Differentiable quantum-trajectory simulation of Lindblad dynamics for QGP transport-coefficient inference

TL;DR

The paper tackles inferring quark-gluon plasma transport coefficients by differentiating Lindblad-based quarkonium suppression simulations, converting a costly Monte Carlo workflow into a gradient-based inference problem. It derives and applies a score-function gradient estimator to the quantum trajectories algorithm, enabling unbiased gradients through discrete quantum-jump sampling and providing variance reduction via a mean baseline. The approach is implemented in the open-source QTraj code and validated on synthetic $R_{AA}$ data to recover the two transport coefficients $(\hat{\kappa}, \hat{\gamma})$. Results show low-variance, scalable gradients and successful end-to-end gradient-based inference at million-trajectory scale, illustrating potential for experimental-data-driven QGP property extraction and future extensions to higher-dimensional parameter spaces.

Abstract

We study parameter estimation for the transport coefficients of the quark-gluon plasma by differentiating open-quantum-system-based Monte Carlo simulations of quarkonium suppression. The underlying simulator requires solving a Lindblad equation in a large Hilbert space, which makes parameter estimation computationally expensive. We approach the problem using gradient-based optimization. Specifically, we apply the score-function gradient estimator to differentiate through discrete jump sampling in the Monte Carlo wave-function algorithm used to solve the Lindblad equation. The resulting stochastic gradient estimator exhibits sufficiently low variance and can still be estimated in an embarrassingly parallel manner, enabling efficient scaling of the simulations. We implement this gradient estimator in the existing open-source quarkonium suppression code QTraj. To demonstrate its utility for parameter estimation, we infer the two transport coefficients $\hatκ$ and $\hatγ$ using gradient-based optimization on synthetic nuclear modification factor data.

Figures (5)

• Figure 1: Waiting-time quantum trajectories algorithm
• Figure 2: The computational graph for the quantum trajectory algorithm. Squares depict deterministic nodes, which compute a function $f$ on their inputs, while circles represent stochastic nodes, which sample a random variable from a distribution $p$, which can depend on input parameters as well. Here $\Theta$ is a set of parameters, and we assume that the Hamiltonian $H$ and the jump operator $C$ both depend on these parameters. For more details on the definition of $p$ and $f$, see the text.
• Figure 3: Comparison of gradient estimators for the $1S$ overlap at the end of the evolution $\braket{1S|\rho(t_\text{max})|1S}$. The underlying simulations are performed for a Bjorken temperature evolution, see text. We compare the score-function gradient estimator (red) against a central finite-difference estimation of the gradient (blue). The black line represents the derivative of a polynomial fit, serving as a smoothed estimate of the ground truth. Left: The derivative with respect to $\hat{\kappa}$ for fixed $\hat{\gamma}=0$. Right: The derivative with respect to $\hat{\gamma}$ for fixed $\hat{\kappa}=4$.
• Figure 4: Results for the nuclear modification factor plotted against the number of participating nucleons. The circular data points show the synthetic dataset drawn from a simulation with $\hat{\kappa}=4$ and $\hat{\gamma}=0$. The solid lines represent the simulation results, at the initial parameters $\hat{\kappa}=2$ and $\hat{\gamma}=-1.5$, while the dashed lines show the simulation results at the last step of the optimization. The shaded bands indicate the statistical uncertainty of the simulations.
• Figure 5: The trajectory of the optimizer in the transport coefficient plane. We perform the optimization using AMSGrad on the synthetic data for the nuclear modification factor, using the loss function \ref{['eq:loss']}. The orange square indicates the initial parameters, open blue circles show the parameter configuration after each step. Black arrows show the estimated gradient of the loss, with the gray cone indicating its statistical uncertainty. Deviations between the gradient and the actual performed step are due to the optimizer. The green star denotes the parameter configuration from which the synthetic data was drawn. The green ellipse around it indicates the $95\%$ confidence interval.