Minimal instances for toric code ground states

TL;DR

This paper investigates the minimal size of toric code systems on square and triangular lattices that retain their unique topological ground state properties, distinguishing them from graph states through local unitary transformations.

Contribution

It introduces a geometric method to transform toric code setups into LC-equivalent graph states and identifies the smallest such configurations on square and triangular lattices.

Findings

Smallest square lattice setup has 5 plaquettes and 16 qubits.

Smallest triangular lattice setup has 4 plaquettes and 9 qubits.

The approach distinguishes toric code ground states from graph states in minimal configurations.

Abstract

A decade ago Kitaev's toric code model established the new paradigm of topological quantum computation. Due to remarkable theoretical and experimental progress, the quantum simulation of such complex many-body systems is now within the realms of possibility. Here we consider the question, to which extent the ground states of small toric code systems differ from LU-equivalent graph states. We argue that simplistic (though experimentally attractive) setups obliterate the differences between the toric code and equivalent graph states; hence we search for the smallest setups on the square- and triangular lattice, such that the quasi-locality of the toric code hamiltonian becomes a distinctive feature. To this end, a purely geometric procedure to transform a given toric code setup into an LC-equivalent graph state is derived. In combination with an algorithmic computation of LC-equivalent graph states, we find the smallest non-trivial setup on the square lattice to contain 5 plaquettes and 16 qubits; on the triangular lattice the number of plaquettes and qubits is reduced to 4 and 9, respectively.