DRAG-Compatible Leakage Suppression in Landau--Zener Control via Isoprobability Twins

This paper introduces the concept of isoprobability twin models to resolve the incompatibility between analytically solvable Landau-Zener pulse shapes and modern leakage-suppression techniques, experimentally validating their equivalence on IBM hardware and demonstrating a leakage reduction of over three orders of magnitude using a custom DRAG implementation.

The Gist

Imagine you are trying to steer a quantum computer's "bit" (a tiny piece of information) from a "sleeping" state to an "awake" state. To do this, physicists use a specific type of control pulse, like a radio wave, to nudge the bit.

For decades, the most famous and reliable way to do this has been using a specific recipe called the Landau-Zener (LMSZ) model. Think of this recipe like a very strict, old-school driving instruction: "Keep your foot on the gas pedal at a perfectly constant speed while you turn the steering wheel."

The Problem:
While this "constant speed" method is mathematically perfect for getting the bit to the right place, it has a major flaw in the real world. Because the speed is constant, the engine never revs up or slows down smoothly. This causes the car to "leak" oil—meaning the quantum bit accidentally spills over into a third, unwanted state (like a "super-awake" state) that ruins the calculation.

Modern technology has a tool to fix leaks called DRAG. However, DRAG works by smoothing out the engine's revs (the changes in speed). If your engine is already running at a perfectly constant speed, there is nothing for DRAG to smooth out. It's like trying to iron a shirt that is already perfectly flat; the tool has nothing to grab onto. So, for years, scientists were stuck: they had a perfect recipe, but it couldn't be fixed with modern leak-prevention tools.

The Solution: "Isoprobability Twins"
The authors of this paper came up with a clever trick. They realized that while the result of a drive (getting the bit awake) depends on the path taken, there are many different paths that lead to the exact same destination.

They call these different paths "Isoprobability Twins."

Imagine you need to drive from City A to City B.

• Route 1 (The Old Way): Drive at a constant 60 mph on a straight highway. (This is the original LMSZ pulse).
• Route 2 (The Twin): Drive on a winding road where you speed up and slow down gently, but you time it perfectly so you arrive at City B at the exact same moment and in the exact same condition as Route 1.

The paper proves mathematically that you can swap the "constant speed" route for a "smooth, speeding-up-and-down" route (like a cosine wave) and still get the exact same result.

The Magic Trick:
Once they swapped the constant-speed route for the smooth, wavy route, they could finally use the DRAG tool.

• Because the new route has smooth changes in speed, DRAG can now "iron out" the rough spots.
• The result? They reduced the "oil leak" (quantum errors) by more than 1,000 times (specifically, over 3 orders of magnitude).

What They Did to Prove It:

1. The Math: They used a mathematical transformation (called Delos-Thorson) to generate a list of 16 different "twin" routes for two famous quantum models.
2. The Test: They took three of these different routes and ran them on a real quantum computer built by IBM (specifically, the ibm_kyiv processor).
3. The Result: The computer showed that all three different routes produced the exact same success rate. The "smooth" route worked just as well as the "constant" route.
4. The Fix: They then applied the DRAG fix to the smooth route. The simulation showed that while the old constant route leaked a lot of energy, the new smooth route with DRAG was almost perfectly clean.

In Summary:
The paper doesn't invent a new way to drive; it invents a new way to describe the drive. It shows that you can trade a rigid, constant-speed pulse for a smooth, wavy one without changing the outcome. This simple swap unlocks the ability to use modern error-correction tools, making quantum computers much more reliable and less prone to "leaking" information.

They demonstrated this on a real IBM quantum chip and showed that by using this "twin" concept, they could stop the leaks almost entirely.

Technical Summary

Technical Summary: DRAG-Compatible Leakage Suppression in Landau–Zener Control via Isoprobability Twins

Problem Statement
Analytically solvable quantum models, specifically the Landau-Majorana-Stückelberg-Zener (LMSZ) and Allen-Eberly-Hioe (AEH) models, are foundational for quantum gate implementations and population transfer protocols. However, their canonical pulse shapes present significant limitations for modern quantum hardware. Most notably, the standard LMSZ pulse features a constant Rabi frequency envelope ($\Omega(t) = \text{const}$). This flat envelope renders the derivative $\dot{\Omega}(t)$ zero, which prevents the application of Derivative Removal by Adiabatic Gate (DRAG) techniques. DRAG is a standard leakage-suppression method that relies on an imaginary correction proportional to $\dot{\Omega}(t)$ to cancel non-adiabatic transitions to non-computational states (leakage). Consequently, the analytically tractable LMSZ model is structurally incompatible with standard leakage suppression, forcing a trade-off between analytic solvability and hardware robustness.

Methodology
The authors address this limitation by formalizing the concept of isoprobability twin models. Utilizing the Delos-Thorson transformation, the authors identify distinct pairs of Rabi frequency $\Omega(t)$ and detuning $\Delta(t)$ that yield identical post-pulse transition probabilities.

1. Theoretical Framework: The method involves a change of the independent variable in the Schrödinger equation from time $t$ to the Delos-Thorson variable $\sigma(t) = \int_0^t \Omega(t') dt'$. This recasts the dynamics such that the transition probability depends solely on the Stückelberg variable $\Theta(\sigma) = \Delta(t(\sigma))/\Omega(t(\sigma))$. Any pair $\{\Omega(t), \Delta(t)\}$ that generates the same function $\Theta(\sigma)$ belongs to the same "isoprobability class" and produces identical transition outcomes.
2. Model Generation: The authors systematically generate 16 distinct models for both the finite LMSZ class and the AEH class. By selecting a smooth pulse envelope (e.g., cosine or hyperbolic secant) instead of the constant envelope, they derive a corresponding detuning function that preserves the original transition probability.
3. Experimental Validation: The equivalence of these twin models was tested on IBM's 127-qubit ibm_kyiv transmon processor. Since the hardware lacks direct time-dependent control of the detuning, the authors emulated the required detuning profiles by applying a time-dependent phase to the Rabi frequency drive ($\phi(t) = \int \Delta(t) dt$). They measured transition probability landscapes for three distinct LMSZ-class models (constant, cosine, and hyperbolic-secant Rabi frequencies) and three AEH-class models.
4. Leakage Suppression Demonstration: A numerical simulation was performed using an 8-level transmon Hamiltonian fitted to qubit 14 of the ibm_kyiv processor. The authors compared the canonical constant-Rabi LMSZ pulse against its "cosine-Rabi" isoprobability twin, to which a DRAG correction ($\beta \dot{\Omega}(t)$) was applied.

Key Results

• Experimental Equivalence: The transition probability landscapes measured on the quantum processor for the different isoprobability twins were nearly identical. The mean squared errors (MSE) between the experimental maps of the constant-Rabi, cosine-Rabi, and hyperbolic-secant-Rabi models were on the order of $10^{-3}$ (e.g., $1.7 \times 10^{-3}$ between pairs 1 and 4), confirming that distinct pulse shapes can produce the same post-pulse population transfer.
• Leakage Reduction: The numerical simulation demonstrated that while the canonical constant-Rabi LMSZ pulse accumulates leakage up to 2.4% (mean 0.53%) due to the inability to apply DRAG, the smooth cosine-Rabi twin combined with DRAG correction reduces leakage to below $1.1 \times 10^{-3}\%$ (mean $2.8 \times 10^{-4}\%$).
• Suppression Factor: This represents a leakage suppression factor of approximately 2200 (peak) and 1900 (mean), exceeding three orders of magnitude, while maintaining the transition probability to the excited state at $P_1 \approx 0.9998$.

Significance and Claims
The paper claims that the isoprobability framework successfully decouples the choice of pulse shape from the choice of the analytical model. This allows researchers to select pulse envelopes that satisfy hardware constraints (such as smoothness for DRAG compatibility) without sacrificing the exact analytic transition probabilities provided by solvable models like LMSZ and AEH.

The authors emphasize that this approach resolves the structural incompatibility between the LMSZ model and DRAG leakage suppression. By switching from a constant envelope to a smooth "twin" within the same isoprobability class, the model becomes compatible with standard derivative-based correction toolkits. The work validates this concept experimentally on a real quantum processor and demonstrates its potential to significantly enhance gate fidelity by suppressing leakage to non-computational states, a critical requirement for fault-tolerant quantum computing. The methodology is presented as applicable to various platforms, including Rydberg atoms, trapped ions, and NMR/MRI systems, wherever analytic solvability is desired but canonical pulse shapes are impractical.