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

In this paper, we report on the resolution of nonlinear differential equations using IBM’s quantum platform. More specifically, we demonstrate that the hybrid classical-quantum algorithm H-DES successfully solves a one-dimensional material deformation problem and the inviscid Burgers’ equation on IBM’s 156-qubit quantum computers. These results constitute a step toward performing physically relevant simulations on present-day Noisy Intermediate-Scale Quantum (NISQ) devices.

💡 Research Summary

The paper presents the first experimental demonstration of solving non‑trivial nonlinear differential equations on real quantum hardware, specifically IBM’s 156‑qubit NISQ processors (ibm_marrakesh and ibm_fez). The authors employ a hybrid classical‑quantum variational algorithm called H‑DES (Hybrid‑Differential‑Equation‑Solver), which encodes the unknown solution functions into the amplitudes of parameterized quantum circuits (VQCs) and extracts function values via expectation values of specially designed observables.

Key methodological innovations include:

1. Spectral observable encoding – instead of embedding collocation points directly into the circuit (as in many prior VQA approaches), the authors define an observable Oₘ(x) that implements a Chebyshev polynomial expansion. The expectation ⟨ψ(θ)|Oₘ(x)|ψ(θ)⟩ yields the function value at any x, allowing the same circuit to be reused for derivatives without additional feature maps.
2. Flexible ansatz design – a hardware‑efficient ansatz (HEA) composed of layers of Ry rotations followed by CNOT (or CZ/ECR) entangling gates, respecting the connectivity of the underlying IBM devices. Shallow depths (2–4 layers) are chosen to balance expressivity against noise accumulation.
3. Boundary‑condition handling – two strategies are explored: (i) penalty terms added to the loss and (ii) a “floating shift” that analytically enforces boundary conditions, the latter being used for the hypoelastic ODE benchmark.
4. Hybrid optimization loop – the loss combines PDE residuals evaluated on a set of collocation points with boundary penalties. Gradients are obtained on‑hardware via the parameter‑shift rule, and classical optimizers such as SLSQP, Adam, L‑BFGS, COBYLA, or CMA‑ES are employed.

Two benchmark problems are tackled:

• 1‑D hypoelastic tensile test – a coupled nonlinear ODE system for displacement u(x) and stress σ(x). The analytical solution is a low‑degree polynomial, providing a precise reference. The authors encode u(x) on a 15‑qubit circuit and σ(x) on a 4‑qubit circuit, enforce boundary conditions through a floating shift, and minimize the summed squared residuals. Using SLSQP with hardware‑evaluated gradients, the loss drops below 10⁻³ and the reconstructed fields match the exact solution within 1–2 % error.

• Inviscid Burgers’ equation – a first‑order quasi‑linear hyperbolic PDE that develops shocks. The initial condition is encoded, and the time evolution is captured by repeatedly evaluating the loss at successive time slices. An 8–10 qubit circuit with the same Chebyshev observable is used. Despite the increased dimensionality and the presence of steep gradients, the quantum solution reproduces the characteristic wave steepening and shock formation, again achieving loss values below 10⁻³.

Hardware‑specific considerations are detailed: circuits are transpiled once using Qiskit’s preset pass manager (optimization level 2, trivial layout) with symbolic parameters to avoid stochastic variations; no advanced error‑mitigation (resilience_level = 0) is required for convergence, demonstrating that shallow HEA circuits can tolerate current noise levels.

The results show that (i) variational quantum circuits can encode nonlinear function families with sufficient fidelity, (ii) spectral observables provide a clean route to evaluate both functions and derivatives, and (iii) realistic NISQ devices can solve coupled ODE systems and hyperbolic PDEs with modest qubit counts and circuit depths. The work establishes a practical pipeline—problem formulation → loss construction → VQC design → hybrid optimization—that can be extended to higher‑dimensional, multi‑field PDEs, and more complex material models. It thus marks a significant step toward physically relevant quantum simulations on near‑term hardware.