Qubitization strategies for bosonic field theories

TL;DR

This paper compares two truncation methods for simulating the 1+1-dimensional O(3) non-linear sigma model on quantum computers, finding that fuzzy sphere truncation better preserves the model's properties than angular momentum cutoff.

Contribution

It introduces and compares two symmetry-preserving truncation strategies for bosonic field theories, demonstrating the effectiveness of fuzzy sphere truncation in quantum simulations.

Findings

Fuzzy sphere truncation aligns with the full theory's spectrum.

Angular momentum cutoff fails to reproduce the sigma model accurately.

Open boundary conditions improve correlation length analysis.

Abstract

Quantum simulations of bosonic field theories require a truncation in field space to map the theory onto finite quantum registers. Ideally, the truncated theory preserves the symmetries of the original model and has a critical point in the same universality class. In this paper, we explore two different truncations that preserve the symmetries of the 1+1-dimensional $O(3)$ non-linear $\sigma$-model - one that truncates the Hilbert space for the unit sphere by setting an angular momentum cutoff and a fuzzy sphere truncation inspired by non-commutative geometry. We compare the spectrum of the truncated theories in a finite box with the full theory. We use open boundary conditions, a novel method that improves on the correlation lengths accessible in our calculations. We provide evidence that the angular-momentum truncation fails to reproduce the $\sigma$-model and that the anti-ferromagnetic fuzzy model agrees with the full theory.