Interpolation of unitaries with time-dependent Hamiltonians via Deep Learning

TL;DR

This work tackles the challenge of predicting unitary time evolution under time-dependent Hamiltonians by training a physics-informed neural network (PINN) that enforces fundamental constraints such as unitarity and the von Neumann dynamics. For systems up to 6 qubits, the PINN directly learns the unitary $U(t)$, while for 7–8 qubits it learns an effective Hamiltonian $H_{ ext{eff}}(t)$ and computes $U(t)$ via the second-order Magnus expansion or Trotterization, enabling scalable interpolation across $t\in[0,1]$. Training uses a dataset generated from simulated dynamics with relatively few time samples (via quantum process tomography-like data), and the model achieves high fidelities, $F(U(t),U_{\theta}(t))$, often exceeding 0.92, even with coarse temporal discretization. The results highlight the potential of PINNs for data-driven quantum dynamics, with implications for quantum simulation, control, and reduced measurement costs, while identifying limitations in Hamiltonian reconstruction that point to future improvements in learning the generator of dynamics.

Abstract

Quantum systems governed by time-dependent Hamiltonians pose significant challenges for the accurate computation of unitary time-evolution operators, which are essential for predicting quantum state dynamics. In this work, we introduce a physics-informed deep learning approach based on Physics-Informed Neural Networks to estimate these operators over the full time domain. By incorporating physical constraints such as unitarity and leveraging the second-order Magnus expansion on the evolution operator, the proposed framework enables the estimation of unitary matrices at different time intervals. The model is trained using simulated unitary operators and evaluated on quantum systems ranging from 2 to 6 qubits. For larger many-body systems, specifically those with 7 and 8 qubits, the same methodology is employed to reconstruct an effective time-dependent Hamiltonian, from which the corresponding time-evolution operator is computed over the entire temporal domain. The proposed framework achieves fidelities exceeding 0.92 using a limited number of unitary samples, indicating a potential reduction in measurement and data acquisition costs. These results highlight the effectiveness of the approach for data-driven simulation and identification of quantum dynamical systems, with direct relevance to quantum computing and quantum simulation applications.

Figures (7)

• Figure 1: Amount of trainable parameters as a function of the number $N$ of qubits. In cases with up to 6 qubits, this relationship is outlined. For systems containing 7 and 8 qubits, we have provided the estimated number of trainable parameters required to apply the same strategy to determine the unitary evolution operator for those systems.
• Figure 2: Illustration of two different methods for simulating the dynamics of quantum systems. (a) The method using PINN is applied to systems up to six qubits, where the model's role is to directly predict the unitary $U(t, t_0)$. (b) For instances involving seven or eight qubits, a different strategy is adopted, in which the model estimates the Pauli components of the effective Hamiltonian $H_{\mathrm{eff}}(t)$, which changes over time, aiding the numerical determination of $U(t, t_0)$. The light blue shade indicates values prior to the model's application, the orange shade signifies the model and the outcomes post-calculation using the model's results, while the red shade denotes backpropagation, which involves comparing the procedure's outcomes with known theoretical values.
• Figure 3: Curves of the loss function are depicted for configurations of 2-qubit to 6-qubit sizes, as well as for each of the four different training setups.
• Figure 4: Loss function curves for 7- and 8-qubit configurations under the four training setups considered. Panel (a): Trotterization. Panel (b): Magnus expansion.
• Figure 5: Fidelity of the predicted unitary evolution for model2–model6, shown (a) before and (b) after SVD-based post-processing error correction.