Generating Generalised Ground-State Ansatzes from Few-Body Examples

Abstract

We introduce a method that generates ground-state ansatzes for quantum many-body systems which are both analytically tractable and accurate over wide parameter regimes. Our approach leverages a custom symbolic language to construct tensor network states (TNS) via an evolutionary algorithm. This language provides operations that allow the generated TNS to automatically scale with system size. Consequently, we can evaluate ansatz fitness for small systems, which is computationally efficient, while favouring structures that continue to perform well with increasing system size. This ensures that the ansatz captures robust features of the ground state structure. Remarkably, we find analytically tractable ansatzes with a degree of universality, which encode correlations, capture finite-size effects, accurately predict ground-state energies, and offer a good description of critical phenomena. We demonstrate this method on the Lipkin-Meshkov-Glick model (LMG) and the quantum transverse-field Ising model (TFIM), where the same ansatz was independently generated for both. The simple structure of the ansatz allows us to obtain exact expressions for the expectation values of local observables as well as for correlation functions. In addition, it permits symmetries that are broken in the ansatz to be restored, which provides a systematic means of improving the accuracy of the ansatz.

Figures (18)

• Figure 1: An overview of our method. A domain-specific-language enables ansatz generation via an evolutionary algorithm. (a) A primitive (cycle, mask or pivot) is an edge generation pattern associated with a tensor. (b) Composition: Sequences of primitives form motifs; sequences of motifs form higher-level motifs. (c) A primitive applied to an initialised state (specified with the init command) forms a tensor network. (d) A specified network, being itself a tensor, can again be associated with an edge generation pattern to form a new primitive. (e) The evolutionary algorithm mutates and crosses over motifs each generation. (f) Once the ansatz is found, broken symmetries can be restored.
• Figure 2: The ansatz generated by our method for the LMG and TFIM models.
• Figure 3: RMS magnetisation of Eq. \ref{['eq:RMS_Magnetisation']} vs $h/J$ for the LMG model. Exact results compared to the symmetrised ansatz.
• Figure 4: Relative ground-state energy error vs $N$ for the LMG model at different field strengths $h$. Compares symmetrised ansatz (solid) and MFT (dashed).
• Figure 5: The density of particular structural classes of discovered ansatzes at different evolutionary steps. Each subplot shows results for a different complexity penalty pair $(l_1,l_2)$(increasing to the right). At each step (vertical slice) the lengths of intervals of a particular colour indicate the fraction of runs which yielded a fittest individual within a specific class. The classes are determined by the structural complexity $S(\psi)$ of the fittest individual $\psi$. White represents the original ansatz $\psi_o$ in Eq. \ref{['eq:ansatz']}. Green and Red denote ansatzes with structural complexity less than the original, with Red being the mean‑field solution. Blue indicate ansatzes with structural complexity equal to the original. Pink and Yellow are more complex: yellow contains the original ansatz as a building block, while pink does not.