Solving The Quantum Many-Body Hamiltonian Learning Problem with Neural Differential Equations

TL;DR

This work proposes a novel method to solve the Hamiltonian learning (HL) problem -inferring quantum dynamics from many-body state trajectories by distilling and filtering an estimator Hamiltonian from a Neural ODE representation of state trajectories.

Abstract

Understanding and characterising quantum many-body dynamics remains a significant challenge due to both the exponential complexity required to represent quantum many-body Hamiltonians, and the need to accurately track states in time under the action of such Hamiltonians. This inherent complexity limits our ability to characterise quantum many-body systems, highlighting the need for innovative approaches to unlock their full potential. To address this challenge, we propose a novel method to solve the Hamiltonian Learning (HL) problem-inferring quantum dynamics from many-body state trajectories-using Neural Differential Equations combined with an Ansatz Hamiltonian. Our method is reliably convergent, experimentally friendly, and interpretable, making it a stable solution for HL on a set of Hamiltonians previously unlearnable in the literature. In addition to this, we propose a new quantitative benchmark based on power laws, which can objectively compare the reliability and generalisation capabilities of any two HL algorithms. Finally, we benchmark our method against state-of-the-art HL algorithms with a 1D spin-1/2 chain proof of concept.

Figures (5)

• Figure 1: The black-box and white-box Hamiltonian Learning (HL) scenarios are shown in (a) and (b) respectively. In both cases, the evolution is unitary, and we may control the length of time, $t$, for which the system evolves. In the white-box scenario (a), the structure of the true Hamiltonian $H_T = \sum_j c_j P_j$ is known, but the coefficients $c_j \in \mathbb{R}$ are not. Whilst in the black-box scenario (b), neither the structure of the Hamiltonian, $P_j$, nor the coefficients $c_j$ are known. As discussed in Sec. \ref{['sec:introduction']}, we choose to formulate the HL problem assuming control on the amount of time evolution, as it requires little optimal control. This makes data from time-evolved states experimentally friendly, provided the chosen evolution times are sufficiently short.
• Figure 2: Architectures of the models used to simulate the time evolution of the parameterised Hamiltonian $H(\theta)$. (a) The numerically exact time evolution under the Hamiltonian. (b) The approximate time evolution through integrating the Schrodinger equation, hereby referred to as the vanilla model. (c) The time evolution through integrating the Schrodinger equation with an added corrective term to the right-hand-side.
• Figure 3: The loss landscapes of for two parameters in the anisotropic Heisenberg Hamiltonian, the PXP Hamiltonian, and the dense nearest-neighbour Hamiltonian respectively from left to right. The center of each plot is the global minimum, and the axes are chosen as two random orthogonal directions in the parameter space. The top row contains the landscapes for the vanilla model architecture from wilde2022scalably, shown in Fig.\ref{['fig:architectures_b']}. The bottom row contains landscapes for the Neural ODE model from Fig.\ref{['fig:architectures_c']}, over the same Hamiltonians, with the same ground-truth parameters. We see clearly in the centre two plots that the Neural ODE model can make significant improvements to the loss landscape in terms of the number of local minima, as well as having a smoothing effect to the ruggedness of the PXP Hamiltonian's landscape. This effect is visible in Table \ref{['tab:success_rate']}, where the convergence rate of the Neural ODE model greatly surpasses the vanilla model. Notice that for the left panels (Heterogeneous Heisenberg), the Neural ODE model has changed the loss landscape in the lower-left corner to have fewer saddle points and a more even loss landscape. This makes it more likely to approach the global minimum when optimising as there are more directions from which one can arrive at the local minimum. However, since the change is relatively small, the Neural ODE causes only a small boost in the success rate, as shown in Table \ref{['tab:success_rate']}. Finally, notice that for the right panels (Dense NN), the Neural ODE makes almost no change to the loss landscape near the global minimum. We see this reflected in the almost equivalent success rates of the vanilla model vs the Neural ODE model in Table \ref{['tab:success_rate']}.
• Figure 4: Infidelities between the simulated time-evolved state of the neural ODE and the true time-evolved state. The boundary of the shaded region on the left represents the training time (i.e. feeding back at this boundary point only), and to the right of it are testing times. Here, the red dashed line marks the 1% error boundary, past which the model has more than a 1% error. (Left) The fidelities as a function of the simulation time for the 6 Hamiltonians indicated, where here we fix the number of spins to $N=8$. (Right) The fidelities as a function of the simulation time for different system sizes, where here we use the PXP Hamiltonian as this was heuristically found to be the most difficult as per Table \ref{['tab:success_rate']}.
• Figure 5: The impact of the number of Pauli measurements on the robustness and the relative error of the learning algorithm using the neural ODE model. Here we use the PXP Hamiltonian on a system of 8 bodies to give a worst-case scaling as Table \ref{['tab:success_rate']} shows this was the most difficult Hamiltonian considered in our examples from Eq.(\ref{['eq:isotropic_heis_ham']} - \ref{['eq:dense_nn_ham']}).