⚛️ quantum physics

Influence of Trotterization error on single-particle tunneling

This paper demonstrates that by adapting the Wentzel-Kramers-Brillouin approximation to the Suzuki-Trotter algorithm, tunneling rates in semiclassical models can be exponentially enhanced through non-perturbative renormalization when resonance is preserved, thereby enabling feasible single-particle tunneling simulations on existing noisy superconducting quantum hardware using large Trotter steps.

Original authors: Anton V. Khvalyuk, Kostyantyn Kechedzhi, Vadim S. Smelyansky, Lev B. Ioffe

Published 2026-08-21

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Anton V. Khvalyuk, Kostyantyn Kechedzhi, Vadim S. Smelyansky, Lev B. Ioffe

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Quantum computers promise to solve problems that are impossible for today's supercomputers, particularly those involving the strange behavior of particles at the smallest scales. One of the most fundamental behaviors in this realm is quantum tunneling, a process where a particle passes through a barrier it seemingly cannot cross, much like a ghost walking through a wall. This phenomenon is not just a curiosity; it drives chemical reactions, powers the sun, and is essential for the operation of modern electronics. However, simulating tunneling on a quantum computer is notoriously difficult. The process is inherently slow and involves particles moving in ways that do not respect the local connections of the computer's physical hardware. To simulate a particle moving from one side of a barrier to the other, the computer must perform a vast number of tiny, sequential steps. If these steps are too small, the simulation takes too long and succumbs to noise. If they are too large, the simulation becomes inaccurate. For years, scientists have struggled to find the right balance, often relying on error estimates that suggested the task was impossible with current technology.

A team of researchers has now re-examined this challenge, focusing on a specific method used to break down complex quantum movements into manageable steps, known as the Suzuki-Trotter algorithm. By applying a classic mathematical technique originally designed for describing particle motion in smooth landscapes, they discovered that the standard way of estimating errors was missing a crucial detail. Their work reveals that for certain symmetric setups, the very errors introduced by taking large steps in the simulation do not ruin the result. Instead, these errors can actually speed up the tunneling process dramatically, making it possible to observe the phenomenon on existing hardware with far fewer steps than previously thought necessary.

The researchers began by modeling a simple system: a single particle moving through a double-well potential, which is essentially a landscape with two dips separated by a hill. In a perfect world, the particle would sit in one dip and occasionally tunnel through the hill to the other. To simulate this on a quantum computer, the researchers had to approximate the continuous flow of time with a series of discrete jumps. The standard approach assumes that if these jumps are small enough, the approximation will be good. However, the team found that this assumption fails to account for how the particle's energy levels shift during the simulation. In many cases, these shifts cause the particle to lose its ability to tunnel, effectively locking it in one well. This happens because the simulation introduces a mismatch, or detuning, between the energy levels of the two wells, preventing the particle from resonating and crossing over.

The study shows that this detuning is the primary obstacle. If the potential landscape is not perfectly symmetric, or if the simulation steps are chosen without care, the particle will remain trapped. The researchers calculated that the required precision to avoid this trapping is so extreme that it would demand more computer steps than any existing device could handle before losing its quantum state to noise. This led to the conclusion that, in a general case, simulating tunneling with current hardware might be impossible using standard methods.

However, the story changes when the potential landscape is perfectly symmetric. In this specific scenario, the energy shifts caused by the simulation steps happen equally in both wells. The particle remains in resonance, and the tunneling process continues. Surprisingly, the researchers found that the errors introduced by the large steps do not just preserve the tunneling; they enhance it. The simulation effectively rewrites the rules of the barrier, making it easier for the particle to cross. This enhancement is not a small improvement; it is exponential. By taking larger steps, the researchers found they could increase the tunneling rate by orders of magnitude, reducing the number of steps required to observe the event from an impossible number to a manageable one.

This discovery relies on a deeper understanding of how the simulation algorithm interacts with the physics of the system. The team used a method called the Wentzel-Kramers-Brillouin approximation, which treats the quantum particle as a wave moving through a smooth landscape, to analyze the algorithm's behavior. They adapted this method to account for the fact that the simulation steps create a repeating pattern in time, a concept known as Floquet theory. This allowed them to see that the large steps create new pathways for the particle to travel, effectively reconstructing the energy spectrum of the system. In some cases, this reconstruction brings high-energy states into resonance with low-energy states, creating new channels for the particle to move through the barrier.

The researchers validated their findings through detailed numerical simulations. They modeled a system of fifty qubits, representing the particle's position, and ran the simulation with various step sizes. The results confirmed that when the resonance was preserved, the tunneling rate increased dramatically as the step size grew. In one specific configuration, they observed a fivefold acceleration in the tunneling process simply by adjusting the step size, without any loss of accuracy in the physical outcome. The simulations showed that the energy shifts, which usually ruin the process, were practically absent in these symmetric cases, allowing the enhancement to dominate.

The implications of this work are significant for the future of quantum simulation. It suggests that existing superconducting quantum devices, which are currently limited by noise and the number of steps they can perform, might be capable of simulating tunneling much sooner than expected. The key is to design the simulation carefully, ensuring that the potential landscape is symmetric and that the step size is chosen to exploit the enhancement rather than suffer from the error. The researchers propose an experimental protocol to test this on real hardware, involving a series of double-well setups where the particle's movement is monitored. They estimate that with current technology, a circuit depth of roughly 120,000 steps would be required, a number that is challenging but potentially achievable given recent improvements in device coherence times.

While the findings are promising, the researchers are careful to note the limitations. The enhancement effect relies on the system remaining in a specific regime where the steps are large but not so large that they destroy the underlying physics. If the steps become too large, the simulation enters a different regime where the energy spectrum is completely reconstructed, leading to complex dynamics that are harder to predict. Furthermore, the current analysis focuses on a single particle. In real-world applications, particles interact with each other, and these interactions could introduce new complications. The researchers suggest that future work will need to explore how these effects play out in more complex, many-body systems and how noise from the environment might interfere with the delicate balance required for the enhancement.

Ultimately, this paper provides a new perspective on the relationship between error and accuracy in quantum computing. It challenges the conventional wisdom that smaller steps always lead to better results. Instead, it shows that in the specific context of quantum tunneling, the "errors" of a coarse approximation can be harnessed to reveal a faster, more efficient path through the quantum landscape. By understanding the precise conditions under which this happens, scientists can design better simulations that work within the constraints of today's hardware, opening the door to studying complex quantum phenomena that were previously out of reach. The work serves as a reminder that in the quantum world, the path to accuracy is not always a straight line, and sometimes, taking a bigger step is the only way to move forward.

Technical Summary: Influence of Trotterization Error on Single-particle Tunneling

Problem Statement
Simulating spatial quantum tunneling on existing superconducting quantum hardware is challenging due to two primary factors: the exponential slowness of the tunneling process relative to barrier strength, and the inherently nonlocal nature of tunneling, which conflicts with the local-only gate capabilities of current devices. While the Suzuki-Trotter algorithm (STA) offers a practical solution by approximating nonlocal evolution with local gate sequences without ancilla qubits, conventional fidelity-based error bounds suggest that the gate counts required to observe tunneling would exceed the coherence limits of real devices. Specifically, standard estimates imply that the simulation time $t$ required to observe tunneling, combined with the error scaling $\epsilon \sim \Lambda t (\Lambda t/M)^{1/2}$ , necessitates a prohibitively large number of Trotter steps $M$ . This work addresses the discrepancy between these rigorous upper bounds and the actual physical observables in the context of tunneling.

Methodology
The authors analyze the second-order Suzuki-Trotter algorithm (STA) applied to a solvable model of semiclassical tunneling: a chain of $L$ spin-1/2 particles with XY nearest-neighbor interactions and a position-dependent $z$ -field. Through the Jordan-Wigner transformation, this system maps to non-interacting fermions, reducing the problem to a single-particle Hamiltonian.

To characterize the STA, the authors adapt the Wentzel-Kramers-Brillouin approximation (WKBA) to the product formula structure of the algorithm. Key methodological steps include:

Effective Hamiltonian Construction: Deriving a time-independent effective single-particle Hamiltonian ( $H_{eff}$ ) that governs the evolution between discrete Trotter steps $\delta t$ . This derivation respects the Floquet theorem, acknowledging that the quasi-energies are defined modulo $\Omega = 2\pi/\delta t$ .
Non-perturbative Extension: Extending WKBA to be non-perturbative in the Trotter step $\delta t$ . This allows for the analysis of regimes where $\Lambda \delta t \sim 1$ , capturing effects missed by standard perturbative expansions.
Semiclassical Analysis: Applying the adapted WKBA to calculate corrections to the classical action, energy level shifts, and tunneling amplitudes. This involves analyzing the complex momentum in forbidden regions and matching boundary conditions, including anomalous turning points arising from lattice discreteness.
Numerical Verification: Validating analytical results using high-precision numerical diagonalization of the single-particle Hamiltonian and the STA evolution operator.

Key Contributions and Results

Resonance Detuning and Suppression:
In the general case, the authors find that STA induces energy shifts in the classically allowed regions. These shifts cause a resonance detuning ( $\epsilon_{eff}$ ) between the metastable states in the two wells. Since the tunneling amplitude $\eta$ is exponentially small, even minute detuning can suppress tunneling. This imposes a strict constraint on the Trotter step size: $(J\delta t)^2 \lesssim \eta n_{cl}/\Delta$ , which is significantly more restrictive than the standard criteria $\Lambda \delta t \ll 1$ .
Exponential Enhancement via Non-perturbative Renormalization:
Crucially, the authors identify a regime where the target resonance is preserved (e.g., in symmetric double-well potentials where symmetry prevents detuning). In this regime, STA does not merely approximate the dynamics but exponentially enhances the tunneling rate. The effective tunneling amplitude $\eta_{eff}$ is renormalized such that $\eta_{eff}/\eta \approx \exp(-\delta S_B/a)$ , where the correction to the under-barrier action $\delta S_B$ accumulates along the barrier. This enhancement can reduce the required gate count by orders of magnitude, making tunneling observable with large Trotter steps ( $\Lambda \delta t \sim 1$ ) that are natural for existing hardware.
Floquet Mixing and Spectrum Reconstruction:
For large Trotter steps ( $\Lambda \delta t \sim 1$ ), the system enters a non-perturbative regime governed by Floquet theory. The periodicity of the quasi-energy spectrum ( $\Omega$ ) allows high-energy states to hybridize with low-energy resonant states. The authors demonstrate that this "Floquet mixing" can reconstruct the low-energy spectrum, creating new allowed regions within the potential barrier and enabling resonant co-tunneling. This leads to a drastic deformation of tunneling dynamics, including the emergence of three-state Rabi oscillations involving hybridized states that would otherwise be decoupled.
Limitations of Rigorous Error Bounds:
The paper demonstrates that rigorous upper bounds on the spectral norm distance between exact and approximate evolution operators overestimate the error for physical observables like tunneling rates. These bounds, which scale with simulation time $t$ , fail to account for the fact that physical observables may be insensitive to the Hilbert space sectors contributing most to the fidelity error. Consequently, the "true" error in tunneling dynamics is often much smaller than the worst-case bounds suggest, particularly when the resonance is preserved.

Significance
The authors conclude that their findings enable direct tunneling simulations on existing superconducting platforms. By characterizing the interplay between tunneling and STA, they show that large Trotter steps can be beneficial rather than detrimental, provided the resonance is preserved. This offers a controlled platform for studying the nontrivial interplay of tunneling and discretization errors. Furthermore, the work provides a framework for investigating the effects of noise, disorder, and interactions in the context of Trotterized dynamics, suggesting that the "error" introduced by STA can be harnessed to accelerate the observation of quantum phenomena that are otherwise inaccessible due to hardware constraints. The paper positions itself as a foundational step toward understanding how to optimize quantum simulation protocols for specific physical processes rather than relying on generic error bounds.

More like this