# Sigma models on quantum computers

## TL;DR

This paper proposes a quantum computing-compatible discretization of sigma models using the fuzzy sphere, preserving symmetries and enabling continuum results from coarse discretizations, with an analysis of computational costs.

## Contribution

It introduces a novel lattice discretization of sigma models on quantum computers using the fuzzy sphere, maintaining exact symmetry and universality class.

## Key findings

- Exact $O(3)$ symmetry preserved in discretization
- Cost of time-evolution scales as $12 L T/ ext{Δt}$
- Discretization allows continuum results from coarse lattices

## Abstract

We formulate a discretization of sigma models suitable for simulation by quantum computers. Space is substituted by a lattice, as usually done in lattice field theory, while the target space (a sphere) is replaced by the "fuzzy sphere", a construction well known from non-commutative geometry. Contrary to more naive discretizations of the sphere, in this construction the exact $O(3)$ symmetry is maintained, which suggests that the discretized model is in the same universality class as the continuum model. That would allow for continuum results to be obtained for very rough discretizations of the target space as long as the space discretization is made fine enough. The cost of performing time-evolution, measured as the number of CNOT operations necessary, is $12 L T/\Delta t $, where $L$ is the number of spatial sites, $T$ the maximum time extent and $\Delta t$ the time spacing.