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

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

TL;DR

This work demonstrates solving nonlinear differential equations on actual IBM quantum hardware using the H-DES hybrid classical–quantum solver. By combining a hardware-efficient ansatz, Chebyshev-based observable encoding, boundary-condition strategies, and noise-aware optimization (including multi-stage shot schedules and CMA-ES), the authors solve a 1-D hypoelastic tensile-test ODE system and the inviscid Burgers' equation on 156-qubit-class hardware. The results show robust convergence without heavy error mitigation, achieving quantitative agreement with analytic references within hardware-imposed limits; this supports H-DES as a flexible, problem- and hardware-aware toolbox for near-term quantum differential equation solvers. The study highlights the importance of jointly tuning ansatz design, observables, loss construction, measurement strategy, and optimizer choice to manage shot noise and device imperfections in practical quantum simulations.

Abstract

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.

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

TL;DR

This work demonstrates solving nonlinear differential equations on actual IBM quantum hardware using the H-DES hybrid classical–quantum solver. By combining a hardware-efficient ansatz, Chebyshev-based observable encoding, boundary-condition strategies, and noise-aware optimization (including multi-stage shot schedules and CMA-ES), the authors solve a 1-D hypoelastic tensile-test ODE system and the inviscid Burgers' equation on 156-qubit-class hardware. The results show robust convergence without heavy error mitigation, achieving quantitative agreement with analytic references within hardware-imposed limits; this supports H-DES as a flexible, problem- and hardware-aware toolbox for near-term quantum differential equation solvers. The study highlights the importance of jointly tuning ansatz design, observables, loss construction, measurement strategy, and optimizer choice to manage shot noise and device imperfections in practical quantum simulations.

Abstract

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.
Paper Structure (13 sections, 21 equations, 8 figures, 3 tables)

This paper contains 13 sections, 21 equations, 8 figures, 3 tables.

Table of Contents

  1. Model and analytical reference.
  2. Ansatz selection (real-hardware constraints).
  3. Boundary-condition strategy and loss.
  4. Optimizer and gradients.
  5. Observable encoding.
  6. Hardware execution and results.
  7. Model and analytical solution.
  8. Optimization strategy under hardware noise.
  9. Ansatz selection — Case 1:
  10. Results — Case 1: $a=0.5$, $b=0.25$.
  11. Ansatz selection — Case 2:
  12. Results — Case 2: $a=1.0$, $b=1.0$.
  13. Effect of initial-condition parameters.

Figures (8)

  • Figure 1: H-DES workflow and hybrid loop: The input PDEs and BCs define the loss function that guides the optimization process. The solution is encoded in the VQC. The loop stops when the algorithm reaches an acceptable value of the loss function which indicates a potentially acceptable approximate solution of the problem.
  • Figure 2: Hardware-efficient ansatz used in H-DESjaffali2024h.
  • Figure 3: Real-hardware H-DES results for the hypoelastic 1-D tensile test: convergence and comparison to the analytical reference.
  • Figure 4: Left: HEA structure pre-stacking two-qubit CNOT and CZ gates. The last block represents additional operations required for the estimation of expectation values of the observables. Right: The full circuit to obtain satisfactory solutions for the inviscid Burgers' equation with 40 qubits.
  • Figure 5: H-DES results for inviscid Burgers' equation on ibm_fez (Heron R2) using 40 qubits and a 4-stage optimization strategy. Left: loss convergence across stages. Center: reconstructed solution $u(x,t)$. Right: absolute error along selected $x$-cuts.
  • ...and 3 more figures