Trotter Extrapolation for High-Fidelity Quantum Simulation

• Trotter extrapolation is a technique that systematically reduces discretization errors in digital quantum simulations by combining outcomes from multiple step sizes.
• It employs methodologies like Richardson extrapolation and Chebyshev interpolation to cancel leading error terms, achieving exponential improvements in precision.
• The approach adapts step sizes based on commutator structures, significantly lowering circuit depth and resource requirements for high-fidelity quantum simulations.

Trotter extrapolation in quantum computing is a class of algorithmic error mitigation techniques designed to systematically reduce the discretization error—known as Trotter error—in digital quantum simulations that approximate continuous-time evolution via a product formula (Trotterization). By leveraging Richardson extrapolation and polynomial interpolation over families of Trotterized evolutions, these methods enable exponential improvements in precision, lower circuit depth, and offer practical pathways to achieving algorithmic error rates near theoretical bests without requiring error-prone ancillary qubit constructions. This article synthesizes the principles, practical realization, and analytical guarantees of Trotter extrapolation, drawing specifically on recent advancements (Watson et al., 2024, Rendon et al., 2022, Childs et al., 2019, Rendon, 2023, Chen, 2024, Zylberman et al., 21 Aug 2025, Ikeda et al., 2023).

1. Trotterization and the Origin of Error

Digital quantum simulation of many-body dynamics typically relies on a decomposition of the system Hamiltonian $H = \sum_{\gamma=1}^\Gamma H_\gamma$ into composable segments—often local terms or physically meaningful interactions. The exact time evolution $U(T) = e^{-i H T}$ is approximated by splitting the total time $T$ into $r$ steps of size $\tau = T/r$ and applying a product formula $P(\tau)$ in each step:

$$P(\tau) = \prod_{\gamma=1}^\Gamma e^{-i H_\gamma \tau}\,,$$

$U_\text{approx}(T) = P(\tau)^r \approx e^{-i H T}.$

The Trotter error arises because $P(\tau) \neq e^{-i H \tau}$ for non-commuting $H_\gamma$. For a $p$th-order product formula, the local error scales as $O(\tau^{p+1})$. When propagated over $r$ steps, the leading-order global operator-norm error for observables such as $$\langle O(T) \rangle = \text{Tr}[O U(T) \rho_0 U(T)^\dagger]$$ is $O(\tau^p T)$, resulting in the well-known scaling (Childs et al., 2019, Layden, 2021):

$$\| U_\text{approx}(T) - e^{-i H T} \| = O(T^{p+1}/r^p)\,.$$

Consequently, achieving precision $\epsilon$ requires $r = O\bigl((T^{p+1}/\epsilon)^{1/p}\bigr)$ steps, yielding circuit depth growth that is polynomial in $1/\epsilon$.

2. Algorithmic Structure of Trotter Extrapolation

The key insight underlying Trotter extrapolation is that the leading (and systematically structured) error can be canceled by performing Trotterized simulations at multiple step sizes and suitably post-processing the outcomes. Central methodologies include:

• Richardson Extrapolation: If the estimate $f(\tau)$ of some observable or amplitude admits an expansion $$f(\tau) = f(0) + c_p \tau^{p} + c_{p+1}\tau^{p+1} + \dots$$, then a linear combination of values at distinct step sizes $\tau_k$ can be formed to cancel the first $m-1$ terms, e.g.,

$$F^{(m)}(\tau) = \sum_{k=1}^m b_k f(\tau_k), \quad \text{with}\ \sum_{k} b_k \tau_k^j = 0\ (j=1,\ldots, m-1),$$

resulting in an error $O(\tau^{p+m-1})$ (Watson et al., 2024, Childs et al., 2019, Layden, 2021).

• Polynomial (Chebyshev) Interpolation: By viewing the step-size dependence as an analytic function over a complex domain, observables are evaluated at Chebyshev nodes $\tau_k$ and extrapolated to $\tau = 0$ by fitting a Lagrange-form polynomial interpolant. The Chebyshev strategy is numerically stable and avoids the Runge phenomenon, enabling exponential convergence in the interpolation order $n$ (Rendon et al., 2022, Rendon, 2023).
• Trotter$(m,n)$ Adaptive-Order Protocol: Direct measurement of the difference between $m$th- and $n$th-order product formula outcomes at fixed step size $dt$ gives a robust, device-compatible error estimator for adaptivity and resource control (Ikeda et al., 2023).

3. Analytical Guarantees and Scaling Results

Trotter extrapolation yields an exponential improvement in circuit depth scaling with precision $\epsilon$. For a $p$th-order formula and a total simulation time $T$, Richardson or Chebyshev-based extrapolation gives the following resource scalings (Watson et al., 2024, Rendon, 2023, Rendon et al., 2022):

Method	Max Depth $D_\mathrm{max}$	Total Trotter Steps
Raw $p$th-order Trotter	$O((\lambda T)^{1+1/p}\epsilon^{-1/p})$	$O((\lambda T)^{1+1/p}\epsilon^{-1/p})$
Extrapolated (Richardson/Poly.)	$$O((\lambda T)^{1+1/p}\operatorname{polylog}(1/\epsilon))$$	$$O((\lambda T)^{1+1/p}\epsilon^{-2}\operatorname{polylog}(1/\epsilon))$$ (incoherent), $$O((\lambda T)^{1+1/p}\epsilon^{-1}\operatorname{polylog}(1/\epsilon))$$ (coherent)

Here, $\lambda \le 4 \sum_\gamma \|H_\gamma\|$, and the commutator structure can yield tighter problem-dependent scalings.

For Chebyshev-extrapolation protocols, error bounds of the form

$$|f(0) - P_n(0)| \le \frac{4C}{\rho - 1} \rho^{-(n+1)}$$

are attained for analytic $f(\tau)$ in a Bernstein ellipse parameterized by $\rho > 1$, supporting $n = O(\log(1/\epsilon))$ for desired error $\epsilon$ (Rendon et al., 2022, Rendon, 2023).

Trotter$(m, n)$ adaptivity protocols further enable the dynamic maximization of step size at each simulation segment, resulting in typical resource savings of an order of magnitude over worst-case-commutator bound-based step size selection (Ikeda et al., 2023).

4. Practical Protocols and Measurement Considerations

A typical Trotter extrapolation protocol proceeds as follows (Watson et al., 2024, Rendon, 2023, Rendon et al., 2022):

1. Choice of Product Formula: Fix product formula order $p$ and simulation time $T$. Compute commutator norm $\lambda$, or a tighter model-based variant.
2. Extrapolation Order/Nodes: Choose $m = \Theta(\log(1/\epsilon))$. For Richardson, select integer step counts $r_k$ and weights $b_k$ solving a Vandermonde system; for Chebyshev, define nodes $s_k$ and map these to nearest implementable step sizes.
3. Circuit Execution: For each $k$, prepare the simulation with step size $s_k T$ (corresponding to $r_k$ steps), compute the observable/expectation value to precision $O(\epsilon / 2 \|b\|_1)$.
4. Extrapolation/Post-processing: Form the linear combination $\sum_k b_k f(s_k)$ (Richardson) or polynomial extrapolate $P_{m-1}f(0)$ (Chebyshev).

For expectation-value or amplitude estimation, either incoherent sampling ($O(\epsilon^{-2})$ cost per node) or coherent strategies such as amplitude estimation ($O(\epsilon^{-1})$ cost per node) may be employed (Watson et al., 2024, Rendon et al., 2022).

Fractional power simulations required for certain step sizes (e.g., in Chebyshev node implementation) can be efficiently realized via classical linear combinations of integer powers (cardinal sine methods), avoiding the need for signal-processing ancillas (Rendon, 2023).

5. State-Dependent and Model-Specific Error Scalings

Several recent works have emphasized that naive operator-norm error bounds can be strongly pessimistic. By tracking the action of commutators on the simulated state or leveraging model-specific commutator decay/locality, both error and resource costs can be drastically reduced (Zylberman et al., 21 Aug 2025, Childs et al., 2019, Chen, 2024). For example, in quantum algorithms for transport equations, vector-norm error bounds eliminate the exponential in system-size cost, yielding $L=O(T^2/\epsilon)$ Trotter steps compared to $O(T^2 4^n/\epsilon)$ under operator norm (Zylberman et al., 21 Aug 2025).

Commutant decomposition techniques yield two error contributions with distinct scaling—the $O(\tau^p t)$ "secular" term and an $O(\tau^p)$ constant-in-time term—allowing Richardson-type schemes to efficiently cancel dominant contributions (Chen, 2024).

6. Benchmarking, Limitations, and Best Practices

Empirical evidence across various platforms and model classes demonstrates that Trotter extrapolation yields substantial reductions in required step count and circuit depth. Adaptive order protocols such as Trotter$(m,n)$ routinely find step sizes an order of magnitude larger than commutator-norm-based prescriptions, without sacrificing precision guarantees (Ikeda et al., 2023).

Best practices for maximizing performance and accuracy include:

• Choosing error bounds in terms of nested commutator structures relevant to the target Hamiltonian (Childs et al., 2019).
• Using low-order ($p = 2,4$) product formulas with careful extrapolation often outperforms high-order schemes in near-term regimes.
• Balancing truncation and Trotter error, especially for long-range interacting systems or large lattices.
• Adopting Chebyshev/interpolative schemes when numerical stability at high extrapolation order is required (Rendon et al., 2022, Rendon, 2023).

A plausible implication is that the ultimate practical scaling of digital quantum simulation for many physically relevant tasks may be essentially quasi-linear in $T$ and only polylogarithmic in $1/\epsilon$, up to hardware noise and state-preparation limits.

7. Future Directions and Extensions

Recent developments suggest several avenues for further progress:

• Sharper model-specific error analysis leveraging state fidelity and local observable propagation, potentially further reducing quantum resource requirements (Zylberman et al., 21 Aug 2025, Chen, 2024).
• Integration of on-the-fly error estimation with error mitigation for hardware noise, possibly via hybrid extrapolation combining Trotter error and device error cancellation.
• Extension to time-dependent Hamiltonians and open system simulation, where commutant structures and error separation may remain exploitable (Ikeda et al., 2023).

In summary, Trotter extrapolation methods—through rigorous analytical foundations and practical adaptability—now constitute a central toolset for high-fidelity algorithmic error reduction in quantum simulation, providing a blueprint for continual resource savings and scalability improvements as quantum hardware and algorithmic sophistication advance (Watson et al., 2024, Childs et al., 2019, Rendon, 2023, Rendon et al., 2022, Zylberman et al., 21 Aug 2025, Ikeda et al., 2023, Chen, 2024, Layden, 2021).