⚛️ quantum physics

Solving nonlinear differential equations on noisy $156$-qubit quantum computers

This paper demonstrates the successful application of a hybrid classical-quantum algorithm called H-DES to solve nonlinear differential equations, such as the inviscid Burgers' equation, on IBM's 156-qubit noisy quantum computers.

Original authors: Karla Baumann, Youcef Modheb, Roman Randrianarisoa, Roland Katz, Aoife Boyle, Frédéric Holweck

Published 2026-02-11

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Karla Baumann, Youcef Modheb, Roman Randrianarisoa, Roland Katz, Aoife Boyle, Frédéric Holweck

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

The Big Idea: Teaching a "Noisy" Robot to Solve Math Problems

Imagine you are trying to teach a very talented, but incredibly jittery, robot how to solve complex physics equations. This robot is brilliant, but it has a problem: every time it tries to write a number, its hands shake, it gets distracted by static, and it occasionally forgets what it was doing. This is exactly what scientists face with Quantum Computers.

Current quantum computers are in a stage called NISQ (Noisy Intermediate-Scale Quantum). They are powerful, but they are "noisy"—meaning they make lots of tiny errors due to heat, magnetism, or just the general chaos of the subatomic world.

This paper describes a new "toolbox" called H-DES that allows these jittery robots to solve difficult math problems (specifically, differential equations that describe how materials stretch or how fluids move) despite all that shaking.

The Metaphor: The "Sketch Artist" Approach

To understand how the researchers solved this, let’s use an analogy.

The Old Way (The Perfectionist):
Imagine asking an artist to draw a perfect, hyper-realistic portrait of a person in one single stroke. If the artist’s hand shakes even a millimeter, the whole portrait is ruined. This is how many older quantum math theories work—they require perfect, error-free execution. On a noisy quantum computer, this is impossible.

The H-DES Way (The Sketch Artist):
Instead of one perfect stroke, the H-DES method works like a sketch artist.

The Rough Sketch: The artist makes a very quick, messy drawing (the "Variational Circuit"). It’s not accurate, but it gets the general shape down.
The Critic (The Classical Computer): A human critic (a regular, stable computer) looks at the sketch and says, "The nose is too far left, and the eyes are too small."
The Refinement: The artist tries again, adjusting the lines based on the critic's feedback.
The Loop: They go back and forth—sketch, critique, refine—until the drawing looks just like the person.

By using this "Hybrid" approach (half quantum artist, half classical critic), the errors from the "shaky hands" (noise) get smoothed out over time. The math problem isn't solved in one perfect leap; it's "sculpted" into existence through constant correction.

What did they actually solve?

The researchers tested their "Sketch Artist" on two real-world challenges:

The Stretching Bar (Material Science): They simulated a piece of material being pulled until it deforms. This is crucial for engineers designing everything from bridges to smartphone screens. They proved the quantum computer could predict how the stress moves through the material.
The Flowing Fluid (The Burgers’ Equation): They simulated how a fluid (like air or water) moves. This is tricky because fluids can create "shocks"—sudden, violent changes in speed or pressure (like a sonic boom). Even with the "noise" of the quantum computer, the H-DES method successfully mapped out the flow.

Why does this matter?

For a long time, solving these kinds of equations on a quantum computer was purely theoretical—it was like saying, "In a perfect world, a robot could do this."

This paper is a "proof of concept" in the real world. It says, "Even with a shaky, noisy robot, we found a way to get the right answer." It moves us one step closer to a future where quantum computers can help us design new medicines, better airplanes, and stronger materials by simulating the laws of physics with incredible precision.

Technical Summary: Solving Nonlinear Differential Equations on Noisy 156-Qubit Quantum Computers

1. Problem Statement

The paper addresses the challenge of solving nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) on current Noisy Intermediate-Scale Quantum (NISQ) hardware. While theoretical quantum algorithms for solving linear systems (like HHL) exist, they are currently impractical due to high circuit depth requirements. The authors aim to demonstrate that Variational Quantum Algorithms (VQAs)—which are more resilient to noise—can be successfully applied to non-trivial, physically relevant nonlinear problems on real quantum processors, rather than just simulators.

The study focuses on two specific benchmarks:

1D Hypoelastic Tensile Test (ODE): A nonlinear system modeling material deformation (stress and displacement coupling).
Inviscid Burgers’ Equation (PDE): A fundamental equation in fluid dynamics that models non-viscous flow and is known for forming steep gradients and shock waves.

2. Methodology: The H-DES Algorithm

The authors utilize the H-DES (Hybrid Classical-Quantum Differential Equation Solver), a toolbox-based approach that treats the components of the solver as tunable parameters to adapt to specific hardware and problem constraints.

Key methodological components include:

Variational Ansatz: They use a Hardware-Efficient Ansatz (HEA) consisting of parameterized single-qubit rotations ( $R_y$ or $R_x$ ) and entangling gates (CNOT, CZ, or ECR). The choice of gates is optimized to match the specific connectivity and native gate set of the IBM hardware to minimize transpilation overhead and noise.
Solution Encoding: Instead of using feature maps (which can increase circuit depth), H-DES encodes the solution via Chebyshev polynomial expansion within the measurement observables. The function values and their derivatives are obtained as expectation values of these observables.
Loss Function Construction: The loss function $L(\theta)$ $L (θ)$ is a combination of the PDE residual (enforcing the physics) and boundary/initial conditions. They employ two strategies for boundary conditions:
1. Penalty terms added to the loss function.
2. Floating boundary-condition shifts, where the trial function is mathematically modified to satisfy boundary conditions by construction.
Hybrid Optimization Loop: A classical optimizer adjusts the quantum circuit parameters $\theta$ . The authors use different optimizers based on the problem: SLSQP (gradient-based via the parameter-shift rule) for the ODE, and CMA-ES (derivative-free) for the more complex Burgers' equation to handle shot noise.

3. Key Contributions

First Real-Hardware Demonstration: The paper reports the first successful resolution of non-trivial nonlinear differential equations on actual IBM Quantum processors (using the Heron architecture).
Noise-Resilient Framework: They demonstrate that by jointly optimizing the ansatz, observable encoding, shot allocation, and optimizer, the algorithm can "absorb" hardware noise into the variational minimization process without requiring heavy error mitigation.
Advanced Sampling Strategies: To manage the trade-off between accuracy and QPU time, they implement an N-stage sampling strategy (starting with low shots and increasing them as convergence nears) and a parallel stacking strategy (executing identical blocks on disjoint qubits to reduce variance).

4. Results

Hypoelastic 1D Test: Using 15 qubits on the ibm_marrakesh device, the algorithm achieved high accuracy for the stress field ( $\sigma$ ) and a reasonable approximation for the displacement field ( $u$ ). The loss converged by several orders of magnitude without encountering "barren plateaus."
Inviscid Burgers’ Equation:
- Case 1 ( $a=0.5, b=0.25$ ): Using 40 qubits and a 4-stage shot strategy on ibm_fez, the algorithm successfully reconstructed the velocity field $u(x,t)$ with low absolute error across various spatial cuts.
- Case 2 ( $a=1.0, b=1.0$ ): Using 50 qubits and a specialized $R_x+CZ$ ansatz, the algorithm achieved rapid convergence and high accuracy, demonstrating that higher-order gradients can be captured with sufficient qubit counts and expressive architectures.

5. Significance

This work marks a transition from theoretical proof-of-concepts on simulators to practical quantum utility on real, noisy hardware. It proves that the limitations of NISQ devices (gate noise and shot noise) are not insurmountable for solving complex physics problems if the algorithm is designed to be "hardware-aware." By providing a flexible "toolbox" rather than a rigid algorithm, the H-DES framework offers a scalable pathway for performing physically relevant simulations (like fluid dynamics and solid mechanics) on near-term quantum computers.