Ground states of 1D symmetry-protected topological phases and their utility as resource states for quantum computation

TL;DR

This paper generalizes the understanding of ground states in 1D SPT phases, showing their potential as resource states for quantum computation, especially supporting identity and single-qubit gates.

Contribution

It develops a formalism to analyze ground states of SPT phases protected by any finite symmetry group and explores their utility in quantum information processing.

Findings

Ground states of certain SPT phases support quantum gates.

An extended region where the ground state is the AKLT state.

Constructed Hamiltonian invariant under A4 symmetry with useful ground states.

Abstract

The program of classifying symmetry protected topological (SPT) phases in 1D has been recently completed and has opened the doors to study closely the properties of systems belonging to these phases. It was recently found that being able to constrain the form of ground states of SPT order based on symmetry properties also allows to explore novel resource states for processing of quantum information. In this paper, we generalize the consideration of Else et al. [Phys. Rev. Lett. {\bf 108}, 240505 (2012)] where it was shown that the ground-state form of spin-1 chains protected by $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry supports perfect operation of the identity gate, important also for long-distance transmission of quantum information. We develop a formalism to constrain the ground-state form of SPT phases protected by any arbitrary finite symmetry group and use it to examine examples of ground states of SPT phases protected by various finite groups for similar gate protections. We construct a particular Hamiltonian invariant under $A_4$ symmetry transformation which is one of the groups that allows protected identity operation and examine its ground states. We find that there is an extended region where the ground state is the AKLT state, which not only supports the identity gate but also arbitrary single-qubit gates.