Lindbladian Learning with Neural Differential Equations

TL;DR

The method reliably learns open-system dynamics across neutral-atom (with 2D connectivity) and superconducting Hamiltonians, as well as the Heisenberg XYZ, and PXP models on a spin-1/2 chain, and shows robustness over phase noise, thermal noise, and their combination.

Abstract

Inferring the dynamical generator of a many-body quantum system from measurement data is essential for the verification, calibration, and control of quantum processors. When the system is open, this task becomes considerably harder than in the purely unitary case, because coherent and dissipative mechanisms can produce similar measurement statistics and long-time data can be insensitive to coherent couplings. Here we tackle this so-called Lindbladian learning problem of open-system characterisation with maximum-likelihood on Pauli measurements at multiple experimentally friendly \emph{transient} times, exploiting the richer information content of transient dynamics. To navigate the resulting non-convex likelihood loss-landscape, we augment the physical model neural differential-equation term, which is progressively removed during training to distil an interpretable Lindbladian solution. Our method reliably learns open-system dynamics across neutral-atom (with 2D connectivity) and superconducting Hamiltonians, as well as the Heisenberg XYZ, and PXP models on a spin-1/2 chain. For the dissipative part, we show robustness over phase noise, thermal noise, and their combination. Our algorithm can robustly infer these dissipative systems over noise-to-signal ratios spanning four orders of magnitude, and system sizes up to $N=6$ qubits with fewer than $5 \times 10^5$ shots.

Figures (15)

• Figure 1: The white-box Lindbladian Learning (LL) setup. The structure of the true Hamiltonian $H_T = \sum_j c_j P_j$ and open-system terms $\sum_{\alpha}\gamma_\alpha\left(L_\alpha\rho(t)L_\alpha^\dagger-\tfrac{1}{2}\{L_\alpha^\dagger L_\alpha,\rho(t)\}\right)$ structure the ansatz, and the coefficients $c_j, \gamma_\alpha \in \mathbb{R}$ are unknown.
• Figure 2: Computational graphs for automatic differentiation of an ordinary differential equation solver ODEint for a vanilla model (a), and a NDE model, (b). Here by ODE-int we refer to using a numerical solver compatible with automatic differentiation. The subscripts $\theta$ and $\varphi$ indicate the variational parameters assigned to the Lindbladian and Neural component respectively in both (a) and (b). As mentioned in Sec. \ref{['sec:NDEs_open_sys']}, we enforce physical output states by renormalising via the trace and averaging over the conjugate transpose once on the output of the ODEint numerical solver.
• Figure 3: Success rates for a 1D Transverse-Field Ising Hamiltonian model with phase (left), thermal (centre) and combined phase and thermal (right) noise terms over four orders of magnitude of noise-to-unitary ratios, $R$. The top row shows the robustness metric over the Hamiltonian parameters and the shows the robustness metric for Lindbladian parameters. Shaded regions correspond to when the NDE term increases (red) or decreases (green) the fitting error.
• Figure 4: Initial loss landscape of two random orthogonal directions in the Hamiltonian variational subspace of $V$ defined in Eq. (\ref{['eq:variational_subspaces']}) for the vanilla (left) architecture and the NDE architecture (right), for all four noise-to-unitary ratios $R = 0.01, 0.1,1, 10$. In all four cases, notice the heatmap of the contour lines indicating a barren landscape on the vanilla architecture, against a much richer loss-landscape for the NDE. This explains the performance gains of the NDE's robustness from Fig. \ref{['fig:NA_TFIM_robustness']}, corresponding to the experiments in the top-left of Fig. \ref{['fig:NA_TFIM_robustness']}
• Figure 5: Initial loss landscape of two random orthogonal directions in the Lindbladian variational subspace of $V$ defined in Eq. (\ref{['eq:variational_subspaces']}), for the vanilla (left) architecture and the NDE architecture (right), over all four noise-to-unitary ratios $R = 0.01, 0.1,1, 10$. In all four cases, notice the heatmap of the contour lines indicating a barren landscape on the vanilla architecture, against a much richer loss-landscape for the NDE. The legend for each heatmap shows the NDE having a landscape with heights that vary $\sim 14\times$ that of the vanilla model. This explains the performance gains of the NDE's robustness from Fig. \ref{['fig:NA_TFIM_robustness']}, corresponding to the experiments in the bottom-left of Fig. \ref{['fig:NA_TFIM_robustness']}