# Field Theory and The Sum-of-Squares for Quantum Systems

## TL;DR

This paper explores the sum-of-squares hierarchy for quantum spin and fermion systems using quantum field theory techniques, and examines the challenges in approximating the SYK model's ground state energy with various methods.

## Contribution

It introduces new insights into the sum-of-squares hierarchy for quantum systems and analyzes limitations of classical approximation methods for the SYK model.

## Key findings

- Limitations on the Lanczos method starting from Gaussian states
- Constraints on approximating SYK ground state energy with Gaussian wavefunctions
- Application of quantum field theory ideas to sum-of-squares hierarchy

## Abstract

This is a collection of various result and notes, addressing the sum-of-squares hierarchy for spin and fermion systems using some ideas from quantum field theory, including higher order perturbation theory, critical phenomena, nonlocal coupling in time, and auxiliary field Monte Carlo. This paper should be seen as a sequel to Refs. 1,2. Additionally in this paper, we consider the difficulty of approximating the ground state energy of the Sachdev-Ye-Kitaev (SYK) model using other methods. We provide limitations on the power of the Lanczos method, starting with a Gausian wavefunction, and on the power of a sum of Gaussian wavefunctions (in this case under an assumption).