First-Order Trotterization Error in Quantum Simulation

• First-order Trotterization error is the leading error arising when discretizing continuous quantum evolution into sequential exponential steps, due to noncommuting Hamiltonian terms.
• It scales as O(T²/k) per step and accumulates based on the structure of the Hamiltonian, with errors less sensitive for sparse systems.
• Practical guidelines include optimizing the number of Trotter steps and accounting for experimental fluctuations, with self-healing effects enhancing fidelity in adiabatic quantum simulations.

First-order Trotterization error arises from approximating the continuous-time evolution of a quantum system, governed by a Hamiltonian expressed as a sum of noncommuting terms, with a product of exponentials of these terms (the Trotter–Suzuki expansion). In the context of digital quantum simulation—especially for adiabatic state preparation and other gate-based quantum algorithms—first-order Trotterization is the simplest and most widely used digitization method, but it intrinsically introduces a finite error per step due to operator noncommutativity. Both the scaling of this error and its cumulative effects under experimental constraints are central for understanding precision, resource cost, and robustness in quantum simulations.

1. Formal Structure and Quantification of First-Order Trotterization Error

Consider a time-dependent Hamiltonian expressed as $H(t) = f(t) H_0 + g(t) H_p$ with control functions $f(t), g(t)$ (e.g., $f(t) = 1-t/T$, $g(t) = t/T$ for total evolution time $T$). The continuous time-evolution operator is

$$U(T) = \mathcal{T} \exp\left[-i \int_0^T H(t) dt\right]$$

In first-order Trotterization (Trotter–Suzuki), the evolution is discretized into $k$ steps of size $\tau = T/k$:

$$U(T) \approx \prod_{a=0}^{k-1} \exp\left[-i H(a\tau) \tau\right]$$

Each “Trotter” step approximates evolution with the time-independent Hamiltonian $H(a\tau)$ for duration $\tau$.

Expanding the discrete evolution operator, the leading error (after comparing with the ideal continuous evolution) arises from noncommuting Hamiltonian terms. For a generic first-order product formula applied to $H = \sum_{j=1}^{\Gamma} H_j$, the error for a single step is (Childs et al., 2019):

$$\|\mathcal{S}_1(\tau) - e^{-i H \tau}\| = O\left(\left(\sum_{j}\| H_j \| \, \tau \right)^2\right)$$

The dominant term derives from the nested commutator:

$$\mathcal{S}_1(\tau) = e^{-i H \tau} + \frac{(-i \tau)^2}{2} \sum_{j<k} [H_k, H_j] + O(\tau^3)$$

2. Origin, Scaling, and Operator Structure of Cumulative Error

By expanding Trotterized evolution and comparing to the exact evolution (see Eq. (3) and (4) in (Sun et al., 2018)), the cumulative error over $k$ steps is dictated by the sum of commutators evaluated at each discretized time point. The principal error terms can be grouped as:

• $D_1 = (H_p - H_0)\tau / 2$ (from discretization of time-dependence)
• $D_2 = -\frac{1}{2}(-i\tau)^2 \sum H^2(\alpha \tau)$ (self-interaction errors)
• $$D_3 = -\frac{1}{2}(-i\tau)^2 \sum [H_p(\beta\tau), H_0(\beta\tau)]$$ (true Trotter error: noncommutativity)

The cumulative infidelity (one minus the squared overlap between the adiabatic target and Trotterized final state) is estimated by:

$$\Delta \equiv 1 - |\langle \psi_{\text{ad}} | \psi_f \rangle|^2 \approx \frac{T^2}{3k}(A + B \, E_{pg}(N) + C \, E_{pg}^2(N))$$

where $A, B, C$ are $O(1)$ with respect to system size and $E_{pg}(N)$ is the ground state energy of $H_p$. For typical quantum systems, $E_{pg}(N) \sim \sqrt{N}$, so the error can scale as $O(N)$ for fixed $T/k$, but if $H_p$ is sparse, the scaling with $N$ is much weaker.

As $k$ increases (i.e., as $\tau$ decreases), the error falls as $O(T^2 / k)$. Achieving error $\Delta$ thus requires $k \sim T^2/\Delta$, with further dependence on system size entering via $E_{pg}(N)$.

3. Experimental Implementation: Linear-Optical Circuits and Parameter Fluctuations

The first-order Trotterization scheme is mapped to linear-optical networks by decomposing each time-step propagator into a product of diagonal operations (simulated by phase shifters) and off-diagonal interactions (enacted by beam splitters and phase shifters), corresponding to number and bilinear terms, respectively [see Eq. (2) and Appendix D in (Sun et al., 2018)]. The required optical elements scale only with the number of nonzero Hamiltonian matrix elements.

To address real-world parameter fluctuations in the duration $\tau$ of each optical element, the analysis introduces a Randomized Trotter Formula (RTF):

$$U(T) \approx \prod_{a} \exp\left[ -i H(a\tau) \tau_a \right],\quad \tau_a = \tau (1 + g_a)$$

where $g_a$ is a zero-mean random variable. As $\sum_a g_a \to 0$ for large $k$, the dominant error term remains dictated by Eq. (5) above, demonstrating that Trotter error scaling and thus cost estimation are robust against such classical fluctuations.

4. Implications for Resource Scaling and Simulation Cost

Two principal conclusions follow from this error structure:

• For sparse $H_p$ (i.e., with $O(N)$ or fewer nonzero interactions), the required number of Trotter steps $k$ grows very slowly with system size $N$ for fixed accuracy [see numerical results and Eq. (5) in (Sun et al., 2018)].
• For denser Hamiltonians, $k$ can scale less favorably, typically as a weak polynomial (at most $O(N)$ or $O(\sqrt{N})$ in paradigmatic cases), staying within polynomial resource requirements for a broad class of physically relevant instances.

Maintaining fidelity $$|\langle \psi_{\text{ad}} | \psi_f \rangle|^2 \geq 0.9$$ can thus be accomplished with moderate $k$ except for highly connected Hamiltonians.

5. Robustness and Error Cancellations in Digital Adiabatic Quantum Simulation

The fixed-point structure of the Trotter error in digital adiabatic state preparation leads to notable cancellation effects (“self-healing” of the error) at the end of the protocol (Kovalsky et al., 2022). While standard analysis predicts cumulative infidelity $O(T^2 \delta t^2)$ for step size $\delta t = T/k$, adiabatic protocols display

$\mathcal{I}(T) = O(T^{-2} \delta t^2) + O(T^{-2})$

for the total infidelity. As $T$ increases, phase cancellation among error terms induced at intermediate times suppresses final-state deviation, inverting the naive resource scaling expectations.

This result implies that, for a fixed time-step, increasing the total evolution time $T$ generically decreases the digitization error—a major practical advantage for applications such as QAOA and digital quantum annealing.

6. Practical Guidelines for Simulation Design and Error Analysis

The leading-order first-order Trotterization error in adiabatic simulation depends on algorithmic parameters, system structure, and experimental imperfections as follows:

• The error per step is $O(\tau^2)$, and the cumulative error over $k$ steps for total time $T$ is $O(T^2/k)$.
• To achieve error tolerance $\Delta$, select $k$ so that

$$k \gtrsim \frac{T^2}{\Delta} [A + B E_{pg}(N) + C E_{pg}^2(N)]$$

• For sparse $H_p$, the cost is nearly independent of $N$; for denser systems, cost scales as $N$ or $N^{1/2}$.
• Experimental fluctuations in gate times or parameters have effect only in higher orders in $1/k$, so do not alter leading scaling.
• Significant “self-healing” of Trotter error occurs in completed adiabatic protocols, further improving fidelity scaling with $T$ and legitimizing larger time steps.

Parameter	Error dependence	Robustness
Step size $\tau$	$O(\tau^2)$	Insensitive to noise in $\tau$ for large $k$
System size $N$	$O(1)$ to $O(N)$ (depends on $H_p$ structure)	Favorable for sparse $H_p$
Fluctuations	$O(1/k)$ effect	Do not change principal scaling
Evolution time $T$	$O(T^2/k)$	Self-healing in completed adiabatic evolution

The above table summarizes error dependence on key parameters and highlights the core error–robustness features.

7. Significance in the Context of Quantum Simulation

First-order Trotterization in adiabatic quantum simulation enables implementation of complex, time-dependent quantum evolutions with limited circuit complexity. Its error scaling, as made precise in (Sun et al., 2018) and subsequent theoretical analysis, ensures resource efficiency for sparse physical Hamiltonians and is robust to moderate imperfections. The methodology has broader applicability—in adiabatic, variational, and hybrid quantum algorithms—under conditions where high fidelity must be preserved with a manageable number of digital steps. The self-healing mechanism of digitization errors further strengthens its relevance for scalable quantum simulation of ground-state and low-lying excited-state manifolds.

A plausible implication is that, for many experimentally relevant problems in quantum chemistry and condensed matter with sparse interactions, first-order Trotterization suffices for high-precision adiabatic quantum simulation on near-term digital hardware without prohibitive resource or robustness concerns.