Learning out-of-time-ordered correlators with classical kernel methods

Out-of-time-ordered correlators (OTOCs) are widely used to investigate information scrambling in quantum systems. However, directly computing OTOCs with classical computers is an expensive procedure. This is due to the need to classically simulate the dynamics of quantum many-body systems, which entails computational costs that scale rapidly with system size. Similarly, exact simulation of the dynamics with a quantum computer (QC) will either only be possible for short times with noisy intermediate-scale quantum devices or require a fault-tolerant QC which is currently beyond technological capabilities. This motivates a search for alternative approaches to determine OTOCs and related quantities. In this paper, we explore four parameterized sets of Hamiltonians describing local one-dimensional quantum systems of interest in condensed matter physics. For each set, we investigate whether classical kernel methods can accurately learn the $\mathit{XZ}$-OTOC and a particular sum of OTOCs as functions of the Hamiltonian parameters. We frame the problem as a regression task, generating small batches of labeled data with classical tensor network methods for quantum many-body systems with up to 40 qubits. Using this data, we train a variety of standard kernel machines and observe that the Laplacian and radial basis function kernels perform best, achieving a coefficient of determination (${R}^{2}$) on testing sets of at least 0.7167, with averages between 0.8112 and 0.9822 for the various sets of Hamiltonians, together with small root mean squared error and mean absolute error. Hence, after training, the models can replace further uses of tensor networks for calculating an OTOC function of a system within the parameterized sets. Accordingly, the proposed method can assist with extensive evaluations of an OTOC function.