Minimal instances for toric code ground states

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.

14 pages, 11 figures

References in corpus (13)

• Many-Body Physics with Ultracold Gases
• Non-Abelian Anyons and Topological Quantum Computation
• Quantum computing with trapped ions
• An Open-System Quantum Simulator with Trapped Ions
• Multi-party entanglement in graph states
• Experimental entanglement of six photons in graph states
• Complete methods set for scalable ion trap quantum information processing
• Creation, manipulation, and detection of Abelian and non-Abelian anyons in optical lattices
• Anyonic interferometry and protected memories in atomic spin lattices
• Scalable Generation of Graph-State Entanglement through Realistic Linear Optics
• Minimum instances of topological matter in an optical plaquette
• Optimal preparation of graph states
• A scheme for tunable quantum phase gate and effective preparation of graph-state entanglement