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

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.

16 pages, 1 figure, revised version accepted in PRA

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• Qudit quantum computation on matrix product states with global symmetry
• Symmetry-protected topologically ordered states for universal quantum computation
• Measurement-Based Quantum Computation
• Latent Computational Complexity of Symmetry-Protected Topological Order with Fractional Symmetry
• Quantum spin systems for measurement-based quantum computation
• Universal measurement-based quantum computation in two-dimensional SPT phases
• Universal measurement-based quantum computation in a one-dimensional architecture enabled by dual-unitary circuits
• Eigenstate phases with finite on-site non-Abelian symmetry
• Measurement-based quantum computation in finite one-dimensional systems: string order implies computational power
• Universal quantum computing using $({\mathbb{Z}}_d{)}^3$ symmetry-protected topologically ordered states
• Topological Floquet interface states in optical fibre loops
• Symmetric Finite-Time Preparation of Cluster States via Quantum Pumps
• Stable Symmetry-Protected Topological Phases in Systems with Heralded Noise
• Measurement-based quantum simulation of Abelian lattice gauge theories
• Pivoting through the chiral-clock family
• Detection of gapped phases of a 1D spin chain with onsite and spatial symmetries
• Quantum teleportation implies symmetry-protected topological order