Stochastic Geometry of Classical and Quantum Ising Models

Vojtěch Novák, Ivan Zelinka, Swagatam Das +1 more

We benchmark eight classical optimizers for exact-statevector VQE calculations on a controlled hierarchy of frustrated spin models, ranging from a diagonal Ising glass to transverse-field Ising and anisotropic Heisenberg models. The benchmark includes local, stochastic-gradient, evolutionary, covariance-adaptation, and swarm-based optimization methods under matched function-evaluation budgets. To understand their performance beyond final energies, we characterize the underlying Hamiltonian--ansatz landscapes in terms of local minima, gradients, curvature, and ground-state reachability. We use simple variational circuits, from an $R_y$ product-state ansatz for the diagonal model to shallow $R_y$--CNOT hardware-efficient circuits for the noncommuting models, and study how increasing circuit depth changes their expressivity, reachability, and optimization geometry. We find that optimizer performance changes substantially across the model hierarchy and is closely connected to landscape structure, while the variational gap represents a separate source of error. These results show how classical optimization, variational expressivity, and landscape geometry jointly determine VQE performance for frustrated spin models.

Sergey B. Bravyi, Matthew B. Hastings

Kurt Binder

悟 半田

This work is based on the paper [49].In Chapter 4, we will introduce the quantum Ising model, especially focusing on the graphical representations of it.By using the Suzuki-Trotter transformation and a graphical representation based on the classical Ising model, we will analyze the susceptibility of the quantum Ising model.This is an ongoing work with Kamijima and Sakai and based on the preprint [50].

Han-Qing 汉青 Shi 石, Xiao-Yue 小岳 Sun 孙, Ding-Fang 定方 Zeng 曾

Abstract Along the way initiated by Carleo and Troyer [G. Carleo and M. Troyer, Science 355 (2017) 602], we construct the neural-network quantum state of transverse-field Ising model (TFIM) by an unsupervised machine learning method. Such a wave function is a map from the spin-configuration space to the complex number field determined by an array of network parameters. To get the ground state of the system, values of the network parameters are calculated by a Stochastic Reconfiguration (SR) method. We provide for this SR method an understanding from action principle and information geometry aspects. With this quantum state, we calculate key observables of the system, the energy, correlation function, correlation length, magnetic moment, and susceptibility. As innovations, we provide a high efficiency method and use it to calculate entanglement entropy (EE) of the system and get results consistent with previous work very well .

Bruno Nachtergaele