error $\Phi_p(h)$8 obeys $\Phi_p(h)$9 for $U_p(T;h) = [\Phi_p(h)]^n$0-th order formulas (Watson et al., 2024, Childs et al., 2019). Trotter extrapolation produces an exponential improvement to $U_p(T;h) = [\Phi_p(h)]^n$1, as the number of required nodes (distinct step sizes) for Chebyshev or Richardson-based extrapolation scales only polylogarithmically with $U_p(T;h) = [\Phi_p(h)]^n$2 (Watson et al., 2024, Rendon, 2023, Rendon et al., 2022). The depth–precision scaling thus approaches that of block-encoding methods, but with significantly reduced circuit overhead and no requirement for ancillary block-encoding registers.

A summary comparison is given below:

Method	Depth scaling in $U_p(T;h) = [\Phi_p(h)]^n$3	Notes
Unmitigated Trotter	$U_p(T;h) = [\Phi_p(h)]^n$4	Polynomial dependence
Trotter extrapolation (Chebyshev/Richardson)	$U_p(T;h) = [\Phi_p(h)]^n$5	Exponential improvement over standard
Block-encoding (LCU/QSP)	$U_p(T;h) = [\Phi_p(h)]^n$6	Requires ancillas, more complex primitives

5. Simultaneous Physical and Algorithmic Error Mitigation

Zero-noise extrapolation (ZNE) techniques, originally developed for physical error mitigation, can be unified with Trotter extrapolation into a single one-dimensional polynomial interpolation. By simultaneously scaling the Trotter step size $U_p(T;h) = [\Phi_p(h)]^n$7 and noise strength $U_p(T;h) = [\Phi_p(h)]^n$8 with $U_p(T;h) = [\Phi_p(h)]^n$9, interpolation in $h = T/n$0 enables joint mitigation of algorithmic and hardware errors (Mohammadipour et al., 28 Feb 2025). The procedure mirrors the construction of traditional Trotter extrapolation, employing Chebyshev/Lagrange nodes, Richardson weights, and (optionally) least-squares fitting to optimize bias–variance trade-off under shot noise.

Rigorous sample complexity analysis establishes that $h = T/n$1 nodes and $h = T/n$2 shots per node, with constant $h = T/n$3 dependent on problem parameters, suffice for $h = T/n$4-accurate extrapolation (Mohammadipour et al., 28 Feb 2025).

6. State-Dependent Extrapolation and Adaptive Strategies

State-dependent error bounds allow the leading error coefficients to be computed or estimated for specific initial states or eigenstates of $h = T/n$5, refining extrapolation estimates and enabling a posteriori validity checks (Burgarth et al., 2022). Adaptive methods, such as the Trotter$h = T/n$6 algorithm (Ikeda et al., 2023), embed extrapolation "locally" at each simulation step, dynamically choosing time steps based on measured discrepancies between $h = T/n$7-th and $h = T/n$8-th order Trotterizations, and thereby maximizing circuit efficiency while guaranteeing precision.

Contrasted with global (batch) extrapolation—which can be sensitive to interpolation instability and requires multiple full-length simulations—such adaptive local-extrapolation approaches realize precision guarantees with larger and more efficient Trotter steps.

7. Extensions, Limitations, and Algebraic Generality

Trotter extrapolation extends beyond conventional $h = T/n$9-algebras. The methodology has been generalized to non-associative settings such as Jordan–Banach algebras, preserving the key error-cancellation features of Richardson construction and enabling systematically higher-order product formula approximations (Chehade et al., 2024).

The main technical limitation is that coefficients for node selection and weight construction must be chosen to avoid numerical instability (often via Chebyshev grids), and that the method is built on the analyticity and commutator control of the underlying product formula expansion. Nevertheless, practical implementations on quantum hardware (including noisy and resource-constrained platforms) benefit directly from these advances, as confirmed in numerous benchmark settings (Watson et al., 2024, Childs et al., 2019, Rendon, 2023).

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Key references: Watson & Watkins (Watson et al., 2024), Mohammadipour & Li (Mohammadipour et al., 28 Feb 2025), Childs et al. (Childs et al., 2019), Layden (Layden, 2021), Ikeda et al. (Ikeda et al., 2023), Chehade et al. (Chehade et al., 2024), Bachmann & Lange (Bachmann et al., 2021), Gharibian et al. (Rendon et al., 2022), and O’Gorman et al. (Rendon, 2023).