Universality of a truncated sigma-model

TL;DR

This paper investigates a finite-dimensional qubit-based model that captures the essential physics of the 1+1 dimensional $O(3)$ nonlinear sigma-model, potentially simplifying quantum simulations of such field theories.

Contribution

It provides evidence that a proposed qubitization of the sigma-model reproduces its physics across different energy regimes, offering a practical alternative to traditional truncation methods.

Findings

The qubitized model matches the continuum sigma-model in the infrared regime.

It accurately reproduces the ultraviolet behavior of the sigma-model.

The approach simplifies quantum simulations by avoiding double limits.

Abstract

Bosonic quantum field theories, even when regularized using a finite lattice, possess an infinite dimensional Hilbert space and, therefore, cannot be simulated in quantum computers with a finite number of qubits. A truncation of the Hilbert space is then needed and the physical results are obtained after a double limit: one to remove the truncation and another to remove the regulator (the continuum limit). A simpler alternative is to find a model with a finite dimensional Hilbert space belonging to the same universality class as the continuum model (a "qubitization"), so only the space continuum limit is required. A qubitization of the $1+1$ dimensional asymptotically free $O(3)$ nonlinear $\sigma$-model based on ideas of non-commutative geometry was previously proposed arXiv:1903.06577 and, in this paper, we provide evidence that it reproduces the physics of the $\sigma$-model both in the infrared and the ultraviolet regimes.