Generating Low-Dimensional and Scale-Invariant Ansatzes for Interacting Quantum Many-Body States

We present a method for generating low-dimensional ansatzes to obtain mean-field-like theories that incorporate correlations in physical systems. Our technique employs an evolutionary algorithm exploiting a hierarchical and scale-invariant compute-graph [1] to find low-energy states of a given Hamiltonian. This representation helps realize physical symmetries and enables generalization over increasing numbers of degrees of freedom, resulting in a low-dimensional ansatz capturing the fundamental aspects of a model. Remarkably, we find analytically tractable ansatzes with a degree of universality that encode correlations, capture finite-size effects, provide accurate ground state energy predictions and offer a more precise description of critical phenomena. We demonstrate this method on the quantum transverse field Ising model (TFIM) and the Lipkin-Meshkov-Glick (LMG) model, where the same ansatz was independently generated for both. We illustrate its analytical tractability and are able to restore broken symmetries for both models. Interestingly, the generated ansatz is connected to the one-dimensional classical Ising model and a well-studied sequence in signal processing and analytical number theory.

[1] Lourens, M., Sinayskiy, I., Park, D.K. et al. Hierarchical quantum circuit representations for neural architecture search. npj Quantum Inf 9, 79 (2023). https://doi.org/10.1038/s41534-023-00747-z