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Quantum Bayesian Optimization for the Automatic Tuning of Lorenz-96 as a Surrogate Climate Model

Paul J. Christiansen, Daniel Ohl de Mello, Cedric Brügmann, Steffen Hien, Felix Herbort, Martin Kiffner, Lorenzo Pastori, Veronika Eyring, Mierk Schwabe

TL;DR

The paper introduces a quantum-inspired history matching framework to automatically tune the Lorenz-96 surrogate climate model, replacing classical Gaussian process emulators with quantum kernel-based regressors. By benchmarking three quantum kernel architectures (Chebyshev, NPQC, YZ-CX) and employing Optuna-driven hyperparameter optimization, the authors demonstrate that quantum kernels can outperform a classical RBF baseline in locating parameter settings, even in statevector-simulated environments. A convergence criterion based on observational uncertainty enables fully automatic HM, while strategies for transitioning to real hardware (randomized measurements and shot-based readout) address noise and readout challenges. The work highlights the potential of NISQ-friendly quantum kernels to enhance surrogate-model calibration in climate contexts and outlines concrete steps toward hardwareImplementation and future extensions to more complex climate models.

Abstract

In this work, we propose a hybrid quantum-inspired heuristic for automatically tuning the Lorenz-96 model -- a simple proxy to describe atmospheric dynamics, yet exhibiting chaotic behavior. Building on the history matching framework by Lguensat et al. (2023), we fully automate the tuning process with a new convergence criterion and propose replacing classical Gaussian process emulators with quantum counterparts. We benchmark three quantum kernel architectures, distinguished by their quantum feature map circuits. A dimensionality argument implies, in principle, an increased expressivity of the quantum kernels over their classical competitors. For each kernel type, we perform an extensive hyperparameter optimization of our tuning algorithm. We confirm the validity of a quantum-inspired approach based on statevector simulation by numerically demonstrating the superiority of two studied quantum kernels over the canonical classical RBF kernel. Finally, we discuss the pathway towards real quantum hardware, mainly driven by a transition to shot-based simulations and evaluating quantum kernels via randomized measurements, which can mitigate the effect of gate errors. The very low qubit requirements and moderate circuit depths, together with a minimal number of trainable circuit parameters, make our method particularly NISQ-friendly.

Quantum Bayesian Optimization for the Automatic Tuning of Lorenz-96 as a Surrogate Climate Model

TL;DR

The paper introduces a quantum-inspired history matching framework to automatically tune the Lorenz-96 surrogate climate model, replacing classical Gaussian process emulators with quantum kernel-based regressors. By benchmarking three quantum kernel architectures (Chebyshev, NPQC, YZ-CX) and employing Optuna-driven hyperparameter optimization, the authors demonstrate that quantum kernels can outperform a classical RBF baseline in locating parameter settings, even in statevector-simulated environments. A convergence criterion based on observational uncertainty enables fully automatic HM, while strategies for transitioning to real hardware (randomized measurements and shot-based readout) address noise and readout challenges. The work highlights the potential of NISQ-friendly quantum kernels to enhance surrogate-model calibration in climate contexts and outlines concrete steps toward hardwareImplementation and future extensions to more complex climate models.

Abstract

In this work, we propose a hybrid quantum-inspired heuristic for automatically tuning the Lorenz-96 model -- a simple proxy to describe atmospheric dynamics, yet exhibiting chaotic behavior. Building on the history matching framework by Lguensat et al. (2023), we fully automate the tuning process with a new convergence criterion and propose replacing classical Gaussian process emulators with quantum counterparts. We benchmark three quantum kernel architectures, distinguished by their quantum feature map circuits. A dimensionality argument implies, in principle, an increased expressivity of the quantum kernels over their classical competitors. For each kernel type, we perform an extensive hyperparameter optimization of our tuning algorithm. We confirm the validity of a quantum-inspired approach based on statevector simulation by numerically demonstrating the superiority of two studied quantum kernels over the canonical classical RBF kernel. Finally, we discuss the pathway towards real quantum hardware, mainly driven by a transition to shot-based simulations and evaluating quantum kernels via randomized measurements, which can mitigate the effect of gate errors. The very low qubit requirements and moderate circuit depths, together with a minimal number of trainable circuit parameters, make our method particularly NISQ-friendly.
Paper Structure (30 sections, 53 equations, 16 figures, 1 table, 5 algorithms)

This paper contains 30 sections, 53 equations, 16 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lorenz-96 Model
  4. Gaussian Processes
  5. History Matching
  6. Previous work
  7. Semi-Automatic Tuning of the Lorenz-96 Model
  8. Quantum Gaussian Process Regression for Bayesian Optimization
  9. Methods
  10. Automatic Tuning of the Lorenz-96 Model
  11. Quantum Kernel Evaluation Methods
  12. Inversion Test
  13. Randomized Measurements
  14. Quantum Kernel Architectures
  15. Chebyshev Kernel
  16. ...and 15 more sections

Figures (16)

  • Figure 1: Entangling Chebyshev feature map for $N=4$ qubits. In total, the quantum circuit is parameterized by $8(L+1)$ angles $\phi_{i,k} \in [0,2\pi)$, as well as the L96 model parameters $(\theta_1,\theta_2,\theta_3,\theta_4)=(F, h, c, b)$.
  • Figure 2: NPQC feature map for $N=4$ qubits and the maximum number of $L=2^{N/2}=4$ layers. Rotation gates are parameterized by $\left(\bar{\theta}_1^{(\text{y/z})},\bar{\theta}_2^{(\text{y/z})},\bar{\theta}_3^{(\text{y/z})},\bar{\theta}_4^{(\text{y/z})}\right) = \bm{\theta}_r^{(\text{y/z})} + c \, (F, h, c, b)$ according to \ref{['eq:NPQCEncoding']} with reference parameter values \ref{['eq:NPQCReferenceParameter']}. The shift factors evaluate to $a_2=0, a_3=1,a_4=0$; see \ref{['alg:NPQCShiftFactor']}.
  • Figure 3: YZ-CX feature map for $N=4$ qubits. Rotation gates are parameterized by $(\bar{\theta}_1, \bar{\theta}_2, \bar{\theta}_3, \bar{\theta}_4) = \bm{\theta}_r + c (F, h, c, b)$ according to \ref{['eq:NPQCEncoding']} with reference parameters $\bm{\theta}_r$ drawn uniformly at random from $[0,2\pi)$.
  • Figure 4: Comparison of the results for the ideal repetition vs. the average values in the best trial for all kernel architectures. Investigated is the smallest rescaled distance in relation to the mean as well as the corresponding implausibility (averaged over the principal components). The optimal and the mean distances are equal to the values in \ref{['tab:HPO']}. For NPQC, YZ-CX and RBF, the results are in a similar regime. Only the trainable Chebyshev kernel shows qualitative differences.
  • Figure 5: $R^2$ and MSE (mean squared error) scores of the (quantum) GPs fitted during the best HM run of each kernel type. Every box is marked with the number of waves, $N_W$, the respective run performed (and hence (Q)GP fits), accounting for the different number of waves needed to reach convergence. Additionally, the first (initial) scores are colored in red to acknowledge the improved comparability due to fixed input points.
  • ...and 11 more figures