# A diagrammatic approach to variational quantum ansatz construction

## TL;DR

This paper introduces a diagrammatic, size-extensive approach for constructing variational quantum ansatzes that avoids Trotterization, enabling efficient and accurate ground state approximations on near-term quantum devices.

## Contribution

It presents a novel diagrammatic scheme for size-extensive variational ansatzes that do not rely on Trotterization, with methods for ansatz compression and hierarchy construction.

## Key findings

- Effective convergence to ground states in spin chain models.
- Size-extensive ansatzes without Trotterization improve accuracy.
- Hierarchical ansatzes enable systematic improvements.

## Abstract

Variational quantum eigensolvers (VQEs) are a promising class of quantum algorithms for preparing approximate ground states in near-term quantum devices. Minimizing the error in such an approximation requires designing ansatzes using physical considerations that target the studied system. One such consideration is size-extensivity, meaning that the ground state quantum correlations are to be compactly represented in the ansatz. On digital quantum computers, however, the size-extensive ansatzes usually require expansion via Trotter-Suzuki methods. These introduce additional costs and errors to the approximation. In this work, we present a diagrammatic scheme for the digital VQE ansatzes, which is size-extensive but does not rely on Trotterization. We start by designing a family of digital ansatzes that explore the entire Hilbert space with the minimum number of free parameters. We then demonstrate how one may compress an arbitrary digital ansatz, by enforcing symmetry constraints of the target system, or by using them as parent ansatzes for a hierarchy of increasingly long but increasingly accurate sub-ansatzes. We apply a perturbative analysis and develop a diagrammatic formalism that ensures the size-extensivity of generated hierarchies. We test our methods on a short spin chain, finding good convergence to the ground state in the paramagnetic and the ferromagnetic phase of the transverse-field Ising model.